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The impact of media coverage on the transmission dynamics of human influenza

The impact of media coverage on the transmission dynamics of human influenza Background: There is an urgent need to understand how the provision of information influences individual risk perception and how this in turn shapes the evolution of epidemics. Individuals are influenced by information in complex and unpredictable ways. Emerging infectious diseases, such as the recent swine flu epidemic, may be particular hotspots for a media-fueled rush to vaccination; conversely, seasonal diseases may receive little media attention, despite their high mortality rate, due to their perceived lack of newness. Methods: We formulate a deterministic transmission and vaccination model to investigate the effects of media coverage on the transmission dynamics of influenza. The population is subdivided into different classes according to their disease status. The compartmental model includes the effect of media coverage on reporting the number of infections as well as the number of individuals successfully vaccinated. Results: A threshold parameter (the basic reproductive ratio) is analytically derived and used to discuss the local stability of the disease-free steady state. The impact of costs that can be incurred, which include vaccination, education, implementation and campaigns on media coverage, are also investigated using optimal control theory. A simplified version of the model with pulse vaccination shows that the media can trigger a vaccinating panic if the vaccine is imperfect and simplified messages result in the vaccinated mixing with the infectives without regard to disease risk. Conclusions: The effects of media on an outbreak are complex. Simplified understandings of disease epidemiology, propogated through media soundbites, may make the disease significantly worse. Introduction People’s response to the threat of disease is dependent Infectious diseases are responsible for a quarter of all on their perception of risk, which is influenced by public deaths in the world annually, the vast majority occurring and private information disseminated widely by the media. in low- and middle-income countries [1]. There are dis- While government agencies for disease control and pre- eases such as SARS and flu that exhibit some distinct fea- vention may attempt to contain the disease [3], the general tures such as rapid spatial spread and visible symptoms information disseminated to the public is often restricted [2]. These features, associated with the increasing trend to simply reporting the number of infections and deaths. of globalization and the development of information Mass media are widely acknowleged as key tools in risk technology, are expected to be shared by other emerging/ communication [4,5], but have been criticised for making re-emerging infectious diseases. It is therefore important risk a spectacle to capitalise on audience anxiety [6,7]. to refine classical mathematical models to reflect these The original interpretation of media effects in communi- features by adding the dimensions of massive news cov- cation theory was a “hypodermic needle” or “magic bullet” erage that have great influence not only on the individual theory of the mass media. Early communication theorists behaviours but also on the formation and implementa- [8,9] imagined that a particular media message would be tion of public intervention and control policies [2]. directly injected into the minds of media spectators. This theory of media effects, in which the mass media has a direct and rapid influence on everyday understanding, has * Correspondence: rsmith43@uottawa.ca Department of Mathematics and Faculty of Medicine, The University of been substantially revised. Contemporary media studies Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada analyses how media consumers might only partially accept Full list of author information is available at the end of the article © 2011 Tchuenche et al; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 2 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 a particular media message [10], how the media is shaped reporting of both disease dynamics and vaccination is by dominant cultural norms [11,12] and how media con- high. Vaccination is one of the most effective tools for sumers resist dominant media messages [13,14]. It follows reducing the burden of infectious diseases [19]. However, that media effects may sway people into panic (eg swine despite their public-health benefit, vaccination programs flu), especially with a disease where scientific evidence is face obstacles. Individuals often refuse or avoid vaccina- thin or nonexistent. Conversely, media may have little tions they perceive to be risky. Recently, rumours that thepolio vaccinecould cause sterility and spread HIV effect on seasonal diseases (eg regular influenza). have hampered polio eradication in Nigeria [20], while Media reporting plays a key role in the perception, misplaced fears of autism in the developed world have management and even creation of crisis [6]. Since media reports are retrievable and because the messages are stoked vaccination fears [21]. Reporting the number of widely distributed, they gain authority as an intersubjec- individuals who vaccinate may have a positive effect on tive anchorage for personal recollection [4]. At times of the disease transmission by increasing the vaccination crisis, non-state-controlled media thrive, while state- rate. controlled media are usually rewarded for creating an Conversely, behavioural interventions can also have an illusion of normalcy [6]. Media exposure and attention enormous effect on the course of a disease [22,23] Our partially mediate the effects of variables such as demo- model considers the same contact rate after a media alert, graphics and personal experience on risk judgments [5]. as proposed by Liu & Cui [3], but there are fundamental The role of media coverage on disease outbreaks is thus differences in both models. They consider the classical SIR crucial and should be given prominence in the study of type model, while vaccination is included in ours to reflect disease dynamics. transmission dynamics of human influenza. Klein et al., [15] noted that much more research is needed to understand how provision of information Model framework influences individual risk perception and how it shapes We divide the population (N) into four sub-populations, the evolution of epidemics; for example, individuals may according to their disease status: susceptible (S), vaccinated overprotect, which can have additional consequences for (V ), infected (I), and recovered (R). Our model monitors the spread of disease. An example of such complex the dynamics of influenza based on a single strain without dynamics is the 1994 outbreak of plague in a state in effective cross-immunity against the strain. The susceptible India: after the announcement of the disease, many peo- population is increased by recruitment of individuals ple fled the state of Surat in an effort to escape the dis- (either by birth or immigration), and by the loss of immu- ease, thus carrying the disease to other parts of the nity, acquired through previous vaccination or natural country [16]. Even though information on the number of infection. This population is reduced through vaccination cases and deaths can have an adverse effect, the number (moving to class V ), infection (moving to class I)andby of those vaccinated has not been given prominence. natural death or emigration. The population of vaccinated A handful of mathematical models have described the individuals is increased by vaccination of susceptible indivi- impact of media coverage on the transmission dynamics duals. Since the vaccine does not confer immunity to all of infectious diseases. Liu et al.[2] examined the potential vaccine recipients, vaccinated individuals may become for multiple outbreaks and sustained oscillations of emer- infected, but at a lower rate than unvaccinated. The vacci- ging infectious diseases due to the psychological impact nated class is thus diminished by this infection (moving to from reported numbers of infectious and hospitalized class I) by waning of vaccine-based immunity (moving to individuals. Liu & Cui [3] analysed a compartment model class S) and by natural death. The population of infected that described the spread and control of an infectious individuals is increased by infection of susceptibles, includ- disease under the influence of media coverage. Li & Cui ing those who remain susceptible despite being vaccinated. [17] incorporated constant and pulse vaccination in SIS It is diminished by natural death, death due to disease and epidemic models with media coverage. Cui et al., [18] by recovery from the disease (moving to class R). The showed that when the media impact is sufficiently strong, recovered class is increased by individuals recovering from their model – with incidence rate being of the exponen- their infection and is decreased as individuals succumb to tial form capturing the alertness to the disease of each natural death. Media coverage is introduced into the model susceptible individual in the population – exhibits multi- via a saturated incidence function. ple positive equilibria (also see [2]) which poses a chal- A schematic model flowchart is depicted in Figure 1. lenge to the prediction and control of the outbreaks of infectious diseases. Model equations The aim of this study is to investigate the impact of The transmission model with media coverage is given by media coverage on the spread and control of an influenza the following deterministic system of nonlinear ordinary strain when a vaccine is available, and where the media differential equations: SI Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 3 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 S R (1-) SV V I Figure 1 The model Schematic model flow diagram assume this rate to be the same for all sub-populations. b ⎛ ⎞ dS I =+ Λ wqVS −() +m − b −b SI +s R (1) ⎜ ⎟ is the rate at which susceptibles get infected; ω is the rate dt mI + ⎝ I ⎠ at which vaccine-based immunity wanes; g is the vaccine efficacy; a is the death rate due to the infection; and l is dI ⎛ I ⎞ ⎛ I ⎞ =−bb SI+−bb () 1−− g VI (a+m+l)I b ⎜ 12 ⎟ ⎜ 1 3 ⎟ (2) the recovery rate from infection. The terms dt mI + mI + ⎝ II ⎠ ⎝ ⎠ mI + and measure the effect of reduction of the con- mI + ⎛ ⎞ dV I tact rate when infectious and vaccinated individuals are =−qmSV () +w − b −b (1 −g)VI (3) ⎜ ⎟ dt mI + reported in the media [2,3,18]. The half-saturation con- ⎝ ⎠ stant m > 0 reflects the impact of media coverage on the gI () = dR contact transmission. The function is a (4) =−lmIR () +s , mI + dt continuous bounded function which takes into account disease saturation or psychological effects [24]. We note where Λ is the rate at which individuals are recruited that recovered individuals cannot be vaccinated. Also, a into the population (recruitment of infectives is ignored vaccinated individual who gets infected and then recovers for now); θ is the rate at which susceptible individuals will return to the susceptible class with no vaccine protec- receive the vaccine; µ is the the rate at which people leave tion. This is true even if ω is quite small but s and l are the population, through natural death or emigration. We Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 4 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 large. For example, if vaccination lasts three years, but Stability of the equilibrium states recovery and loss of immunity takes 6 months, then we The disease-free equilibrium of the system is given by are assuming this person is subsequently unvaccinated. ⎛ ⎞ ΛΛ () mw + q In the Michaelis-Menten functional response, the rate ES== (,I,V,R) ,,00,. ⎜ ⎟ v0 mq() ++ m w mq() ++ m w at which information is spread by the media rises as ⎝ ⎠ infectives increase, but eventually levels off at a plateau The endemic equilibrium of the system is given by (or asymptote) at which the information (rate) remains constant (i.e. it has reached a maximum number of indi- ^^ ^ ^ ES = (,,IV,R). viduals due to information saturation) regardless of the v1 increase in infections. Such dynamics can easily be     It satisfies and SI>> 00 ,,V>0,R>0 observed in the spread of rumours, gossip and jokes (also known as randomized broadcast) [25,26]. This con-   () Λ++wsVRh(I) stant coverage is extended by examining more complex S = hI () effects which involve more than just reducing contacts 2 down the line. The news in particular is extremely fickle () Λ +swRh( (I)) V = so that what is news one day may be forgotten about hI ()h ()I − w(h (() I ) 23 1 next week; including the media effects in some more lI sophisticated way such as by an impulsive pulsing is R = , also investigated. The limited power of the infection due ms + to contact is accounted for by the saturation incidence. where h (I)= m + I, h (I)=(θ + µ + b I)h (I) – b I , 1 1 2 1 1 2 The first available information is the reported number h (I)= (θ + ω +(1 – g)b I)h (I) – b (1 – g)I . Substitut- 3 1 1 3 of infected individuals when the disease is emerging. We ing the above into the second equation at equilibrium assume that media coverage can slow but not prevent will yield the expression for Î after some rearrangement. disease spread, so b ≥ b and b ≥ b . 1 2 1 3 For illustration, suppose θ = b = b = 0. Then the 2 3 The above model is closely related to those in [27,28] endemic equilibrium satisfies to analyze the transmission dynamics of human influ- enza, but there are some differences. In [27], the authors   consider the inflow of infective immigrants, while in [28] RI = ms + the model includes treatment. Neither of these are con- sidered here. Our model is clearly a crude reflection of () am++l   SI = , the complicated nonlinear phenomena of the transmis- sion dynamics, and it does not incorporate the self-con- trol property due to the change of avoidance patterns of where Î is the positive solution to the quadratic individuals at different stages of the infectious process ma() ++ m l sl [2]. News coverage may have a significant impact on   Λ− II −+() am+l + I = 0. avoidance behaviours at both individual and society b ms + levels, which may reduce the effective contact between The basic reproductive ratio, R , is defined as the susceptible and infectious individuals; we include this v expected number of secondary infections caused by an via a saturation incidence functional response. infective individual upon entering a totally susceptible Since the model monitors human populations, all the population [29-31]. This quantity is not only important in variables and parameters of the model are nonnegative. describing the infectious power of the disease, but can also Based on biological considerations the system of equa- can supply information for controlling the spread of the tions (1)-(4) will be studied in the following region, disease [32]. The linear stability of E is governed by the v0 basic reproductive ratio R . Using the next-generation Ω= {(SI , ,R,V ) ∈ } v method [31], we have which is positively invariant and attracting (thus, the bmΛΛ () +w bg() 1 − q model is mathematically and epidemiologically well- ⎛ ⎞ ⎜ ⎟ posed); it is therefore sufficient to consider solutions mq() ++ m w mq() ++ m w in Ω. Existence, uniqueness and continuation results ⎜ 0 ⎟ F = for model system (1)-(4) hold in this region and all ⎜ ⎟ ⎜ ⎟ solutionsofthissystemstartingin Ω remain in Ω for ⎜ ⎟ ⎝ ⎠ all t ≥ 0. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 5 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 and free equilibrium of the system where ⎛ΛΛ () mw + q ⎞ X = , . ⎜ ⎟ For the set of equations () am++l 000 mq() ++ m w mq() ++ m w ⎛ ⎞ ⎝ ⎠ ⎜ ⎟ in (1)-(4), we set X =(S, V, R)and Z =(I). The con- G G bg() 1 − qΛ ⎜ ⎟ () mw+−q 0 ditions (H1) and (H2) below must be met for global ⎜ ⎟ mq() ++ m w V = ⎜ ⎟ stability. bm Λ() +w dX G * ⎜ 1 ⎟ −+ wq ( m m) 0 (H1) For = FX(, 0),X is globally asymptoti- GG ⎜ ⎟ mq() ++ m w dt cally stable. ⎜ ⎟ ⎜ ⎟ −+ lm 00 ()s (H2) G(X , Z )= A Z – Ĝ(X , Z ), Ĝ(X , Z ) ≥ 0 ⎝ ⎠ G G G G G G G G for (X , Z ) Î Ω where is an M-matrix AD = (,X 0) G G GZG G The basic reproductive ratio is the spectral radius (the off-diagonal elements of A are nonnegative) and Ω –1 r(FV ) which is is the region where the model makes biological sense. If the above two conditions are satisfied, then the fol- bmΛΛ () ++w b(1−g)q lowing theorem holds. R = . (5) ma() ++ l m(q + m+w) Theorem 2 (Castillo-Chavez et al,[33]): The fixed point is a globally stable equilibrium of UX =(, 0) 0GG (2.28) provided that R <1 and that assumptions (H1) and (H2) are satisfied. Local stability of the disease-free equilibrium Lemma 1The disease-free equilibrium E is locally v0 ⎛ ⎞ Λ−+() mqSV +w FX(, 0) = asymptotically stable if R <1, and unstable if R >1. ⎜ ⎟ v v G ⎜ ⎟ qmSV −+()w ⎝ ⎠ Proof. The Jacobian of the system evaluated at E is v0 given by ⎛ ⎞ IS(( bb+− 1 gg )V GX(,Z ) = ⎜ ⎟ GG ⎜ ⎟ mI + ⎛ bm Λ() +w ⎞ ⎝ ⎠ −+() qm − ws ⎜ ⎟ ⎟ mq() ++ m w ⎜ ⎟ ⎜ bmΛΛ () ++w b(1−g)q ⎟ 11 Therefore, E is globally asymptotically stable (GAS) 0 − (() am++l 00 v0 ⎜ ⎟ J mq() ++ m w . E = v 0 ⎜ ⎟ since Ĝ(X ,Z ) > 0. The GAS of E excludes any possi- G G v0 ⎜ ⎟ bg() 1 − qΛ q − −+() wm 0 ⎜ ⎟ bility of the phenomenon of backward bifurcation. We mq() ++ m w ⎜ ⎟ ⎜ ⎟ 00lm−+()s note that the GAS of the DFE E when s =0is ⎝ ⎠ v0 straightforward. The eigenvalues of J are V0 The optimal control model bmΛΛ () ++w b(1−g)q Vm =− , V = −+() am+l, Our objective in this section is to extend the initial mq() ++ m w model to include two intervention methods, called con- Vm =−() +s, V = −−+() mq+w. trols, represented as functions of time and assigned rea- For local stability of the disease-free equilibrium, we sonable upper and lower bounds, each representing a require that all the eigenvalues be negative. Three of the possible method of influenza intervention. Using optimal control theory and numerical simulations, we determine eigenvalues satisfy this condition while ς < 0 implies the benefit of vaccination and media coverage when the that R < 1 and, consequently, all the eigenvalues of the latter has positive or negative effect on the former. Jacobian matrix above have negative real part. Thus, the We will integrate the essential components into one disease-free equilibrium is locally asymptotically stable. SIVR-type model to accommodate the dynamics of an influenza outbreak determined by population-specific para- Global stability of the disease-free equilibrium meters such as the effect of contact reduction when infec- We adopt the method of Castillo-Chavez et al,[33] and tious and vaccinated individuals are reported in the media. we rewrite the set of model equations in the form Let u and u be the control variables for vaccination v m dX and media coverage respectively. Thus, model (1)-(4) = FX(,Z ) GG now reads dt dZ = GX(,Z ), GG dS dt =+ Λwq Vu − ((1− ) +m)S dt (6) with G(X ,0) = 0. X Î ℝ denotes the number of G G ⎛ ⎞ −−bb SI +s R uninfected classes and Z Î ℝ denotes the number of ⎜ ⎟ G 12 () 1−+ um I * mI ⎝ ⎠ infected classes. denotes the disease- UX =(, 0) 0GG Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 6 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 Existence of an optimal control ⎛ ⎞ dI I =−bb SI ⎜ 12 ⎟ The existence of an optimal control can be obtained by dt () 1−+ um I ⎝ mI ⎠ (7) using a result by Joshi [36] and Fister et al.[37]. ⎛ ⎞ Theorem 3Consider the control problem with the sys- +−bb () 1 −g VIII −+() am+l ⎜ 13 ⎟ () 1−+ um I ⎝ mI ⎠ tem of Equations (4.1)-(4.4). There exists an optimal con- * ** trol such that max [Ju( ,) u |(u ,) u ∈= u] Ju( ,) u () u vm vm vm Proof. To prove this theorem on the existence of an dV =−() 1 uSqm −( +w)V optimal control, we use a result from Fleming and dt Rishel [38] (Theorem 4.1 pp. 68-69), where the follow- (8) ⎛ ⎞ ing properties must be satisfied. −−bb () 1 −g VI ⎜ ⎟ 1. The set of controls and corresponding state vari- () 1−+ um I mI ⎝ ⎠ ables is nonempty. 2. The control set U is closed and convex. dR 3. The right-hand side of the state system is bounded =−lmIR () +s . (9) dt above by a linear function in the state and control. 4. The integrand of the functional is concave on U A balance of multiple intervention methods can differ k k and is bounded above by c – c (|u | + |u | ), where c , 2 1 v m 1 between populations. A successful mitigation scheme is c > 0 and k >1. one which reduces influenza-related deaths with mini- An existence result in Lukes [39] (Theorem 9.2.1) for mal cost. A control scheme is assumed to be optimal if the system of equations (6)-(9) for bounded coeffi- it maximizes the objective functional cients is used to give the first condition. The control set is closed and convex by definition. The right-hand Ju((t),u (t)) = vm side of the state system (Equations (4.1)-(4.4)) satisfies tf (10) Condition 3 since the state solutions are a priori [St ()+− V()t B I()t − B (u ()t + u ()t )]dt. 12 vm ∫ ∫ t 0 bounded. The integrand in the objective functional, , is concave on St ()+− V()t B I()t − B (u ()t + u ()t ) 12 vm The first two terms represent the benefit of the sus- U.Furthermore, c , c >0 and k >1,so 1 2 ceptible and vaccinated populations. The parameters B and B represent the weight constraints for the infected St ()+− V()t B I()t − B (u ()t 12 v population and the control, respectively. They can also (11) 2 k k represent balancing coefficients transforming the inte- +≤ ut ()) c −c (|u |+ |u | ). mv 21 m gral into dollars expended over a finite time period of T days [34]. The goal is to maximize the populations of Therefore, the optimal control exists, since the left- susceptible and vaccinated individuals, minimize the hand side of (11) is bounded; consequently, the states population of infectives, maximize the benefits of media are bounded. coverage and vaccination, while minimizing the systemic Since there exists an optimal control for maximizing costs of both media coverage and vaccination. The value the functional (10) subject to equations (6)-(9), we use u (t)= u (t) = 1 represents the maximal control due to v m Pontryagin’s Maximum Principle to derive the necessary vaccination and media coverage, respectively. The terms conditions for this optimal control. Pontryagin’sMaxi- 2 2 and represent the maximal cost of Bu ()t Bu ()t 2 v 2 m mum Principle introduces adjoint functions that allow education, implementation and campaigns on both vac- us to attach our state system (of differential equations), cination and media coverage. S(t)and V(t) account for to our objective functional. After first showing existence the fitness of the susceptible and the vaccinated groups of optimal controls, this principle can be used to obtain as a result of a reduction in the rate at which the vac- the differential equations for the adjoint variables, corre- cine wanes, and vaccination and treatment efforts are sponding boundary conditions and the characterization * * implemented [35]. We thus seek optimal controls ut () v of an optimal control and . This characterization u u v m and such that ut () m gives a representation of an optimal control in terms of the state and adjoint functions. Also, this principle con- ** Ju(,u (t)=∈ max[Ju(,u )|(, u u ) U], vm vm vm verts the problem of minimizing the objective functional subject to the state system into minimizing either the where U ={(u , u )|u , u measurable, 0 ≤ a ≤ u ≤ v m v m 11 v Lagrangian or the Hamiltonian with respect to the con- b ≤ 1, 0 ≤ a ≤ u ≤ b ≤ 1, t Î [0, t ]} is the control 11 22 m 22 f trols (bounded measurable functions) at each time t[40]. set, with t Î [t , t ]. The basic framework of this pro- 0 f The Lagrangian is defined as blem is to characterize the optimal control. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 7 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 optimality technique is utilized. The following cases are LS=+ ()t V()t −BI()t −B (u ()t +u ()t )+lw [Λ+ V− ((1−u )q+m)S 12 vm 1 v considered to determine a specific characterization of ⎛ I ⎞ − bbb − SI +s R] ⎜ 12 ⎟ () 1−+ um I the optimal control. ⎝ mI ⎠ ⎛ ⎞ Case 1: Optimality of I u +− lb [ b SI ⎜ ⎟ 21 2 −+ um I () 1 ⎝ mI ⎠ 1. On the set . {|ta << u (t) b },w =w = 0 11 v 11 11 12 ⎛ I ⎞ Hence, the optimal control is + bbb − () 1−− g VI (a+m+l)I] ⎜ 13 ⎟ () 1−+ um I ⎝ mI ⎠ () ll − qS ⎛ ⎞ I * +−lq [(1 uS ) − (m+w)V V−−bb () 1 −g VI] ⎜ ⎟ ut () = 3 v 13 () 1−+ um I mI ⎝ ⎠ 2B +− ll[(IR m+s) ] +w ()ta ( −−+ut )( w )(u −b ) 11 11 vv 12 11 2. On the set .We have {|ta== u (t)},w 0 11 v 11 +−ww ()ta ( u )+ ()tu ( −b ), 21 22 mm 22 22 () ll−+qSw (t) 13 12 where w (t) ≥ 0, w (t) ≥ 0 are penalty multipliers ut () = 11 12 2B satisfying w (t)(a – u (t)) + w (t)(u (t) – b )at the 11 11 v 12 v 11 optimal , and w (t) ≥ 0, w (t) ≥ 0 are penalty multi- u 21 22 or pliers satisfying w (t)(a – u (t)) + w (t)(u (t) – b ) 21 22 m 22 m 22 at the optimal . () ll − qS * * ut () = ≤ a v 11 Given optimal controls and , and solutions of the u u v m 2B corresponding state system (6)-(9),there exist adjoint variables l , for i = 1, 2, 3, 4 satisfying the following i since w ≥ 0. equations 3. On the set . Hence {|tb== u (t)},w 0 11 v 12 dl ∂L =− () ll−−qSw(t) * 13 11 dt ∂S ut () = 2B =−1 +() ll − (b −b )(Iu +−ll )(1− )q+l m 12 1 2 1 3 v 1 1 () 1−+ um I mI dl ∂L or =− dt ∂I ⎛ ⎞ I () 1 −um mI =+ B () ll− (b −b )S −b IS () ll − qS ⎜ ⎟ 11 2 1 2 2 * 13 ⎜ ⎟ () 1−+ um I ((1 −−+ um))I ut () = ≥ b . mI ⎝ mI ⎠ v 11 2B +−() ll (b −b )(1 −g )V 32 1 3 () 1−+ um I mI Combining all the three sub-cases in a compact form () 1 −um mI −b () 1 −gl VI ++(a m+l)−ll 3 3 24 gives ((1−+ um ) I) mI ⎠ dl ∂L =− ⎧ ⎫ ⎧ ⎫ dt ∂V V ⎪ () ll − qS ⎪ * 13 ut () = min max a , ,. b (12) I v ⎨ ⎨ 11 ⎬ 11 ⎬ =−1 +() ll − (b −b )(1−+ g )I lm+ (ll− )w 2B 32 1 3 3 3 1 ⎪ ⎩ 2 ⎭ ⎪ ⎩ ⎭ () 1−+ um I mI dl ∂L L =− dt ∂R Case 2: Optimality of =−() ll s+lm * 41 4 1. On the set . {( ta | << u t) b},w = w = 0 22 m 22 21 22 We have with transversality conditions l [t ] = 0, for i =1, 2, 3, i f 4. To determine the interior maximum of our Lagran- b mSI * 2 I gian, we take the partial derivatives of L with respect to ut ()=− (ll ) m 12 21Bu ((−+ )m I) u and u , respectively, and set it to zero. Thus, 2 mI v m bgmV () 1 − I 3 I ∂L +−() ll . * 23 =−2Bu ()t + (ll − )qS − w ()t + w ()t 21 v 3 11 12 2BBu ((1−+ )m I) 2 mI ∂ut () ∂L b mSI * * 2 I 2. On the set . We have {|ta== u (t)},w 0 =−2Bu () t +− (ll ) 22 m 21 2 m 12 ∂ut () ((1−+ um ) I) m mI b mSI bgmV () 1 − I * 2 I 3 I +−() ll −+ wt () w ()t ut ()=− (ll ) m 12 23 21 22 2 2 ((1 − u)) mI + 21Bu ((−+ )m I) mmI 2 mI bgmV () 1 − I wt () 3 I 22 To determine an explicit expression for our controls +−() ll + * * 2B 2BBu ((1−+ )m I) , (without w ,w , w , w ), a standard u u 2 mI 2 11 12 21 22 m m Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 8 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 or obtain the uniqueness of the optimal control for small [t ] [36]. The uniqueness of the optimal control follows b mSI from the uniqueness of the optimality system. * 2 I ut ()=− (ll ) m 12 The state system of differential equations and the 21Bu ((−+ )m I) 2 mI adjoint system of differential equations together with bgmV () 1 − I 3 I the control characterization above form the optimality +−() ll ≤ a 23 22 2BBu ((1−+ )m I) 2 mI system solved numerically and depicted in Figures 2, 3, 4, 5. since w ≥ 0. 3. On the set . Hence {|tb== u (t)},w 0 22 m 22 The model with pulse vaccination The general model with pulse vaccination is given as b mSI * 2 I ut ()=− (ll ) m 12 21Bu ((−+ )m I) ⎛ ⎞ dS I 2 mI =+ Λ wm VS − − b −b SI +s R ⎜ 12 ⎟ dt mI + ⎝ I ⎠ bgmV () 1 − I wt () 3 I 21 +−() ll − ⎛ ⎞ ⎛ ⎞ 23 dI I I 2B =−bb SI + bbb − () 1−− g VI (a+m+l)I 2BBu ((1−+ )m I) ⎜ 12 ⎟ ⎜ 13 ⎟ 2 mI 2 dt mI + mI + ⎝ I ⎠ ⎝ I ⎠ ⎛ ⎞ dV I or =−() mw + V − b −b () 1 − g VI ⎜ ⎜ 13 ⎟ dt mI + ⎝ I ⎠ 2 2 dR b mSI bgmV () 1 − I 2 I 3 I ut ()=− (ll ) +−() ll ≤ b . =−lmIR () +s , m 12 23 22 2 2 21Bu ((−+ )m I) 2BBu ((1−+ )m I) dt 2 mI 2 mI Combining all the three sub-cases in a compact form for t ≠ t ,where t is the time of the kth vaccination. k k gives We may have t – t either constant or not, as we k+1 k choose. The impulsive effect is given by 2 2 ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ () ll − b mSI () l −l b m() 1 − g VI ⎪ ⎪ * 12 2 I 23 3 II ut () = min max a , + ,. b ⎨ ⎨ ⎬ ⎬ (13) m 22 2 2 22 ΔS = –θS 21Bu ((−+ )m I) 21Bu ((−+ )m I) ⎪ ⎪ 2 mI 2 mI ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ ΔV = θS +− when t = t . Here, is thechangein Δ≡yy()t −y(t ) kk state at the impulse time. The optimal system In this model, vaccination occurs at fixed times, not The optimality system consists of the state system continuously. This is closer to reality, since vaccination coupled with the adjoint system, with the initial condi- centres are only open at certain times, when people may tions, the transversality conditions and the characteriza- get vaccinated in waves. Similarly, media stories tend to tion of the optimal control: clump together, so that a big news story occurs on one day, which may trigger a short period of intense vacci- ⎛ ⎞ dS I =+ Λwq Vu − ((1− ) +m)S−b −b SI +s R ⎜ ⎟ nation. We shall use a simplified version of this model v 12 ⎜ * ⎟ dt () 1−+ um I ⎝ mI ⎠ to illustrate the possibility that media may have an ⎛ ⎞ ⎛ ⎞ dI I I ==− ⎜ bb ⎟ SI+− ⎜ bb ⎟ (1 − gga )( VI−+m+l)I 12 1 3 ** ⎜ ⎟ ⎜ ⎟ dt adverse effect. ()11 −+ um I ()−+ um I ⎝ mI ⎠ ⎝ mI ⎠ ⎛ ⎞ dV I =−() 1 uSqm −( +w)V− b −b () 1 − g VI v ⎜ 1 3 ⎟ ⎜ ⎟ dt () 1−+ um I ⎝ mI ⎠ ⎠ Adverse effects dR =−lmIR () +s Consider the following scenario. At the onset of the out- dt dl I 1 * break, the media - and hence the general population - is =−1 +() ll − (b −b )(Iu +−ll )(1− )q+lm 12 1 2 13 v 1 dt () 1−+ um II mI unaware of the disease and thus nobody gets the vac- ⎛ ⎞ d I () −um l 1 2 mI =+ B () ll− (b −b ) )S − b IS ⎜ ⎟ 11 2 1 2 2 cine, allowing the disease to spread in its initial stages. ⎜ * * 2 ⎟ dt () 1−+ um I ((1−+ um ) I) ⎝ mI mI ⎠ * At some point, there is a critical number of infected ⎛ ⎞ I () 1 −um mI +−() ll ⎜(b −b )(1−−gb )V () 1 −g VI ⎟ 32 1 3 3 * * 2 ⎜ ⎟ (1 − u )mI + ((1−+ um ) I) individuals, whereupon people are sufficiently aware of mmI mI ⎝ ⎠ +l() am++l −l l 224 the infection to change their behaviour. We suppose dl I =−1 +() ll − (b −b )(1−+ g )I lm ++−() ll w that, initially, new infected people arrive at fixed times. 32 1 3 3 31 dt () 1−+ um I mI We further assume that vaccinated people mix more dl =−() ll s+lm 41 4 dt than susceptibles. In this case, people who are vacci- nated feel confident enough to mix with the infected, * * where and are given by expressions (12) ut () ut () v m even though they may still have the possibility to con- and (13), respectively, with S(0) = S , I(0) = I , V(0) = 0 0 tract the virus. This might be the case for health-care V , R(0) = R and l [t ]= 0for i =1,··· ,4. Due to the a 0 0 i f workers, for instance, who get vaccinated and then have priori boundedness of the state and adjoint functions to tend to the sick. and the resulting Lipschitz structure of the ODEs, we Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 9 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 40 18 i1 v1 i2 v2 i3 v3 10 4 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 2 Optimality effect when the weight constraint for the infected population varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint for the infected population varies. (a) Infected individuals. (b) Vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) =50.0, R(0) = 40.0. The value of the weights used are (i) B1=0.0025 corresponds to variables with subscript 1 (++), (ii) B1=25.0 corresponds to variables with subscript 2 (xx), (iii) B1= 250000.0 corresponds to variables with subscript 3 (**). The value B2=0.0025 is kept constant in all three cases. Mathematically, we have a threshold for the critical For I >I , the model becomes crit number of infectives, I . crit dS For I <I , this model would look like crit =+ Λ wqVS −() +m −(b −b)SI+sR dt dS dI =+ Λ wm VS − −bSI+sR 1 =−() bb SI+b(1−g)VI−(a+m + + l)I 46 5 dt dt dI dV =+bb SI () 1−g VI−(a+m+l)I 45 =−qmSV () +w −b(1−g)VI dt dt dV dR =−(mw + ))(VV −−bg 1 )I 5 =−lmIR () +s , dt dt dR =−lmIR () +s . with b – b ≥ 0. 4 6 dt 40 50 i1 v1 i2 v2 i3 v3 25 25 10 0 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 3 Optimality effect when the weight constraint for the control varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint for the control varies. (a) Graph of infectives, (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B2=25.0 corresponds to variables with subscript 1 (++), (ii) B2 = 2500.0 corresponds to variables with subscript 2 (xx), (iii) B2 = 250000.0 corresponds to variables with subscript 3 (**). The value B1=0.0025 is kept constant in all three cases. Infectives Infectives Vaccinated Vaccinated Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 10 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 40 5 i1 v1 i2 v2 i3 v3 35 4.5 30 4 25 3.5 20 3 15 2.5 10 2 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 4 Optimality effect when the weight constraint for the infected population varies and media has an adverse effect on the vaccine. Graphical representation of the optimality system when media coverage has an adverse effect on the vaccination rate and when the weight constraint for the infected population varies. (a) Graph of infectives. (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) =25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B1=0.0025 corresponds to variables with subscript 1 (++), (ii) B1=25.0 corresponds to variables with subscript 2 (xx), (iii) B1 = 250000.0 corresponds to variables with subscript 3 (**). The value B2=0.0025 is kept constant in all three cases. However, to illustrate the adverse affect, we shall sim- The model then becomes plify the model even further. For a short timescale, we dS can assume recovery is permanent, so s =0.Thus,we (14) =+ Λ wm VS − t≠t dt can ignore the R equation. For I <I , we assume that there is no mixing, but crit dI rather that new infectives arrive impulsively into the sys- (15) =−() am + +lIt ≠t i i k tem at fixed times t and in numbers I , where I ≪ I . dt k crit (If the new infectives arrive at irregular times, then the broad results will be unchanged.) dV (16) =−() mw +Vt ≠t For I >I , fear of the disease keeps susceptibles from crit dt mixing with the infected, but the vaccinated will. Thus b = b = 0. Since I ≪ I , we can assume that, 4 6 crit (17) Δ=II t =t for I >I , the effects of new infectives are negligible. crit 40 25 i1 v1 i2 v2 i3 v3 10 0 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 5 Optimality effect when the weight constraint for the control varies and media has an adverse effect on the vaccine. Graphs of the optimality system when media coverage has an adverse effect on the vaccination rate and when the weight constraint for the control population varies. (a) Graph of infectives. (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B2= 25.0 corresponds to variables with subscript 1 (++), (ii) B2 = 2500.0 corresponds to variables with subscript 2 (xx), (iii) B2 = 250000.0 corresponds to variables with subscript 3 (**). The value B1=0.0025 is kept constant in all three cases. Infectives Infectives Vaccinated Vaccinated Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 11 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 for I <I and The Jacobian is crit −−qm 0 w ⎡ ⎤ dS (18) =+ Λ wqVS −() +m ⎢ ⎥ JV =01 −+(am+l)+b() −g b()1−gI . ⎢ ⎥ dt ⎢ ⎥ qb−−()11 g V −(m+w)−b()−g II ⎣ 22 ⎦ dI At the endemic equilibrium, bg() 1−= V (a+m+l) . (19) =−bg() 1 VI−(a+m+l)I dt Thus, we have ⎡ ⎤ −−qm 0 w ⎢ ⎥ dV (20)    =−qmSV () +w −b(1−g)VI ⎢  ⎥ 5 JSI (,,V) = 00 . bg() 1 − I dt ⎢ ⎢ ⎥ qa−+()m+l  ⎢ ⎥ −+() mw −b(1−g)I ⎣ 2 ⎦ for I >I . crit Thus, the effects of the media are to trigger a vacci- The characteristic equation is nating panic whenever the number of infectives is det(Jx−=I) −x −x [mw+ +b (1−g )I+q+m] large enough. We kept the model with impulse vacci- 1   −+xI [(qm b (1−g) )+ m(m+w ++−bg()11II )+−bg()(a+m+l)] 2 22 nation as simple as possible since even this simplified −−bg() 1 (a+m+l)(q+m)I. version shows that media reports could have an adverse effect. It follows that the endemic equilibrium is stable if Î Suppose new infectives appear regularly, so that t – k+1 >I . Thus, even in an extremely simplified version of crit t = τ. (If not, the analysis generalizes quite easily.) For the model, the media may make things significantly t <t <t , we have k k+1 worse than if no media effect were included. We kept this model deliberately simple, partly for mathematical +−() am + +l t It () = I e , tractability and partly to show that the media effects apply even in this idealised scenario. ++ where is the value immediately after the II ≡ ()t kk Note that, in reality, the fluctuations would apply in kth impulse. Then, since the period is constant, we the upper region as well, making the actual value even have larger. In the lower region, we ignored interaction between susceptibles and infectives (ie we assume b = −+−() am++l t 4 II = e kk +1 b = 0). The effect of including these terms would be to −− i () am++l t =+() II e . slow the exponential decay between impulses (or possi- bly cause it to increase). This would only increase the This is a recursion relation with solution effect seen here. In summary, a small series of outbreaks that would i −+() am+l t Ie − equilibrate at some maximum level m >I will, as a crit m = . −+() am+l t 1 − e result of the media, instead equilibrate at a much larger value I >m >I . The driving factor here is if an imper- crit Consequently, fect vaccine causes overconfidence, so that people who have been vaccinated mix significantly more with infec- tives than susceptibles do. If this happens (as would be m = . −+() am+l t 1 − e quite likely; most people who have been vaccinated feel invulnerable, even if the vaccine is not perfect, largely Thus, if m >I , then eventually the system will crit thanks to media oversimplifications), then the media switch from model (14)-(17) to model (18)-(20). The effect is likely to be adverse. A simplified version of the endemic equilibrium in model (18)-(20) satisfies model with pulse vaccination shows that the media can make things worse, if the vaccine is imperfect because () am++l V = the vaccinated mix over-confidently with the infectives. bg () 1 − Λ wa() ++ m l  Numerical simulations S = + qm + bg () 1−+(q m) We now return to model (6)-(9) and illustrate some of the properties discussed in the previous sections. The Λqb() 1−− g mmq() ++ m w(a ++ m l) I = . parameter values that we use for numerical simulations () qm++(a m+l)b(1−g) 2 Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 12 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 are in Table 1. Initial conditions: S(0) = 200.0, I(0) = 1.0, 4, we vary the cost weight, while in Figure 5, we vary V(0) = 10.0, R(0) = 0.0. The parameter θ varies between the weight of minimizing infectives. 0.3 and 0.7withanaverage of0.5 [41]. We consider an We note from Figure 2 that, during the initial days, imperfect flu vaccine for which the waning rate is about there is a very sharp drop in the population of infected 0.15. The relationship between b and b is not very individuals, while other populations show increases. 2 3 obvious; consequently, we can either assume equality or Increasing minimization of infectives, while keeping that the former is slightly greater than the latter. Trans- costs low, can lead to the disease being controllable. The slight rise and fall, after the initial 20 days, in the mission dynamics of infectious diseases with and with- population of infectives could be attributed to compla- out media coverage have already been carried out in previous studies, but these models do not account for cency on the part of some individuals (or may be due to the vaccination coverage. Therefore, we illustrate some oscillations in the system independent of external fac- numerical results for the model with optimal control tors). We find that people tend to relax after the initial when media coverage has (i) a beneficial effect (see Fig- shock of the disease threat. However, we note that this ures 2 and 3) (ii) and adverse effect on the vaccination is not for long, and this could be attributed to the fact rate (see Figures 4 and 5). that vaccination levels continue to rise, so as people The optimality system is solved using an iterative continue to receive vaccination, infection is controlled. method with a fourth order Runge-Kutta scheme. Start- Thus, if costs are kept minimal, and more people are ing with a guess for the adjoint variables, the state equa- able to access media and vaccination, then infection can tions are solved forward in time. Then the state values be controlled. Both vaccination and media coverage con- obtained are used to solve the adjoint equations back- tinue at optimum levels as a result of the low costs and ward in time; the iterations continue until convergence. minimization of infectives. Simulations are carried out to determine how maximiz- From Figure 3, as costs are increased, few people have ing media coverage enhances vaccination. The effects of access to media and vaccination; as a result, low num- costs that can be incurred, which include education, bers get vaccinated against the disease. In the long run, implementation and campaigns on media coverage, are the infection levels rise. The degree of media coverage also studied to evaluate how these costs can affect the and vaccination also decrease as a result of the exorbi- transmission of human influenza. We increase the value tant costs. With the little available media coverage and of B2 (the cost weight) in Figure 2 to assess how the the few vaccinated individuals, we find that, due to populations of susceptibles, infectives, vaccinated and information filtration, there is a jump in the vaccination levels, though these only last briefly; as the degree of recovered individuals are altered. In Figure 3, we investi- gate how increasing minimization of infectives through media coverage and vaccination decrease, so do the vac- increasing the weight B1 affects the control of human cination levels. influenza transmission. We do the same in Figures 4 From Figure 4, even though costs are kept at minimal and 5, respectively, to see how, if media coverage has an levels, the negative reports concerning vaccination result adverse effect, the various populations behave. In Figure in a drastic reduction in the vaccination levels. After some time, we note a slight increase in the vaccination levels; however, these numbers remain very low. This Table 1 Parameter values could be due to the fact that, as infection rises, a few will risk getting vaccinated in the hope of being cured. Parameter Symbol Value Units Reference Thus, media coverage can have adverse effects if peo- Recruitment rate Λ 5.0 People day [3] ple’s perception towards the vaccine is negatively influ- –1 Rate at which vaccine ω 0.15 day Assumed enced by the media. wanes In Figure 5, both media coverage and vaccination are –1 Vaccine uptake rate θ 0.3- day [41] eventually withdrawn. Very low numbers get vaccinated. 0.7 It is only when infection escalates that vaccination levels –1 Natural death rate µ 0.02 day [3] also increase as some might find it better to try to pre- Infection rate b 0.02 people [3] 1 –1 vent the infection, despite the negativity towards vacci- day –1 nation in the media. Figure 6 illustrates other potential Loss of immunity s 0.01 day [3] adverse effects that media may have, if the effect is to Vaccine efficacy g 0.8 (unitless) [3] –1 trigger a vaccinating panic where vaccinated individuals Infection death rate a 0.1 day [3] –1 are not fully protected and mix with infected individuals Recovery rate of Infectives l 0.05 day [3] but susceptible individuals do not. In this case, the num- Reaction due to media m 10.0 people Assumed coverage ber of infected individuals may increase sharply as a result of the media. Figure 7 illustrates the long-term Media coverage model parameters, their interpretations and values Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 13 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 3.2 2.8 With media effects 2.6 16.5 17 17.5 18 18.5 Vaccination panic threshold Without media effects 0 5 10 15 20 25 time (days) Figure 6 The vaccination panic threshold The effect of the vaccination panic threshold using the simplified model (14)-(20). Without media triggering a vaccinating panic, the number of infected individuals remains low (solid purple curve). However, if the media triggers a vaccinating panic, then the number of infected individuals rises sharply (dashed green curve). Inset: Comparison of the two outcomes around the vaccination threshold. results of a media-induced vaccination panic. Without Conclusion media effects, the result is a low-level infection. When Media simplifications can lead to overconfidence in the the media triggers a vaccinating panic, there is a large idea of a vaccine as a cure-all. The result is not just a outbreak, followed by an endemic level of infected indi- vaccinating panic and a blow-out epidemic, but a net viduals significantly higher than the level of infected increase in the endemic equilibrium. Thus, media cover- individuals without the media effects. Note that these age of an emerging epidemic can fan the flames of fear examples assume no post-vaccination mixing of suscep- and also implicitly reinforce an imperfect solution as the tible and infected individuals. only answer. Figure 8 illustrates the cases when post-vaccination We have formulated and investigated a simple deter- mixing of susceptible and infected individuals is maxi- ministic vaccination model describing the effects of mal (b = b , b = 0), 50% (b = b , b = b /2) or zero 4 5 6 4 5 6 4 media coverage on the transmission dynamics of influ- (b = b = 0). Thus, if susceptible and infected indivi- 4 6 enza. The media effect due to reporting the number of duals mix after a vaccinating panic has occurred, the infections as well as the number of individuals success- effect is an earlier outbreak and a larger number of fully vaccinated is introduced into the compartmental infected individuals. model via a saturated incidence-type function. The Infected individuals Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 14 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 90 95 100 With postvaccination fluctuations Without media effects Without postvaccination fluctuations 0 20 40 60 80 100 time (days) Figure 7 A media-induced epidemic Long-term dynamics for the simplified model (15)-(20). Without media effects, the maximum number of infected individuals remains low (solid purple curve). When media effects trigger a vaccinating panic where partially protected vaccinated individuals mix with infectives significantly more than susceptible individuals, a significant outbreak may occur, with the final number of infected individuals much higher than if no media effects had been included (dashed green curve, assuming no fluctuations after the vaccinating panic occurs). The result of including fluctuations above the vaccination panic threshold are also shown (solid blue curve). Inset: magnified view of the fluctuation versus nonfluctuation cases. impact of costs that can be incurred, which include vac- effects of media reporting on the transmission dynamics cination, education, implementation and campaigns on of infectious diseases for which a vaccine exists. The media coverage, are also investigated using optimal con- present study is in no way exhaustive and can be trol theory applied via the Pontryagin’smaximum prin- extended in various ways: for example, to investigate the ciple. A simplified version of the model with pulse case in which there is media coverage but people ignore vaccination shows that the media can have an adverse it (in which case the vaccination rate is unchanged effect if the vaccine is imperfect and the vaccinated mix despite the control). Thus, the effects of media on an over-confidently with the infectives. Numerical simula- outbreak of influenza with a partially effective vaccine tions are carried out to support the analytical results. may be complicated. While the media may encourage We note, however, that our caricature model is not more people to get vaccinated, they may also trigger a complete; a more comprehensive study will require vaccinating panic or promote overconfidence in the abil- interdisciplinary research across traditional boundaries ity of a vaccine to fully protect against the disease. This of social, natural, medical sciences and mathematics [2]. may have potentially disastrous consequences in the Nevertheless, our work provides some insights into the face of a new pandemic. Infected individuals Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 15 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 Maximal postvaccination mixing of susceptibles and infecteds 50% postvaccination mixing of susceptibles and infecteds No postvaccination mixing of susceptibles and infecteds 0 20 40 60 80 100 time (days) Figure 8 The effect of different post-panic mixing rates Comparison of mixing rates between susceptibles and infected individuals after a vaccinating panic. The more susceptibles mix with infecteds, the earlier the outbreak occurs and the larger the number of infected individuals. Author details Authors contributions Department of Mathematics and Statistics, University of Guelph, Guelph, JMT, RJS and CTB developed the model. JMT and CTB Ontario N1G 2W1, Canada. Department of Applied Mathematics, National designed and formulated the study framework and ana- University of Science and Technology, Box AC 939 Ascot, Bulawayo, Zimbabwe. Department of Mathematics and Faculty of Medicine, The lyzed the model. CPB and ND carried out the optimal University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada. control analysis and numerical simulations for Figures 2, 3, 4, 5. RJS wrote the section on adverse effects, some of Competing interests The authors declare that they have no competing interests. the introduction, performed numerical simulations for Figures 16,7,8, and edited the manuscript. All authors Published: 25 February 2011 read and approved the final manuscript. References 1. Laxminarayan R, Mills AJ, Breman JG, Measham AR, Alleyne G, Claeson M, Acknowledgements Jha P, Musgrove P, Chow J, Shahid-Salles S, Jamison DT: Advancement of We thank Shoshana Magnet, Penelope Ironstone-Catterall and Tim Reluga global health: key messages from the Disease Control Priorities Project. for technical discussions. JMT was supported through a postdoctoral Lancet 2006, 367:1193-1208. appointment funded by a grant from the Ontario Ministry of Research and 2. Liu R, Wu J, Zhu H: Media/psychological impact on multiple outbreaks of Innovation awarded to CTB. RJS is supported by an NSERC Discovery Grant, emerging infectious disease. Comput. Math. Meth. Med. 2007, 8(3):153-164. an Early Researcher Award and funding from MITACS. Handling editor for 3. Liu Y, Cui J: The impact of media coverage on the dynamics of infectious this manuscript was Jane Heffernan. disease. Int. J. Biomath. 2008, 1:65-74. This article has been published as part of BMC Public Health Volume 11 4. Simpson CR: Nature as News: Science Reporting in The New York Times Supplement 1, 2011: Mathematical Modelling of Influenza. The full contents 1898 to 1983. The International Journal of Politics, Culture and Society 1987, of the supplement are available online at http://www.biomedcentral.com/ 1(2):218-241. 1471-2458/11?issue=S1. 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Interface 2005, 2(4):281-293. 31. van den Driessche P, Watmough J: Reproduction numbers and sub- • Convenient online submission threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002, 180:29-48. • Thorough peer review 32. Cui J, Tao X, Zhu H: An SIS infection model incorporating media • No space constraints or color figure charges coverage. Rocky Mountain J. Math 2008, 38(5):1323-1334. • Immediate publication on acceptance 33. Castillo-Chavez C, Feng Z, Huang W: On the computation of R0 and its role on global stability. In Mathematical Approaches for Emerging and • Inclusion in PubMed, CAS, Scopus and Google Scholar Reemerging Infectious Diseases: An Introduction. Berlin-Heidelberg-New York: • Research which is freely available for redistribution Springer-Verlag;Castillo-Chavez C, van den Driessche P, Kirschner D, Yakubu AA 2002:229-250. 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The impact of media coverage on the transmission dynamics of human influenza

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Springer Journals
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Copyright © 2011 by Tchuenche et al; licensee BioMed Central Ltd.
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Medicine & Public Health; Public Health; Medicine/Public Health, general; Epidemiology; Environmental Health; Biostatistics; Vaccine
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1471-2458
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10.1186/1471-2458-11-S1-S5
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21356134
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Abstract

Background: There is an urgent need to understand how the provision of information influences individual risk perception and how this in turn shapes the evolution of epidemics. Individuals are influenced by information in complex and unpredictable ways. Emerging infectious diseases, such as the recent swine flu epidemic, may be particular hotspots for a media-fueled rush to vaccination; conversely, seasonal diseases may receive little media attention, despite their high mortality rate, due to their perceived lack of newness. Methods: We formulate a deterministic transmission and vaccination model to investigate the effects of media coverage on the transmission dynamics of influenza. The population is subdivided into different classes according to their disease status. The compartmental model includes the effect of media coverage on reporting the number of infections as well as the number of individuals successfully vaccinated. Results: A threshold parameter (the basic reproductive ratio) is analytically derived and used to discuss the local stability of the disease-free steady state. The impact of costs that can be incurred, which include vaccination, education, implementation and campaigns on media coverage, are also investigated using optimal control theory. A simplified version of the model with pulse vaccination shows that the media can trigger a vaccinating panic if the vaccine is imperfect and simplified messages result in the vaccinated mixing with the infectives without regard to disease risk. Conclusions: The effects of media on an outbreak are complex. Simplified understandings of disease epidemiology, propogated through media soundbites, may make the disease significantly worse. Introduction People’s response to the threat of disease is dependent Infectious diseases are responsible for a quarter of all on their perception of risk, which is influenced by public deaths in the world annually, the vast majority occurring and private information disseminated widely by the media. in low- and middle-income countries [1]. There are dis- While government agencies for disease control and pre- eases such as SARS and flu that exhibit some distinct fea- vention may attempt to contain the disease [3], the general tures such as rapid spatial spread and visible symptoms information disseminated to the public is often restricted [2]. These features, associated with the increasing trend to simply reporting the number of infections and deaths. of globalization and the development of information Mass media are widely acknowleged as key tools in risk technology, are expected to be shared by other emerging/ communication [4,5], but have been criticised for making re-emerging infectious diseases. It is therefore important risk a spectacle to capitalise on audience anxiety [6,7]. to refine classical mathematical models to reflect these The original interpretation of media effects in communi- features by adding the dimensions of massive news cov- cation theory was a “hypodermic needle” or “magic bullet” erage that have great influence not only on the individual theory of the mass media. Early communication theorists behaviours but also on the formation and implementa- [8,9] imagined that a particular media message would be tion of public intervention and control policies [2]. directly injected into the minds of media spectators. This theory of media effects, in which the mass media has a direct and rapid influence on everyday understanding, has * Correspondence: rsmith43@uottawa.ca Department of Mathematics and Faculty of Medicine, The University of been substantially revised. Contemporary media studies Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada analyses how media consumers might only partially accept Full list of author information is available at the end of the article © 2011 Tchuenche et al; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 2 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 a particular media message [10], how the media is shaped reporting of both disease dynamics and vaccination is by dominant cultural norms [11,12] and how media con- high. Vaccination is one of the most effective tools for sumers resist dominant media messages [13,14]. It follows reducing the burden of infectious diseases [19]. However, that media effects may sway people into panic (eg swine despite their public-health benefit, vaccination programs flu), especially with a disease where scientific evidence is face obstacles. Individuals often refuse or avoid vaccina- thin or nonexistent. Conversely, media may have little tions they perceive to be risky. Recently, rumours that thepolio vaccinecould cause sterility and spread HIV effect on seasonal diseases (eg regular influenza). have hampered polio eradication in Nigeria [20], while Media reporting plays a key role in the perception, misplaced fears of autism in the developed world have management and even creation of crisis [6]. Since media reports are retrievable and because the messages are stoked vaccination fears [21]. Reporting the number of widely distributed, they gain authority as an intersubjec- individuals who vaccinate may have a positive effect on tive anchorage for personal recollection [4]. At times of the disease transmission by increasing the vaccination crisis, non-state-controlled media thrive, while state- rate. controlled media are usually rewarded for creating an Conversely, behavioural interventions can also have an illusion of normalcy [6]. Media exposure and attention enormous effect on the course of a disease [22,23] Our partially mediate the effects of variables such as demo- model considers the same contact rate after a media alert, graphics and personal experience on risk judgments [5]. as proposed by Liu & Cui [3], but there are fundamental The role of media coverage on disease outbreaks is thus differences in both models. They consider the classical SIR crucial and should be given prominence in the study of type model, while vaccination is included in ours to reflect disease dynamics. transmission dynamics of human influenza. Klein et al., [15] noted that much more research is needed to understand how provision of information Model framework influences individual risk perception and how it shapes We divide the population (N) into four sub-populations, the evolution of epidemics; for example, individuals may according to their disease status: susceptible (S), vaccinated overprotect, which can have additional consequences for (V ), infected (I), and recovered (R). Our model monitors the spread of disease. An example of such complex the dynamics of influenza based on a single strain without dynamics is the 1994 outbreak of plague in a state in effective cross-immunity against the strain. The susceptible India: after the announcement of the disease, many peo- population is increased by recruitment of individuals ple fled the state of Surat in an effort to escape the dis- (either by birth or immigration), and by the loss of immu- ease, thus carrying the disease to other parts of the nity, acquired through previous vaccination or natural country [16]. Even though information on the number of infection. This population is reduced through vaccination cases and deaths can have an adverse effect, the number (moving to class V ), infection (moving to class I)andby of those vaccinated has not been given prominence. natural death or emigration. The population of vaccinated A handful of mathematical models have described the individuals is increased by vaccination of susceptible indivi- impact of media coverage on the transmission dynamics duals. Since the vaccine does not confer immunity to all of infectious diseases. Liu et al.[2] examined the potential vaccine recipients, vaccinated individuals may become for multiple outbreaks and sustained oscillations of emer- infected, but at a lower rate than unvaccinated. The vacci- ging infectious diseases due to the psychological impact nated class is thus diminished by this infection (moving to from reported numbers of infectious and hospitalized class I) by waning of vaccine-based immunity (moving to individuals. Liu & Cui [3] analysed a compartment model class S) and by natural death. The population of infected that described the spread and control of an infectious individuals is increased by infection of susceptibles, includ- disease under the influence of media coverage. Li & Cui ing those who remain susceptible despite being vaccinated. [17] incorporated constant and pulse vaccination in SIS It is diminished by natural death, death due to disease and epidemic models with media coverage. Cui et al., [18] by recovery from the disease (moving to class R). The showed that when the media impact is sufficiently strong, recovered class is increased by individuals recovering from their model – with incidence rate being of the exponen- their infection and is decreased as individuals succumb to tial form capturing the alertness to the disease of each natural death. Media coverage is introduced into the model susceptible individual in the population – exhibits multi- via a saturated incidence function. ple positive equilibria (also see [2]) which poses a chal- A schematic model flowchart is depicted in Figure 1. lenge to the prediction and control of the outbreaks of infectious diseases. Model equations The aim of this study is to investigate the impact of The transmission model with media coverage is given by media coverage on the spread and control of an influenza the following deterministic system of nonlinear ordinary strain when a vaccine is available, and where the media differential equations: SI Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 3 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 S R (1-) SV V I Figure 1 The model Schematic model flow diagram assume this rate to be the same for all sub-populations. b ⎛ ⎞ dS I =+ Λ wqVS −() +m − b −b SI +s R (1) ⎜ ⎟ is the rate at which susceptibles get infected; ω is the rate dt mI + ⎝ I ⎠ at which vaccine-based immunity wanes; g is the vaccine efficacy; a is the death rate due to the infection; and l is dI ⎛ I ⎞ ⎛ I ⎞ =−bb SI+−bb () 1−− g VI (a+m+l)I b ⎜ 12 ⎟ ⎜ 1 3 ⎟ (2) the recovery rate from infection. The terms dt mI + mI + ⎝ II ⎠ ⎝ ⎠ mI + and measure the effect of reduction of the con- mI + ⎛ ⎞ dV I tact rate when infectious and vaccinated individuals are =−qmSV () +w − b −b (1 −g)VI (3) ⎜ ⎟ dt mI + reported in the media [2,3,18]. The half-saturation con- ⎝ ⎠ stant m > 0 reflects the impact of media coverage on the gI () = dR contact transmission. The function is a (4) =−lmIR () +s , mI + dt continuous bounded function which takes into account disease saturation or psychological effects [24]. We note where Λ is the rate at which individuals are recruited that recovered individuals cannot be vaccinated. Also, a into the population (recruitment of infectives is ignored vaccinated individual who gets infected and then recovers for now); θ is the rate at which susceptible individuals will return to the susceptible class with no vaccine protec- receive the vaccine; µ is the the rate at which people leave tion. This is true even if ω is quite small but s and l are the population, through natural death or emigration. We Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 4 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 large. For example, if vaccination lasts three years, but Stability of the equilibrium states recovery and loss of immunity takes 6 months, then we The disease-free equilibrium of the system is given by are assuming this person is subsequently unvaccinated. ⎛ ⎞ ΛΛ () mw + q In the Michaelis-Menten functional response, the rate ES== (,I,V,R) ,,00,. ⎜ ⎟ v0 mq() ++ m w mq() ++ m w at which information is spread by the media rises as ⎝ ⎠ infectives increase, but eventually levels off at a plateau The endemic equilibrium of the system is given by (or asymptote) at which the information (rate) remains constant (i.e. it has reached a maximum number of indi- ^^ ^ ^ ES = (,,IV,R). viduals due to information saturation) regardless of the v1 increase in infections. Such dynamics can easily be     It satisfies and SI>> 00 ,,V>0,R>0 observed in the spread of rumours, gossip and jokes (also known as randomized broadcast) [25,26]. This con-   () Λ++wsVRh(I) stant coverage is extended by examining more complex S = hI () effects which involve more than just reducing contacts 2 down the line. The news in particular is extremely fickle () Λ +swRh( (I)) V = so that what is news one day may be forgotten about hI ()h ()I − w(h (() I ) 23 1 next week; including the media effects in some more lI sophisticated way such as by an impulsive pulsing is R = , also investigated. The limited power of the infection due ms + to contact is accounted for by the saturation incidence. where h (I)= m + I, h (I)=(θ + µ + b I)h (I) – b I , 1 1 2 1 1 2 The first available information is the reported number h (I)= (θ + ω +(1 – g)b I)h (I) – b (1 – g)I . Substitut- 3 1 1 3 of infected individuals when the disease is emerging. We ing the above into the second equation at equilibrium assume that media coverage can slow but not prevent will yield the expression for Î after some rearrangement. disease spread, so b ≥ b and b ≥ b . 1 2 1 3 For illustration, suppose θ = b = b = 0. Then the 2 3 The above model is closely related to those in [27,28] endemic equilibrium satisfies to analyze the transmission dynamics of human influ- enza, but there are some differences. In [27], the authors   consider the inflow of infective immigrants, while in [28] RI = ms + the model includes treatment. Neither of these are con- sidered here. Our model is clearly a crude reflection of () am++l   SI = , the complicated nonlinear phenomena of the transmis- sion dynamics, and it does not incorporate the self-con- trol property due to the change of avoidance patterns of where Î is the positive solution to the quadratic individuals at different stages of the infectious process ma() ++ m l sl [2]. News coverage may have a significant impact on   Λ− II −+() am+l + I = 0. avoidance behaviours at both individual and society b ms + levels, which may reduce the effective contact between The basic reproductive ratio, R , is defined as the susceptible and infectious individuals; we include this v expected number of secondary infections caused by an via a saturation incidence functional response. infective individual upon entering a totally susceptible Since the model monitors human populations, all the population [29-31]. This quantity is not only important in variables and parameters of the model are nonnegative. describing the infectious power of the disease, but can also Based on biological considerations the system of equa- can supply information for controlling the spread of the tions (1)-(4) will be studied in the following region, disease [32]. The linear stability of E is governed by the v0 basic reproductive ratio R . Using the next-generation Ω= {(SI , ,R,V ) ∈ } v method [31], we have which is positively invariant and attracting (thus, the bmΛΛ () +w bg() 1 − q model is mathematically and epidemiologically well- ⎛ ⎞ ⎜ ⎟ posed); it is therefore sufficient to consider solutions mq() ++ m w mq() ++ m w in Ω. Existence, uniqueness and continuation results ⎜ 0 ⎟ F = for model system (1)-(4) hold in this region and all ⎜ ⎟ ⎜ ⎟ solutionsofthissystemstartingin Ω remain in Ω for ⎜ ⎟ ⎝ ⎠ all t ≥ 0. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 5 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 and free equilibrium of the system where ⎛ΛΛ () mw + q ⎞ X = , . ⎜ ⎟ For the set of equations () am++l 000 mq() ++ m w mq() ++ m w ⎛ ⎞ ⎝ ⎠ ⎜ ⎟ in (1)-(4), we set X =(S, V, R)and Z =(I). The con- G G bg() 1 − qΛ ⎜ ⎟ () mw+−q 0 ditions (H1) and (H2) below must be met for global ⎜ ⎟ mq() ++ m w V = ⎜ ⎟ stability. bm Λ() +w dX G * ⎜ 1 ⎟ −+ wq ( m m) 0 (H1) For = FX(, 0),X is globally asymptoti- GG ⎜ ⎟ mq() ++ m w dt cally stable. ⎜ ⎟ ⎜ ⎟ −+ lm 00 ()s (H2) G(X , Z )= A Z – Ĝ(X , Z ), Ĝ(X , Z ) ≥ 0 ⎝ ⎠ G G G G G G G G for (X , Z ) Î Ω where is an M-matrix AD = (,X 0) G G GZG G The basic reproductive ratio is the spectral radius (the off-diagonal elements of A are nonnegative) and Ω –1 r(FV ) which is is the region where the model makes biological sense. If the above two conditions are satisfied, then the fol- bmΛΛ () ++w b(1−g)q lowing theorem holds. R = . (5) ma() ++ l m(q + m+w) Theorem 2 (Castillo-Chavez et al,[33]): The fixed point is a globally stable equilibrium of UX =(, 0) 0GG (2.28) provided that R <1 and that assumptions (H1) and (H2) are satisfied. Local stability of the disease-free equilibrium Lemma 1The disease-free equilibrium E is locally v0 ⎛ ⎞ Λ−+() mqSV +w FX(, 0) = asymptotically stable if R <1, and unstable if R >1. ⎜ ⎟ v v G ⎜ ⎟ qmSV −+()w ⎝ ⎠ Proof. The Jacobian of the system evaluated at E is v0 given by ⎛ ⎞ IS(( bb+− 1 gg )V GX(,Z ) = ⎜ ⎟ GG ⎜ ⎟ mI + ⎛ bm Λ() +w ⎞ ⎝ ⎠ −+() qm − ws ⎜ ⎟ ⎟ mq() ++ m w ⎜ ⎟ ⎜ bmΛΛ () ++w b(1−g)q ⎟ 11 Therefore, E is globally asymptotically stable (GAS) 0 − (() am++l 00 v0 ⎜ ⎟ J mq() ++ m w . E = v 0 ⎜ ⎟ since Ĝ(X ,Z ) > 0. The GAS of E excludes any possi- G G v0 ⎜ ⎟ bg() 1 − qΛ q − −+() wm 0 ⎜ ⎟ bility of the phenomenon of backward bifurcation. We mq() ++ m w ⎜ ⎟ ⎜ ⎟ 00lm−+()s note that the GAS of the DFE E when s =0is ⎝ ⎠ v0 straightforward. The eigenvalues of J are V0 The optimal control model bmΛΛ () ++w b(1−g)q Vm =− , V = −+() am+l, Our objective in this section is to extend the initial mq() ++ m w model to include two intervention methods, called con- Vm =−() +s, V = −−+() mq+w. trols, represented as functions of time and assigned rea- For local stability of the disease-free equilibrium, we sonable upper and lower bounds, each representing a require that all the eigenvalues be negative. Three of the possible method of influenza intervention. Using optimal control theory and numerical simulations, we determine eigenvalues satisfy this condition while ς < 0 implies the benefit of vaccination and media coverage when the that R < 1 and, consequently, all the eigenvalues of the latter has positive or negative effect on the former. Jacobian matrix above have negative real part. Thus, the We will integrate the essential components into one disease-free equilibrium is locally asymptotically stable. SIVR-type model to accommodate the dynamics of an influenza outbreak determined by population-specific para- Global stability of the disease-free equilibrium meters such as the effect of contact reduction when infec- We adopt the method of Castillo-Chavez et al,[33] and tious and vaccinated individuals are reported in the media. we rewrite the set of model equations in the form Let u and u be the control variables for vaccination v m dX and media coverage respectively. Thus, model (1)-(4) = FX(,Z ) GG now reads dt dZ = GX(,Z ), GG dS dt =+ Λwq Vu − ((1− ) +m)S dt (6) with G(X ,0) = 0. X Î ℝ denotes the number of G G ⎛ ⎞ −−bb SI +s R uninfected classes and Z Î ℝ denotes the number of ⎜ ⎟ G 12 () 1−+ um I * mI ⎝ ⎠ infected classes. denotes the disease- UX =(, 0) 0GG Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 6 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 Existence of an optimal control ⎛ ⎞ dI I =−bb SI ⎜ 12 ⎟ The existence of an optimal control can be obtained by dt () 1−+ um I ⎝ mI ⎠ (7) using a result by Joshi [36] and Fister et al.[37]. ⎛ ⎞ Theorem 3Consider the control problem with the sys- +−bb () 1 −g VIII −+() am+l ⎜ 13 ⎟ () 1−+ um I ⎝ mI ⎠ tem of Equations (4.1)-(4.4). There exists an optimal con- * ** trol such that max [Ju( ,) u |(u ,) u ∈= u] Ju( ,) u () u vm vm vm Proof. To prove this theorem on the existence of an dV =−() 1 uSqm −( +w)V optimal control, we use a result from Fleming and dt Rishel [38] (Theorem 4.1 pp. 68-69), where the follow- (8) ⎛ ⎞ ing properties must be satisfied. −−bb () 1 −g VI ⎜ ⎟ 1. The set of controls and corresponding state vari- () 1−+ um I mI ⎝ ⎠ ables is nonempty. 2. The control set U is closed and convex. dR 3. The right-hand side of the state system is bounded =−lmIR () +s . (9) dt above by a linear function in the state and control. 4. The integrand of the functional is concave on U A balance of multiple intervention methods can differ k k and is bounded above by c – c (|u | + |u | ), where c , 2 1 v m 1 between populations. A successful mitigation scheme is c > 0 and k >1. one which reduces influenza-related deaths with mini- An existence result in Lukes [39] (Theorem 9.2.1) for mal cost. A control scheme is assumed to be optimal if the system of equations (6)-(9) for bounded coeffi- it maximizes the objective functional cients is used to give the first condition. The control set is closed and convex by definition. The right-hand Ju((t),u (t)) = vm side of the state system (Equations (4.1)-(4.4)) satisfies tf (10) Condition 3 since the state solutions are a priori [St ()+− V()t B I()t − B (u ()t + u ()t )]dt. 12 vm ∫ ∫ t 0 bounded. The integrand in the objective functional, , is concave on St ()+− V()t B I()t − B (u ()t + u ()t ) 12 vm The first two terms represent the benefit of the sus- U.Furthermore, c , c >0 and k >1,so 1 2 ceptible and vaccinated populations. The parameters B and B represent the weight constraints for the infected St ()+− V()t B I()t − B (u ()t 12 v population and the control, respectively. They can also (11) 2 k k represent balancing coefficients transforming the inte- +≤ ut ()) c −c (|u |+ |u | ). mv 21 m gral into dollars expended over a finite time period of T days [34]. The goal is to maximize the populations of Therefore, the optimal control exists, since the left- susceptible and vaccinated individuals, minimize the hand side of (11) is bounded; consequently, the states population of infectives, maximize the benefits of media are bounded. coverage and vaccination, while minimizing the systemic Since there exists an optimal control for maximizing costs of both media coverage and vaccination. The value the functional (10) subject to equations (6)-(9), we use u (t)= u (t) = 1 represents the maximal control due to v m Pontryagin’s Maximum Principle to derive the necessary vaccination and media coverage, respectively. The terms conditions for this optimal control. Pontryagin’sMaxi- 2 2 and represent the maximal cost of Bu ()t Bu ()t 2 v 2 m mum Principle introduces adjoint functions that allow education, implementation and campaigns on both vac- us to attach our state system (of differential equations), cination and media coverage. S(t)and V(t) account for to our objective functional. After first showing existence the fitness of the susceptible and the vaccinated groups of optimal controls, this principle can be used to obtain as a result of a reduction in the rate at which the vac- the differential equations for the adjoint variables, corre- cine wanes, and vaccination and treatment efforts are sponding boundary conditions and the characterization * * implemented [35]. We thus seek optimal controls ut () v of an optimal control and . This characterization u u v m and such that ut () m gives a representation of an optimal control in terms of the state and adjoint functions. Also, this principle con- ** Ju(,u (t)=∈ max[Ju(,u )|(, u u ) U], vm vm vm verts the problem of minimizing the objective functional subject to the state system into minimizing either the where U ={(u , u )|u , u measurable, 0 ≤ a ≤ u ≤ v m v m 11 v Lagrangian or the Hamiltonian with respect to the con- b ≤ 1, 0 ≤ a ≤ u ≤ b ≤ 1, t Î [0, t ]} is the control 11 22 m 22 f trols (bounded measurable functions) at each time t[40]. set, with t Î [t , t ]. The basic framework of this pro- 0 f The Lagrangian is defined as blem is to characterize the optimal control. Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 7 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 optimality technique is utilized. The following cases are LS=+ ()t V()t −BI()t −B (u ()t +u ()t )+lw [Λ+ V− ((1−u )q+m)S 12 vm 1 v considered to determine a specific characterization of ⎛ I ⎞ − bbb − SI +s R] ⎜ 12 ⎟ () 1−+ um I the optimal control. ⎝ mI ⎠ ⎛ ⎞ Case 1: Optimality of I u +− lb [ b SI ⎜ ⎟ 21 2 −+ um I () 1 ⎝ mI ⎠ 1. On the set . {|ta << u (t) b },w =w = 0 11 v 11 11 12 ⎛ I ⎞ Hence, the optimal control is + bbb − () 1−− g VI (a+m+l)I] ⎜ 13 ⎟ () 1−+ um I ⎝ mI ⎠ () ll − qS ⎛ ⎞ I * +−lq [(1 uS ) − (m+w)V V−−bb () 1 −g VI] ⎜ ⎟ ut () = 3 v 13 () 1−+ um I mI ⎝ ⎠ 2B +− ll[(IR m+s) ] +w ()ta ( −−+ut )( w )(u −b ) 11 11 vv 12 11 2. On the set .We have {|ta== u (t)},w 0 11 v 11 +−ww ()ta ( u )+ ()tu ( −b ), 21 22 mm 22 22 () ll−+qSw (t) 13 12 where w (t) ≥ 0, w (t) ≥ 0 are penalty multipliers ut () = 11 12 2B satisfying w (t)(a – u (t)) + w (t)(u (t) – b )at the 11 11 v 12 v 11 optimal , and w (t) ≥ 0, w (t) ≥ 0 are penalty multi- u 21 22 or pliers satisfying w (t)(a – u (t)) + w (t)(u (t) – b ) 21 22 m 22 m 22 at the optimal . () ll − qS * * ut () = ≤ a v 11 Given optimal controls and , and solutions of the u u v m 2B corresponding state system (6)-(9),there exist adjoint variables l , for i = 1, 2, 3, 4 satisfying the following i since w ≥ 0. equations 3. On the set . Hence {|tb== u (t)},w 0 11 v 12 dl ∂L =− () ll−−qSw(t) * 13 11 dt ∂S ut () = 2B =−1 +() ll − (b −b )(Iu +−ll )(1− )q+l m 12 1 2 1 3 v 1 1 () 1−+ um I mI dl ∂L or =− dt ∂I ⎛ ⎞ I () 1 −um mI =+ B () ll− (b −b )S −b IS () ll − qS ⎜ ⎟ 11 2 1 2 2 * 13 ⎜ ⎟ () 1−+ um I ((1 −−+ um))I ut () = ≥ b . mI ⎝ mI ⎠ v 11 2B +−() ll (b −b )(1 −g )V 32 1 3 () 1−+ um I mI Combining all the three sub-cases in a compact form () 1 −um mI −b () 1 −gl VI ++(a m+l)−ll 3 3 24 gives ((1−+ um ) I) mI ⎠ dl ∂L =− ⎧ ⎫ ⎧ ⎫ dt ∂V V ⎪ () ll − qS ⎪ * 13 ut () = min max a , ,. b (12) I v ⎨ ⎨ 11 ⎬ 11 ⎬ =−1 +() ll − (b −b )(1−+ g )I lm+ (ll− )w 2B 32 1 3 3 3 1 ⎪ ⎩ 2 ⎭ ⎪ ⎩ ⎭ () 1−+ um I mI dl ∂L L =− dt ∂R Case 2: Optimality of =−() ll s+lm * 41 4 1. On the set . {( ta | << u t) b},w = w = 0 22 m 22 21 22 We have with transversality conditions l [t ] = 0, for i =1, 2, 3, i f 4. To determine the interior maximum of our Lagran- b mSI * 2 I gian, we take the partial derivatives of L with respect to ut ()=− (ll ) m 12 21Bu ((−+ )m I) u and u , respectively, and set it to zero. Thus, 2 mI v m bgmV () 1 − I 3 I ∂L +−() ll . * 23 =−2Bu ()t + (ll − )qS − w ()t + w ()t 21 v 3 11 12 2BBu ((1−+ )m I) 2 mI ∂ut () ∂L b mSI * * 2 I 2. On the set . We have {|ta== u (t)},w 0 =−2Bu () t +− (ll ) 22 m 21 2 m 12 ∂ut () ((1−+ um ) I) m mI b mSI bgmV () 1 − I * 2 I 3 I +−() ll −+ wt () w ()t ut ()=− (ll ) m 12 23 21 22 2 2 ((1 − u)) mI + 21Bu ((−+ )m I) mmI 2 mI bgmV () 1 − I wt () 3 I 22 To determine an explicit expression for our controls +−() ll + * * 2B 2BBu ((1−+ )m I) , (without w ,w , w , w ), a standard u u 2 mI 2 11 12 21 22 m m Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 8 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 or obtain the uniqueness of the optimal control for small [t ] [36]. The uniqueness of the optimal control follows b mSI from the uniqueness of the optimality system. * 2 I ut ()=− (ll ) m 12 The state system of differential equations and the 21Bu ((−+ )m I) 2 mI adjoint system of differential equations together with bgmV () 1 − I 3 I the control characterization above form the optimality +−() ll ≤ a 23 22 2BBu ((1−+ )m I) 2 mI system solved numerically and depicted in Figures 2, 3, 4, 5. since w ≥ 0. 3. On the set . Hence {|tb== u (t)},w 0 22 m 22 The model with pulse vaccination The general model with pulse vaccination is given as b mSI * 2 I ut ()=− (ll ) m 12 21Bu ((−+ )m I) ⎛ ⎞ dS I 2 mI =+ Λ wm VS − − b −b SI +s R ⎜ 12 ⎟ dt mI + ⎝ I ⎠ bgmV () 1 − I wt () 3 I 21 +−() ll − ⎛ ⎞ ⎛ ⎞ 23 dI I I 2B =−bb SI + bbb − () 1−− g VI (a+m+l)I 2BBu ((1−+ )m I) ⎜ 12 ⎟ ⎜ 13 ⎟ 2 mI 2 dt mI + mI + ⎝ I ⎠ ⎝ I ⎠ ⎛ ⎞ dV I or =−() mw + V − b −b () 1 − g VI ⎜ ⎜ 13 ⎟ dt mI + ⎝ I ⎠ 2 2 dR b mSI bgmV () 1 − I 2 I 3 I ut ()=− (ll ) +−() ll ≤ b . =−lmIR () +s , m 12 23 22 2 2 21Bu ((−+ )m I) 2BBu ((1−+ )m I) dt 2 mI 2 mI Combining all the three sub-cases in a compact form for t ≠ t ,where t is the time of the kth vaccination. k k gives We may have t – t either constant or not, as we k+1 k choose. The impulsive effect is given by 2 2 ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ () ll − b mSI () l −l b m() 1 − g VI ⎪ ⎪ * 12 2 I 23 3 II ut () = min max a , + ,. b ⎨ ⎨ ⎬ ⎬ (13) m 22 2 2 22 ΔS = –θS 21Bu ((−+ )m I) 21Bu ((−+ )m I) ⎪ ⎪ 2 mI 2 mI ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ ΔV = θS +− when t = t . Here, is thechangein Δ≡yy()t −y(t ) kk state at the impulse time. The optimal system In this model, vaccination occurs at fixed times, not The optimality system consists of the state system continuously. This is closer to reality, since vaccination coupled with the adjoint system, with the initial condi- centres are only open at certain times, when people may tions, the transversality conditions and the characteriza- get vaccinated in waves. Similarly, media stories tend to tion of the optimal control: clump together, so that a big news story occurs on one day, which may trigger a short period of intense vacci- ⎛ ⎞ dS I =+ Λwq Vu − ((1− ) +m)S−b −b SI +s R ⎜ ⎟ nation. We shall use a simplified version of this model v 12 ⎜ * ⎟ dt () 1−+ um I ⎝ mI ⎠ to illustrate the possibility that media may have an ⎛ ⎞ ⎛ ⎞ dI I I ==− ⎜ bb ⎟ SI+− ⎜ bb ⎟ (1 − gga )( VI−+m+l)I 12 1 3 ** ⎜ ⎟ ⎜ ⎟ dt adverse effect. ()11 −+ um I ()−+ um I ⎝ mI ⎠ ⎝ mI ⎠ ⎛ ⎞ dV I =−() 1 uSqm −( +w)V− b −b () 1 − g VI v ⎜ 1 3 ⎟ ⎜ ⎟ dt () 1−+ um I ⎝ mI ⎠ ⎠ Adverse effects dR =−lmIR () +s Consider the following scenario. At the onset of the out- dt dl I 1 * break, the media - and hence the general population - is =−1 +() ll − (b −b )(Iu +−ll )(1− )q+lm 12 1 2 13 v 1 dt () 1−+ um II mI unaware of the disease and thus nobody gets the vac- ⎛ ⎞ d I () −um l 1 2 mI =+ B () ll− (b −b ) )S − b IS ⎜ ⎟ 11 2 1 2 2 cine, allowing the disease to spread in its initial stages. ⎜ * * 2 ⎟ dt () 1−+ um I ((1−+ um ) I) ⎝ mI mI ⎠ * At some point, there is a critical number of infected ⎛ ⎞ I () 1 −um mI +−() ll ⎜(b −b )(1−−gb )V () 1 −g VI ⎟ 32 1 3 3 * * 2 ⎜ ⎟ (1 − u )mI + ((1−+ um ) I) individuals, whereupon people are sufficiently aware of mmI mI ⎝ ⎠ +l() am++l −l l 224 the infection to change their behaviour. We suppose dl I =−1 +() ll − (b −b )(1−+ g )I lm ++−() ll w that, initially, new infected people arrive at fixed times. 32 1 3 3 31 dt () 1−+ um I mI We further assume that vaccinated people mix more dl =−() ll s+lm 41 4 dt than susceptibles. In this case, people who are vacci- nated feel confident enough to mix with the infected, * * where and are given by expressions (12) ut () ut () v m even though they may still have the possibility to con- and (13), respectively, with S(0) = S , I(0) = I , V(0) = 0 0 tract the virus. This might be the case for health-care V , R(0) = R and l [t ]= 0for i =1,··· ,4. Due to the a 0 0 i f workers, for instance, who get vaccinated and then have priori boundedness of the state and adjoint functions to tend to the sick. and the resulting Lipschitz structure of the ODEs, we Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 9 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 40 18 i1 v1 i2 v2 i3 v3 10 4 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 2 Optimality effect when the weight constraint for the infected population varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint for the infected population varies. (a) Infected individuals. (b) Vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) =50.0, R(0) = 40.0. The value of the weights used are (i) B1=0.0025 corresponds to variables with subscript 1 (++), (ii) B1=25.0 corresponds to variables with subscript 2 (xx), (iii) B1= 250000.0 corresponds to variables with subscript 3 (**). The value B2=0.0025 is kept constant in all three cases. Mathematically, we have a threshold for the critical For I >I , the model becomes crit number of infectives, I . crit dS For I <I , this model would look like crit =+ Λ wqVS −() +m −(b −b)SI+sR dt dS dI =+ Λ wm VS − −bSI+sR 1 =−() bb SI+b(1−g)VI−(a+m + + l)I 46 5 dt dt dI dV =+bb SI () 1−g VI−(a+m+l)I 45 =−qmSV () +w −b(1−g)VI dt dt dV dR =−(mw + ))(VV −−bg 1 )I 5 =−lmIR () +s , dt dt dR =−lmIR () +s . with b – b ≥ 0. 4 6 dt 40 50 i1 v1 i2 v2 i3 v3 25 25 10 0 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 3 Optimality effect when the weight constraint for the control varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint for the control varies. (a) Graph of infectives, (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B2=25.0 corresponds to variables with subscript 1 (++), (ii) B2 = 2500.0 corresponds to variables with subscript 2 (xx), (iii) B2 = 250000.0 corresponds to variables with subscript 3 (**). The value B1=0.0025 is kept constant in all three cases. Infectives Infectives Vaccinated Vaccinated Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 10 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 40 5 i1 v1 i2 v2 i3 v3 35 4.5 30 4 25 3.5 20 3 15 2.5 10 2 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 4 Optimality effect when the weight constraint for the infected population varies and media has an adverse effect on the vaccine. Graphical representation of the optimality system when media coverage has an adverse effect on the vaccination rate and when the weight constraint for the infected population varies. (a) Graph of infectives. (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) =25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B1=0.0025 corresponds to variables with subscript 1 (++), (ii) B1=25.0 corresponds to variables with subscript 2 (xx), (iii) B1 = 250000.0 corresponds to variables with subscript 3 (**). The value B2=0.0025 is kept constant in all three cases. However, to illustrate the adverse affect, we shall sim- The model then becomes plify the model even further. For a short timescale, we dS can assume recovery is permanent, so s =0.Thus,we (14) =+ Λ wm VS − t≠t dt can ignore the R equation. For I <I , we assume that there is no mixing, but crit dI rather that new infectives arrive impulsively into the sys- (15) =−() am + +lIt ≠t i i k tem at fixed times t and in numbers I , where I ≪ I . dt k crit (If the new infectives arrive at irregular times, then the broad results will be unchanged.) dV (16) =−() mw +Vt ≠t For I >I , fear of the disease keeps susceptibles from crit dt mixing with the infected, but the vaccinated will. Thus b = b = 0. Since I ≪ I , we can assume that, 4 6 crit (17) Δ=II t =t for I >I , the effects of new infectives are negligible. crit 40 25 i1 v1 i2 v2 i3 v3 10 0 0 20 40 60 80 100 0 20 40 60 80 100 Time/Days Time/Days (a)(b) Figure 5 Optimality effect when the weight constraint for the control varies and media has an adverse effect on the vaccine. Graphs of the optimality system when media coverage has an adverse effect on the vaccination rate and when the weight constraint for the control population varies. (a) Graph of infectives. (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V(0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B2= 25.0 corresponds to variables with subscript 1 (++), (ii) B2 = 2500.0 corresponds to variables with subscript 2 (xx), (iii) B2 = 250000.0 corresponds to variables with subscript 3 (**). The value B1=0.0025 is kept constant in all three cases. Infectives Infectives Vaccinated Vaccinated Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 11 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 for I <I and The Jacobian is crit −−qm 0 w ⎡ ⎤ dS (18) =+ Λ wqVS −() +m ⎢ ⎥ JV =01 −+(am+l)+b() −g b()1−gI . ⎢ ⎥ dt ⎢ ⎥ qb−−()11 g V −(m+w)−b()−g II ⎣ 22 ⎦ dI At the endemic equilibrium, bg() 1−= V (a+m+l) . (19) =−bg() 1 VI−(a+m+l)I dt Thus, we have ⎡ ⎤ −−qm 0 w ⎢ ⎥ dV (20)    =−qmSV () +w −b(1−g)VI ⎢  ⎥ 5 JSI (,,V) = 00 . bg() 1 − I dt ⎢ ⎢ ⎥ qa−+()m+l  ⎢ ⎥ −+() mw −b(1−g)I ⎣ 2 ⎦ for I >I . crit Thus, the effects of the media are to trigger a vacci- The characteristic equation is nating panic whenever the number of infectives is det(Jx−=I) −x −x [mw+ +b (1−g )I+q+m] large enough. We kept the model with impulse vacci- 1   −+xI [(qm b (1−g) )+ m(m+w ++−bg()11II )+−bg()(a+m+l)] 2 22 nation as simple as possible since even this simplified −−bg() 1 (a+m+l)(q+m)I. version shows that media reports could have an adverse effect. It follows that the endemic equilibrium is stable if Î Suppose new infectives appear regularly, so that t – k+1 >I . Thus, even in an extremely simplified version of crit t = τ. (If not, the analysis generalizes quite easily.) For the model, the media may make things significantly t <t <t , we have k k+1 worse than if no media effect were included. We kept this model deliberately simple, partly for mathematical +−() am + +l t It () = I e , tractability and partly to show that the media effects apply even in this idealised scenario. ++ where is the value immediately after the II ≡ ()t kk Note that, in reality, the fluctuations would apply in kth impulse. Then, since the period is constant, we the upper region as well, making the actual value even have larger. In the lower region, we ignored interaction between susceptibles and infectives (ie we assume b = −+−() am++l t 4 II = e kk +1 b = 0). The effect of including these terms would be to −− i () am++l t =+() II e . slow the exponential decay between impulses (or possi- bly cause it to increase). This would only increase the This is a recursion relation with solution effect seen here. In summary, a small series of outbreaks that would i −+() am+l t Ie − equilibrate at some maximum level m >I will, as a crit m = . −+() am+l t 1 − e result of the media, instead equilibrate at a much larger value I >m >I . The driving factor here is if an imper- crit Consequently, fect vaccine causes overconfidence, so that people who have been vaccinated mix significantly more with infec- tives than susceptibles do. If this happens (as would be m = . −+() am+l t 1 − e quite likely; most people who have been vaccinated feel invulnerable, even if the vaccine is not perfect, largely Thus, if m >I , then eventually the system will crit thanks to media oversimplifications), then the media switch from model (14)-(17) to model (18)-(20). The effect is likely to be adverse. A simplified version of the endemic equilibrium in model (18)-(20) satisfies model with pulse vaccination shows that the media can make things worse, if the vaccine is imperfect because () am++l V = the vaccinated mix over-confidently with the infectives. bg () 1 − Λ wa() ++ m l  Numerical simulations S = + qm + bg () 1−+(q m) We now return to model (6)-(9) and illustrate some of the properties discussed in the previous sections. The Λqb() 1−− g mmq() ++ m w(a ++ m l) I = . parameter values that we use for numerical simulations () qm++(a m+l)b(1−g) 2 Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 12 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 are in Table 1. Initial conditions: S(0) = 200.0, I(0) = 1.0, 4, we vary the cost weight, while in Figure 5, we vary V(0) = 10.0, R(0) = 0.0. The parameter θ varies between the weight of minimizing infectives. 0.3 and 0.7withanaverage of0.5 [41]. We consider an We note from Figure 2 that, during the initial days, imperfect flu vaccine for which the waning rate is about there is a very sharp drop in the population of infected 0.15. The relationship between b and b is not very individuals, while other populations show increases. 2 3 obvious; consequently, we can either assume equality or Increasing minimization of infectives, while keeping that the former is slightly greater than the latter. Trans- costs low, can lead to the disease being controllable. The slight rise and fall, after the initial 20 days, in the mission dynamics of infectious diseases with and with- population of infectives could be attributed to compla- out media coverage have already been carried out in previous studies, but these models do not account for cency on the part of some individuals (or may be due to the vaccination coverage. Therefore, we illustrate some oscillations in the system independent of external fac- numerical results for the model with optimal control tors). We find that people tend to relax after the initial when media coverage has (i) a beneficial effect (see Fig- shock of the disease threat. However, we note that this ures 2 and 3) (ii) and adverse effect on the vaccination is not for long, and this could be attributed to the fact rate (see Figures 4 and 5). that vaccination levels continue to rise, so as people The optimality system is solved using an iterative continue to receive vaccination, infection is controlled. method with a fourth order Runge-Kutta scheme. Start- Thus, if costs are kept minimal, and more people are ing with a guess for the adjoint variables, the state equa- able to access media and vaccination, then infection can tions are solved forward in time. Then the state values be controlled. Both vaccination and media coverage con- obtained are used to solve the adjoint equations back- tinue at optimum levels as a result of the low costs and ward in time; the iterations continue until convergence. minimization of infectives. Simulations are carried out to determine how maximiz- From Figure 3, as costs are increased, few people have ing media coverage enhances vaccination. The effects of access to media and vaccination; as a result, low num- costs that can be incurred, which include education, bers get vaccinated against the disease. In the long run, implementation and campaigns on media coverage, are the infection levels rise. The degree of media coverage also studied to evaluate how these costs can affect the and vaccination also decrease as a result of the exorbi- transmission of human influenza. We increase the value tant costs. With the little available media coverage and of B2 (the cost weight) in Figure 2 to assess how the the few vaccinated individuals, we find that, due to populations of susceptibles, infectives, vaccinated and information filtration, there is a jump in the vaccination levels, though these only last briefly; as the degree of recovered individuals are altered. In Figure 3, we investi- gate how increasing minimization of infectives through media coverage and vaccination decrease, so do the vac- increasing the weight B1 affects the control of human cination levels. influenza transmission. We do the same in Figures 4 From Figure 4, even though costs are kept at minimal and 5, respectively, to see how, if media coverage has an levels, the negative reports concerning vaccination result adverse effect, the various populations behave. In Figure in a drastic reduction in the vaccination levels. After some time, we note a slight increase in the vaccination levels; however, these numbers remain very low. This Table 1 Parameter values could be due to the fact that, as infection rises, a few will risk getting vaccinated in the hope of being cured. Parameter Symbol Value Units Reference Thus, media coverage can have adverse effects if peo- Recruitment rate Λ 5.0 People day [3] ple’s perception towards the vaccine is negatively influ- –1 Rate at which vaccine ω 0.15 day Assumed enced by the media. wanes In Figure 5, both media coverage and vaccination are –1 Vaccine uptake rate θ 0.3- day [41] eventually withdrawn. Very low numbers get vaccinated. 0.7 It is only when infection escalates that vaccination levels –1 Natural death rate µ 0.02 day [3] also increase as some might find it better to try to pre- Infection rate b 0.02 people [3] 1 –1 vent the infection, despite the negativity towards vacci- day –1 nation in the media. Figure 6 illustrates other potential Loss of immunity s 0.01 day [3] adverse effects that media may have, if the effect is to Vaccine efficacy g 0.8 (unitless) [3] –1 trigger a vaccinating panic where vaccinated individuals Infection death rate a 0.1 day [3] –1 are not fully protected and mix with infected individuals Recovery rate of Infectives l 0.05 day [3] but susceptible individuals do not. In this case, the num- Reaction due to media m 10.0 people Assumed coverage ber of infected individuals may increase sharply as a result of the media. Figure 7 illustrates the long-term Media coverage model parameters, their interpretations and values Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 13 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 3.2 2.8 With media effects 2.6 16.5 17 17.5 18 18.5 Vaccination panic threshold Without media effects 0 5 10 15 20 25 time (days) Figure 6 The vaccination panic threshold The effect of the vaccination panic threshold using the simplified model (14)-(20). Without media triggering a vaccinating panic, the number of infected individuals remains low (solid purple curve). However, if the media triggers a vaccinating panic, then the number of infected individuals rises sharply (dashed green curve). Inset: Comparison of the two outcomes around the vaccination threshold. results of a media-induced vaccination panic. Without Conclusion media effects, the result is a low-level infection. When Media simplifications can lead to overconfidence in the the media triggers a vaccinating panic, there is a large idea of a vaccine as a cure-all. The result is not just a outbreak, followed by an endemic level of infected indi- vaccinating panic and a blow-out epidemic, but a net viduals significantly higher than the level of infected increase in the endemic equilibrium. Thus, media cover- individuals without the media effects. Note that these age of an emerging epidemic can fan the flames of fear examples assume no post-vaccination mixing of suscep- and also implicitly reinforce an imperfect solution as the tible and infected individuals. only answer. Figure 8 illustrates the cases when post-vaccination We have formulated and investigated a simple deter- mixing of susceptible and infected individuals is maxi- ministic vaccination model describing the effects of mal (b = b , b = 0), 50% (b = b , b = b /2) or zero 4 5 6 4 5 6 4 media coverage on the transmission dynamics of influ- (b = b = 0). Thus, if susceptible and infected indivi- 4 6 enza. The media effect due to reporting the number of duals mix after a vaccinating panic has occurred, the infections as well as the number of individuals success- effect is an earlier outbreak and a larger number of fully vaccinated is introduced into the compartmental infected individuals. model via a saturated incidence-type function. The Infected individuals Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 14 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 90 95 100 With postvaccination fluctuations Without media effects Without postvaccination fluctuations 0 20 40 60 80 100 time (days) Figure 7 A media-induced epidemic Long-term dynamics for the simplified model (15)-(20). Without media effects, the maximum number of infected individuals remains low (solid purple curve). When media effects trigger a vaccinating panic where partially protected vaccinated individuals mix with infectives significantly more than susceptible individuals, a significant outbreak may occur, with the final number of infected individuals much higher than if no media effects had been included (dashed green curve, assuming no fluctuations after the vaccinating panic occurs). The result of including fluctuations above the vaccination panic threshold are also shown (solid blue curve). Inset: magnified view of the fluctuation versus nonfluctuation cases. impact of costs that can be incurred, which include vac- effects of media reporting on the transmission dynamics cination, education, implementation and campaigns on of infectious diseases for which a vaccine exists. The media coverage, are also investigated using optimal con- present study is in no way exhaustive and can be trol theory applied via the Pontryagin’smaximum prin- extended in various ways: for example, to investigate the ciple. A simplified version of the model with pulse case in which there is media coverage but people ignore vaccination shows that the media can have an adverse it (in which case the vaccination rate is unchanged effect if the vaccine is imperfect and the vaccinated mix despite the control). Thus, the effects of media on an over-confidently with the infectives. Numerical simula- outbreak of influenza with a partially effective vaccine tions are carried out to support the analytical results. may be complicated. While the media may encourage We note, however, that our caricature model is not more people to get vaccinated, they may also trigger a complete; a more comprehensive study will require vaccinating panic or promote overconfidence in the abil- interdisciplinary research across traditional boundaries ity of a vaccine to fully protect against the disease. This of social, natural, medical sciences and mathematics [2]. may have potentially disastrous consequences in the Nevertheless, our work provides some insights into the face of a new pandemic. Infected individuals Tchuenche et al. BMC Public Health 2011, 11(Suppl 1):S5 Page 15 of 16 http://www.biomedcentral.com/1471-2458/11/S1/S5 Maximal postvaccination mixing of susceptibles and infecteds 50% postvaccination mixing of susceptibles and infecteds No postvaccination mixing of susceptibles and infecteds 0 20 40 60 80 100 time (days) Figure 8 The effect of different post-panic mixing rates Comparison of mixing rates between susceptibles and infected individuals after a vaccinating panic. The more susceptibles mix with infecteds, the earlier the outbreak occurs and the larger the number of infected individuals. Author details Authors contributions Department of Mathematics and Statistics, University of Guelph, Guelph, JMT, RJS and CTB developed the model. JMT and CTB Ontario N1G 2W1, Canada. Department of Applied Mathematics, National designed and formulated the study framework and ana- University of Science and Technology, Box AC 939 Ascot, Bulawayo, Zimbabwe. Department of Mathematics and Faculty of Medicine, The lyzed the model. CPB and ND carried out the optimal University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada. control analysis and numerical simulations for Figures 2, 3, 4, 5. RJS wrote the section on adverse effects, some of Competing interests The authors declare that they have no competing interests. the introduction, performed numerical simulations for Figures 16,7,8, and edited the manuscript. All authors Published: 25 February 2011 read and approved the final manuscript. References 1. Laxminarayan R, Mills AJ, Breman JG, Measham AR, Alleyne G, Claeson M, Acknowledgements Jha P, Musgrove P, Chow J, Shahid-Salles S, Jamison DT: Advancement of We thank Shoshana Magnet, Penelope Ironstone-Catterall and Tim Reluga global health: key messages from the Disease Control Priorities Project. for technical discussions. 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