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Journal of Dynamics and Differential Equations, Vol. 20, No. 1, March 2008 (© 2007) DOI: 10.1007/s10884-007-9075-0 The Impact of Media on the Control of Infectious Diseases 1,2 3 4,5,6 Jingan Cui, Yonghong Sun, and Huaiping Zhu We develop a three dimensional compartmental model to investigate the impact of media coverage to the spread and control of infectious diseases (such as SARS) in a given region/area. Stability analysis of the model shows that the disease-free equilibrium is globally-asymptotically stable if a certain threshold quantity, the basic reproduction number (R ), is less than unity. On the other hand, if R > 1, it is shown that a unique endemic equilibrium appears and a Hopf bifurcation can occur which causes oscillatory phenom- ena. The model may have up to three positive equilibria. Numerical simu- lations suggest that when R > 1 and the media impact is stronger enough, the model exhibits multiple positive equilibria which poses challenge to the prediction and control of the outbreaks of infectious diseases. KEY WORDS: Infectious disease; SEI model; media impact; Hopf bifurca- tion; multiple outbreaks. 1. INTRODUCTION In recent years, attempts have been made to develop realistic math- ematical models for the transmission dynamics of infectious diseases. In modelling of communicable diseases, the incidence function has been considered to play a key role in ensuring that the models indeed give School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China. Research supported by the NNSF of China (10471066). Department of Information and Technology, Jiangsu Institute of Economic & Trade Technology Jiangning, Nanjing, 211168 China. Department of Mathematics and Statistics, York University, Toronto, Canada, M3P 1P3. E-mail: huaiping@mathstat.yorku.ca. Research supported by NSERC, MITACS and CFI/OIT of Canada. To whom correspondence should be addressed. 1040-7294/08/0300-0031/0 © 2007 Springer Science+Business Media, LLC 32 Cui, Sun, and Zhu reasonable qualitative description of the transmission dynamics of the dis- eases [3,9]. Some factors, such as media coverage, density of population and life style, may affect the incidence rate directly or indirectly. In the classical endemic models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by βSI , where β is the probability of transmission per contact (a positive constant), and S and I represent the susceptible and infected populations, respectively. However, there are several reasons for using non-linear incidence rates such as saturating and nearly bilinear. For instance, Yorke and London [20] showed that the incidence rate β(1 − cI )I S with positive C and time dependent β is consistent with the results of the simulations for measles outbreaks. In order to avoid the unboundedness of the contact rate, Cap- βSI asso and Serio [4] used a saturated incidence function of the form , 1+βδI δ> 0. To incorporate the effect of the behavioral changes of the suscep- tible individuals, Liu and coworkers [10,11] used a non-linear incidence kI S rate given by with k, l, α, h > 0. Ruan and Wang, [14] showed that 1+αI endemic models with such non-linear incidence rates exhibit various bifur- cations include Hopf, homoclinic, and even Bogdanov-Takens bifurcations. There have been many models using variety of different non-linear inci- dence functions to study the disease transmission, we refer the reader to Levin et al. [9] for a more detailed summarization. The aim of this paper is to investigate the impact of media cover- age to the spread and control of infectious diseases in a given region. In [12], the authors consider a model with the compartments of exposed (E), infectious (I) and hospitalized (H) individuals to explore the possible mechanism for multiple outbreaks of emerging infectious diseases due to the psychological impact of the reported numbers of infectious and hos- pitalized individuals. The model was simpliﬁed by assuming that the total population size remain a constant. In this paper, we extend the classical SEI model and the ideas in [12] to consider a new incidence functional which reﬂects the impact of the media coverage to the spreading and con- trol of the disease. This study was also originated from the observation of the spread of SARS coronavirus in Asia and some other regions of the world. SARS [7,15,19] as a new emerging infection disease, it was ﬁrst appeared in Guangdong province, China in November, 2002. Then in the follow- ing year the SARS coronavirus spread rapidly throughout Asia and cer- tain other part of the world [16,18]. For SARS in the cities of Beijing, Hongkong and Toronto, the spreading and outbreaks all experienced a typical process for people to see how the media coverage and the public alerting plays a role in the whole course of the spreading. For the case The Impact of Media on the Control of Infectious Diseases 33 in Beijing, it was not clear of the existence and type of such disease until April 21, 2003 [16]. During this period, more susceptible individuals might have been exposed to and infected with the disease unconsciously due to the luck of knowledge of the disease. This fact suggests us to consider the following question: How does the media coverage affect the spreading and control of the infectious diseases like SARS? The media coverage is obviously not the most important factor responsible for the transmission of the infectious disease, but it is a very important issue which has to be taken care of seriously. In the case of a large number of infected cases, on one hand, the media coverage may cause the panic of the society, while on the other hand, it can certainly reduce the opportunity and probability of contact transmission among the alerted susceptible populations, which in turn helps to control and prevent the disease from further spreading. In this paper, we use a compartmental model to address the impact of media coverage on the transmission of infectious diseases. In the SEI −mI model, the incidence rate is assumed to be in the form µe . This paper is organized as follows. In Section 2 we develop a SEI model to incor- porate the media impact to the spreading of the infectious diseases such as SARS. We calculate the reproduction number in Section 3 and prove the local and global stability of the disease free equilibrium. The model in general can have up to three positive equilibria, we shall restrict our- selves to the case when the media impact is small enough that there exists at most one endemic equilibrium. In Section 5 we shall study the local and global stability of the unique endemic equilibrium when the reproduction number R > 1 and m is small. We also study the Hopf bifurcation of the endemic equilibrium when the reproduction number is larger enough. The paper ends with a brief discussion of the results on the impact of media and related control and prediction issues. 2. A SEI MODEL WITH MEDIA IMPACT Consider the transmission of certain infectious disease (such as SARS) in a given region/area. We classify the population into the follow- ing categories: • S(t), the number of susceptible individuals; • E(t), the number of individuals exposed to the infected but not infectious; • I(t), the infected who are infectious. We assume that the infectious individuals I receive medical treatment in hospital settings as soon as they are identiﬁed from the category of 34 Cui, Sun, and Zhu exposed. Once they are recovered, they no longer impose risk to the sus- ceptible individuals. In most of the studies, the compartmental models were built by either assuming the total population to be a constant or sat- isfy exponential growth [1,2,5,8]. It is more reasonable to assume that the population of a given region obey the Logistic growth. Then we have the model dS S −mI = bS 1 − − µe SI, ⎪ dt K dE −mI (2.1) = µe SI − (c + d)E, dt dI = cE − γI, dt where all the parameters are positive, and • b, the intrinsic growth rate of the human population, K is the car- rying capacity for the human population of a given region/area. −mI • β(I ) = µe is the contact and transmission term, it measures the spreading of the virus from the infected to the susceptible indi- vidual. If m = 0, the transmission rate is a constant. Naturally the contact transmission rate is not only related to the spreading abil- ity of the virus or disease, but also closely related to the alertness to the disease of each susceptible individual of the population. Here we use the parameter m> 0 to reﬂect the impact of media coverage to the contact transmission. Since the media coverage and alertness are not the intrinsic deterministic factor responsible for the transmission, hence it is reasonable to assume that m> 0 is a small parameter. Also for simplicity, the mass action law is assumed in the model [8]. As one can see that if m> 0 but com- paratively small enough, this incidence term β(I ) is close to the constant µ. Also as m> 0 increases or the media coverage and the alertness to the public is comprehensive and in time, the general public will be more alert and aware of the virus/diseases. Hence the transmission rate will be decreasing as I increases. • c is the rate per unit time (day) that infected individuals become infectious. • d is natural death rate for the susceptible population. • γ is the removed rate from the infected compartment, which include the recovery rate of the hospitalized infectious individuals and natural death. Hence we have γ>d. Model (2.1) involves the interaction of both the population dynam- ics of logistic type and the transmission dynamics of disease epidemiology. Hence the dynamics of the system (2.1) can be very complicated. In this paper, we are going to study the impact of the media coverage/alert to the The Impact of Media on the Control of Infectious Diseases 35 spreading of the disease by assuming that m> 0 is small. We will show that if the media coverage fails to report the real situation of disease to alert and educate the public, then there will be an outbreak or even mul- tiple outbreaks of such a disease. 3. DISEASE FREE EQUILIBRIUM (DFE), STABILITY AND REPRODUCTION NUMBER Let the right hand side of (2.1) be zero, one can verify that the origin E = (0, 0, 0) is an equilibrium with eigenvalues b, −(c + d), −γ . Hence E 0 0 is a hyperbolic saddle point. The model (2.1) has one disease free equilibrium (DFE) at E = (K, 0, 0). The local stability of E can be obtained through a straightfor- ward calculation for the eigenvalues. It follows from [17] that for the compartmental models, the local sta- bility of the disease free equilibrium is governed by the reproduction num- ber of the model. Using the notations in [17], we have two vectors F and V to represent the new infection term and remaining transfer terms, respectively: ⎛ ⎞ ⎛ ⎞ −mI (c + d)E µe SI ⎝ ⎠ ⎝ ⎠ −cE + γI F = 0 , V = . (3.1) −mI 0 −bS(1 − ) + µe SI The infected compartments are E and I , hence a straightforward calcula- tion gives 0 µK (c + d) 0 F = ,V = , (3.2) 00 −cγ where F is non-negative and V is a non-singular M-matrix, therefore −1 FV is non-negative, and µKc µK(c + d) −1 FV = . (3.3) γ(c + d) −1 Hence the reproduction number is given by ρ(F V ), and µcK R = . (3.4) γ(c + d) It follows from [17] that we have Proposition 3.1. For the model (2.1), the disease free equilibrium E is locally asymptotically stable if R < 1, and unstable if R > 1. 0 0 36 Cui, Sun, and Zhu Note that for the characteristic equation of (2.1) at E (λ + b)[λ + (c + d + γ)λ + γ(c + d) − µcK] = 0, (3.5) it follows from the Routh-Hurwitz criteria [13] that all the eigenvalues have negative real parts if and only if R < 1. Theorem 3.2. For the model (2.1), the disease free equilibrium E is globally asymptotically stable whenever R < 1. dS S Proof. From (2.1) we have ≤ bS(1 − ).For S = K is the glob- dt K dS S ally asymptotically stable equilibrium of = bS(1 − ), so for any ε> 0, dt K when t →+∞ we have S(t) ≤ K + ε. (3.6) Then we have dE −mI ≤ µe (K + ε)I − (c + d)E, dt (3.7) dI = cE − γI . dt Now we consider dE ⎪ −mI = µe (K + ε)I − (c + d)E = P(E, I), dt (3.8) dI = cE − γI = Q(E, I ). dt System (3.8) has a unique equilibrium (0, 0) and the corresponding eigen- values are determined by λ + (c + d + γ)λ + γ(c + d) − µc(K + ε) = 0. (3.9) For ε> 0 sufﬁciently small, since R < 1, hence γ(c + d) − µc(K + ε) > 0. Thus all the eigenvalues of (3.9) have negative real parts. Hence (0, 0) is ∂P (E, I ) ∂Q(E, I ) locally asymptotically stable. Since + =−(c + d + γ)< ∂E ∂I 0, system (3.8) has no close orbit. Let bK D = (S,E,I ) S, E, I ≥ 0,S + E + I ≤ K, K = ,l = min{b, d, γ } . We ﬁrst prove that D is positively invariant. By (2.1), for (S,E,I ) ∈ D we dS dE dI dS S −mI have = 0, = µe SI ≥ 0, ≥ 0, and ≤ bS(1 − ). S=0 E=0 I =0 dt dt dt dt K The Impact of Media on the Control of Infectious Diseases 37 dS S For = bS(1 − ) with S< K,wehave lim S(t) = K, and S(t) ≤ K. t →∞ dt K Note that if we let N(t) = S(t) + E(t) + I(t), then dN | = bS 1 − − dE − γI N =K,(S,E,I )∈D K S+E+I =K dt ≤ [bK − bS − dE − γI ] S+E+I =K ≤ bK − lN | = bK − lK = 0. N =K Hence D is positively invariant. Therefore, (0, 0) is globally asymptotically stable for (3.8). Consequently, for system (3.7) there holds lim E(t) = 0, lim I(t) = 0. t →∞ t →∞ Then for the above ε> 0, there exists T> 0 such that for all t>T , I (t)<ε. By (2.1), we have dS S >bS 1 − − µεS. dt K µε Note that for ε> 0 sufﬁciently small, S = K(1 − ) is a globally asymp- totically stable equilibrium of dS µε S = bS 1 − − , dt b K thus we have S(t) ≥ K − ε.(t →∞). (3.10) It follows from (3.6) and (3.10) that we have lim S(t) = K. t →∞ Hence E is the globally asymptotically stable equilibrium of (2.1). 4. EXISTENCE OF THE ENDEMIC EQUILIBRIUM (EE) First note that if m = 0, i.e., if the media impact is not considered, one can verify that when R > 1, system (2.1) has a unique endemic equi- ∗ ∗ ∗ librium (EE) (S ,E ,I ) where 0 0 0 γ(c + d) K bγ (c + d) bγ (c + d) ∗ ∗ ∗ S = = ,E = (R − 1), I = (R − 1). 0 0 0 0 0 2 2 2 µc R µ c K µ cK (4.1) 38 Cui, Sun, and Zhu But if the media and psychological impact are incorporated, system (2.1) can have up to three equilibria. Let −mI g(I ) = K 1 − Ie . (4.2) Then the model (2.1) becomes dS b = S(g(I ) − S), dt K dE b (4.3) = (K − g(I ))S − (c + d)E, dt K dI = cE − γI. dt Let the right hand side of (4.3) be zero. If a positive equilibrium exists, it is a positive solution of S = g(I ), Kγ (c+d) (4.4) S = = h(I ), bc K−g(I ) cE − γI = 0, where using the expression of g(I ) in (4.2) was used, h(I ) can be simpli- ﬁed to γ(c + d) mI h(I ) = e . (4.5) µc Then if a positive equilibrium, an endemic equilibrium exists, its (S, I ) coordinates must satisfy S = g(I ), S = h(I ), (4.6) and the E coordinate is given by E = I. One can verify that if R > 1, then g(0)>h(0). Note lim g(I ) = K I →∞ and lim h(I ) =∞. Hence if R > 1, the two curves S = g(I ) and S = h(I ) I →∞ have at least one positive intersection which gives at least one endemic equilibrium. As shown in Fig. 1(a)–(c), the two planar curves S = g(I ) and S = h(I ) can have up to three intersections in the SI -plane. Now we develop conditions to decide the tangency of the two curves in order to determine the number of positive equilibria. If the two curves The Impact of Media on the Control of Infectious Diseases 39 (a) (b) 600 600 500 500 400 400 300 300 200 200 100 100 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 –100 –100 –200 One EE –200 Two EE (c) 20 40 60 80 100 120 140 160 180 200 –100 –200 Three EE Figure 1. Three possible cases of the intersection of the curves S = g(I ) and S = h(I ) indi- cating the existence of up to three positive equilibria. The curves were plotted using Maple. S = g(I ) and S = h(I ) are tangent at some positive points, we must have S = g(I ) = h(I ), g (I ) = h (I ). Or equivalently, µ γ(c + d) −mI mI ⎨ K 1 − Ie = e , b µc (4.7) Kµ γ(c + d)m −mI mI − (1 − mI )e = e . b µc Eliminating the exponential terms in (4.7), if the two curves are tangent, the I coordinate must satisfy the quadratic equation R m(mI − 1) = (2mI − 1) . (4.8) It follows from (4.7) and (4.8) that if the tangency occurs at some point, its I coordinate must satisfy mI > 1. 40 Cui, Sun, and Zhu Let µ 8µ 8δ δ := ,m := = . (4.9) b bR R 0 0 A straightforward calculation can verify that (4.8) has two distinct positive roots satisfying mI > 1 if and only if R > 1 and m>m .For m>m , solv- 0 0 0 ing (4.8) in terms of I,wehave mR + 4δ ± I = , (4.10) 8mδ where = mR (mR − 8δ). 0 0 In summary, regarding the existence and the number of the endemic equilibria, we have: Proposition 4.1. Consider the model (2.1) with all parameters positive. Let m be deﬁned in (4.9).If R > 1, then model (2.1) has at least one and 0 0 at most three positive equilibrium (endemic equilibria). Furthermore, • if 0 <m<m , the model has a unique endemic equilibrium; • if m>m , the model has three endemic equilibria; • if m = m , the model has one endemic equilibria of multiplicity at least two. By the above proposition, if m = m , the model can have a more degenerate endemic equilibrium with multiplicity three (both the eigen- values are zero), and model can have a Bogdanov-Takens bifurcation of codimension two, even codimension three [6,21]. The study of the Bogdanov-Takens bifurcations is certainly out of the scope of this paper. 5. STABILITY AND HOPF BIFURCATION OF THE ENDEMIC EQUILIBRIUM (EE) In this section, we shall study the stability and Hopf bifurcation of the endemic equilibria and determine how the media impact can inﬂuence the periods of the oscillations of virus/disease transmission. 5.1. m= 0 ∗ ∗ ∗ When m = 0, model (2.1) has a unique endemic equilibrium (S ,E ,I ). 0 0 0 A straightforward calculation yields the associate characteristic equation: b b 1 3 2 λ + c + d + γ + λ + (c + d + γ)λ + bγ (c + d) 1 − = 0. R R R 0 0 0 (5.1) The Impact of Media on the Control of Infectious Diseases 41 Let 1 (c + d + γ) R = 1 + 2 γ(c + d) 2(c + d + γ)(2b + c + d + γ) (c + d + γ) + 1 + + . (5.2) 2 2 γ(c + d) γ (c + d) Obviously, for any positive parameters we have R > 1. Next propo- ∗ ∗ ∗ sition is about the local stability of the equilibrium (S ,E ,I ). 0 0 0 Proposition 5.1. For the model (2.1) with m = 0, the endemic equilib- ∗ ∗ ∗ rium (S ,E ,I ) is locally asymptotically stable if 1 < R < R . 0 H 0 0 0 Proof. To prove, we only need to show that all roots of the charac- teristic equation (5.1) have negative real parts. Since R > 1, all the coefﬁcients of the cubic polynomial (5.1) are pos- itive. If we also have R < R , then we have R γ(c + d) − R [γ(c + d) + 0 H 0 (c + d + γ) ] − b(c + d + γ) < 0. This is equivalent to (c + d + γ )(c + d + b 1 γ + ) − bγ (c + d)(1 − )> 0. R R 0 0 So if 1 < R < R ,wehave c + d + γ + > 0,bγ(c + d)(1 − R )> 0 0 b b 1 0, (c + d + γ )(c + d + γ + ) − bγ (c + d)(1 − )> 0. R R R 0 0 0 It follows from the Routh-Hurwitz criteria [13] that all eigenvalues of ∗ ∗ ∗ (5.1) have negative real parts, hence (S ,E ,I ) is locally asymptotically 0 0 0 stable if 1 < R < R . 0 H From (3.4) one can see that the reproduction number is linearly dependent on the parameter µ, hence solving R = R in terms of µ, one 0 H gets a threshold condition on the parameter µ for the endemic equilibrium to be locally asymptotically stable: γ(c + d) µ = R . (5.3) H H 0 0 cK Hence it follows from Proposition 5.1 that if µ<µ , then the endemic equilibrium is locally asymptotically stable. Theorem 5.2. For the model (2.1) with m = 0, when R = R or equiv- 0 H ∗ ∗ ∗ alently when µ = µ , (S ,E ,I ) becomes unstable and model (2.1) under- 0 0 0 0 goes a Hopf bifurcation. 42 Cui, Sun, and Zhu Proof. First note that if R = R or µ = µ ,wehave 0 H H 0 0 b b 1 (c + d + γ )(c + d + γ + ) = bγ (c + d)(1 − ), R R R 0 0 0 then one can verify that equation (5.1) has a negative root and a pair of purely imaginary roots λ =±ω i, where b(c + d + γ) ω = . (5.4) For the the characteristic equation (5.1), we consider the characteris- tic root λ as a function of R or a function of µ. Differentiating equation (5.1) with respect to µ,weget b b dλ 3λ + 2 c + d + γ + λ + (c + d + γ) R R dµ 0 0 b b b dR = λ + (c + d + γ)λ − γ(c + d) . 2 2 2 dµ R R R 0 0 0 This gives 2 b b −1 3λ + 2(c + d + γ + )λ + (c + d + γ) dλ cK R R 0 0 = · . dµ γ(c + d) [λ + (c + d + γ)λ − γ(c + d)] Thus d(Reλ) dλ −1 sign = sign Re( ) dµ λ=iω dµ λ=iω 0 0 b b 3λ + (c + d + γ) + 2(c + d + γ + )λ R R 0 0 = sign Re λ=iω λ − γ(c + d) + (c + d + γ)λ 2 b b −3ω + (c + d + γ) + 2i(c + d + γ + )ω 0 R R 0 0 = sign Re −ω − γ(c + d)] + i(c + d + γ)ω 2b(c + d + γ) b = sign (c + d + γ) + γ(c + d) R R 0 0 2b b + (c + d + γ) c + d + γ + R R 0 0 > 0. Therefore, as µ> 0 increases, the real part of a pair of characteristic roots changes from negative to positive through zero, the transversality condi- tion holds. Hence, the model with m = 0 undergoes an Hopf bifurcation when R = R . This completes the proof. 0 H 0 The Impact of Media on the Control of Infectious Diseases 43 5.2. m is Sufﬁciently Small When R > 1 and 0 ≤ m<m , the model (2.1) has a unique endemic 0 0 ∗ ∗ ∗ equilibrium (S ,E ,I ). Evaluating the Jacobian of (2.1) at the equilib- rium gives ⎛ ⎞ b γ(c + d) S ∗ ∗ − S 0 − + mbS 1 − ⎜ ⎟ K c K ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∗ ∗ ∗ ∗ ∗ S γ(c + d) S J(S ,E ,I ) =⎜ ⎟ . ⎜ b 1 − −(c + d) − mbS 1 − ⎟ ⎜ ⎟ K c K ⎝ ⎠ 0 c −γ ∗ ∗ ∗ The characteristic equation about (S ,E ,I ) is given by 3 2 λ + a λ + a λ + a = 0, (5.5) 2 1 0 where a = c + d + γ + S > 0, ∗ ∗ S bS a = cmbS 1 − + (c + d + γ), (5.6) K K ∗ ∗ S 2S a = b 1 − γ(c + d) + cmbS − 1 . K K Since we do not have a closed form for the endemic equilibrium, it is not easy to study the bifurcations analytically for the general case of m.We are going to use the fact that m> 0 is small to study the Hopf bifurcation of the endemic equilibrium. ∗ ∗ ∗ The coordinates of the endemic equilibrium (S ,E ,I ) are smooth functions of m. When m> 0 is sufﬁciently small, or if 0 <m<m ,wecan ∗ ∗ ∗ expand the coordinates for the unique endemic equilibrium (S ,E ,I ) as ∗ ∗ ∗ 2 S = S + mS + O(m ), ⎪ 0 1 ∗ ∗ ∗ 2 E = E + mE + O(m ), (5.7) 0 1 ∗ ∗ ∗ 2 I = I + mI + O(m ), 0 1 where particularly, by (4.4) we have K b 1 ∗ ∗ ∗ S = ,S = S 1 − . (5.8) 0 1 0 R µ R 0 0 44 Cui, Sun, and Zhu Note that the cubic polynomial (5.5) reduces to (5.1) when m = 0. Similar to the case of m = 0 in the above subsection, we will now study how the media coverage has an impact on the dynamics of the disease transmission by the method of perturbation. It is not difﬁcult to verify that (5.5) has a pair of purely imaginary roots if and only if a a = a . Let 1 2 0 H = a a − a . (5.9) 1 2 0 If H = 0, the endemic equilibrium has a pair of purely imaginary roots. Using the expressions in (5.6), (5.7) and (5.8), one can verify that H = 0 is equivalent to H (m, R ) = 0, where H (m, R ) = R γ(c + d) − γ(c + d)(b + c + d + γ) bm bm 2 2 +R − 1 γ(c + d) − 1 + (c + d + γ) µ µ + γ(c + d)(c + d + γ) 2b m bm +R −b(c + d + γ) − (c + d + γ) + (c + d + γ) µ µ 2b m + (c + d + γ). (5.10) Note that when H (m, R ) = 0, the endemic equilibrium has a pair of purely imaginary eigenvalues λ =±ωi, where ∗ ∗ S bS 2 ∗ ω = cmbS 1 − + (c + d + γ). (5.11) K K Hence if the parameters m and R satisfy H (m, R ) = 0, an Hopf bifurca- 0 0 tion may occur. Now we develop the function determined by H (m, R ) = 0. Proposition 5.3. Consider H (m, R ) = 0 for 0 ≤ m<m and R > 1.In 0 0 0 the neighborhood of (0, R ), there exists a unique smooth function R = H 0 R (m) such that H (m, R (m)) = 0 for 0 ≤ m<m sufﬁciently small. Further- 0 0 0 more, we have R (m) = R + mR + O(m ), (5.12) 0 H H 0 1 where R is deﬁned as in (5.2) and 2 2 2 (c + d + γ) (2b + c + d + γ)R + b (c + d + γ)(3R − 2) H 0 R = . µ[γ(c + d)R + b(c + d + γ)] (5.13) The Impact of Media on the Control of Infectious Diseases 45 Proof. Note that H(0, R ) = 0 and in the neighborhood of (0, R ), H H 0 0 we have ∂H 2 2 = 3R γ(c + d) − 2R [γ(c + d) + (c + d + γ) ] H 0 ∂R m=0,R =R 0 0 −b(c + d + γ) = R γ(c + d) + b(c + d + γ) = 0, then by the Implicit Function Theorem, there exists a unique function R = R (m) such that H (m, R (m)) =0for m ≥ 0 sufﬁciently small. 0 0 0 If we write the Taylor expansion for R (m) in terms of m as in (5.12) and plug it into (5.10), we have H (m, R (m)) = H (m, R + mR + O(m )) H H 0 1 γ(c + d) 3 2 = (R + 3R R m) γ(c + d) − m (b + c + d + γ) H H 1 0 0 2 2 +(R + 2R R m) − γ(c + d) − (c + d + γ) H H H 0 1 γ(c + d) b +m (b + c + d + γ) − (c + d + γ) µ µ +(R + mR ) − b(c + d + γ) H H 0 1 2 2 b 2b 2b +m (c + d + γ) − (c + d + γ) + (c + d + γ)m µ µ µ = 0. (5.14) Equalizing the terms of same power of m on both sides of the above equation, from the constant term, we have 3 2 2 R γ(c + d) − R [γ(c + d) + (c + d + γ) ] − R b(c + d + γ) = 0, H H 0 0 0 which is the same as the equation to deﬁne R . For the coefﬁcients for the ﬁrst term, we have γ(c + d) 2 3 3R R γ(c + d) − R (b + c + d + γ) H 1 H 0 0 γ(c + d) b 2 2 + R (b + c + d + γ) − (c + d + γ) µ µ + 2R R [−γ(c + d) − (c + d + γ) ] H H 0 1 + R (c + d + γ) 2 2 2b 2b − (c + d + γ) − bR (c + d + γ) + (c + d + γ) µ µ = 0. (5.15) 46 Cui, Sun, and Zhu Solving equation (5.15) in terms of R we obtain (5.13). Theorem 5.4. If 1 < R < R (m), where R (m) is deﬁned in (5.12) for 0 0 0 ∗ ∗ ∗ m ≥ 0 sufﬁciently small, then the endemic equilibrium (S ,E ,I ) is locally- asymptotically stable. Proof. When R > 1, consider the characteristic equation for the ∗ ∗ ∗ equilibrium (S ,E ,I ) in (5.5). Obviously, a > 0. We need to prove a > 0 2 0 and a a − a > 0 in order to use Routh-Hurwitz criteria [13] to conclude. 2 1 0 By (5.8), we have for m> 0 small that 2S a = γ(c + d) + mbcS − 1 K 2 K = γ(c + d) + mbc − 1 + O(m ). (5.16) R K R 0 0 8γ(c + d) mbcK 2 Since m<m = ,wehave γ(c + d) + ( − 1)> 0, hence bcK R R 0 0 a > 0. Next we prove a a − a > 0. By (5.8) and R γ(c + d) = µcK,a 2 1 0 0 straightforward calculation gives ∗ ∗ b S bS ∗ ∗ a a − a = (c + d + γ + S ) mbcS 1 − + (c + d + γ) 2 1 0 K K K ∗ ∗ S 2S −b 1 − γ(c + d) + mbcS − 1 K K b bm 2 2 = R 1 + (c + d + γ) + (b + c + d + γ)cKm − µcK 2b m +R µcK + b(c + d + γ) + (c + d + γ) bm − (c + d + γ) − (b + c + d + γ)cKm 2b m − (c + d + γ)}+ O(m )> 0. (5.17) Then it follows from Routh-Hurwitz criteria [13] that all eigenvalues of (5.5) have negative real parts. Hence E is locally-asymptotically stable when 1 < R < R (m) and m> 0 is sufﬁciently small. 0 0 Theorem 5.5. When 0 <m<m and R = R (m), the system undergoes 0 0 0 a Hopf bifurcation. The Impact of Media on the Control of Infectious Diseases 47 Proof. It follows from Proposition 5.3 and Theorem 5.4, we only need to prove the transversality to conclude the existence of the bifurca- tion. Differentiating Eq.(5.5) with respect µ,weget dλ dS B = · , (5.18) dµ dµ B where b 2S b B =− λ − cmb 1 − + (c + d + γ) λ K K K ∗ ∗ ∗ b 2S S 4S ∗ 2 + γ(c + d) + cmbS − 1 − 1 − cmb − 1 , K K K K ∗ ∗ bS S b 2 ∗ ∗ B = 3λ + 2 c + d + γ + λ + cmbS 1 − + S (c + d + γ) . K K K (5.19) Recall that when R = R (m), or equivalently when 0 0 ∗ ∗ ∗ bS S bS c + d + γ + cmbS (1 − ) + (c + d + γ) K K K ∗ ∗ S 2S = b 1 − γ(c + d) + cmbS − 1 , (5.20) K K Equation (5.5) has a pair of purely imaginary roots λ =±ωi with ∗ ∗ S bS 2 ∗ ω = cmbS 1 − + (c + d + γ) K K b(c + d + γ) bck 1 b (c + d + γ) 1 = + m 1− + 1 − + O(m ). R R R µR R 0 0 0 0 0 Note that mc cmb S K ∗ γ(c + d) ∗ E S (1− ) γ γ(c+d) K S = e = e , R µc so we get ∗ ∗ dS γ(c + d) S = · . 2S dµ µ cmbS (1 − ) − γ(c + d) 2S By (5.20), we have γ(c + d) + cmbS ( − 1)> 0. Hence we always have dS < 0. dµ 48 Cui, Sun, and Zhu Therefore, it follows from (5.18) that we have d(Reλ) sign dµ λ=iω dλ = sign Re dµ λ=iω ∗ ∗ ∗ b 2S S 4S 2 ∗ 2 [ω + γ(c + d) + cmbS ( −1)] −(1− )cmb ( −1) K K K K = sign −Re S b b 2 ∗ ∗ ∗ −3ω + cmbS (1− ) + S (c+ d+ γ)+ 2i(c+ d+ γ + S )ω K K K 2S b i[cmb(1 − ) + (c + d + γ)]ω K K S b b 2 ∗ ∗ ∗ −3ω + cmbS (1 − ) + S (c + d + γ) + 2i(c + d + γ + S )ω K K K ∗ ∗ ∗ S 3S 2S = sign cmb (1 − )(1 − ) + cmb(1 − )(c + d + γ) K K K b b 2b 2 ∗ + γ(c + d) + (c + d + γ) + S (c + d + γ) K K > 0. (5.21) Therefore, the transversality condition holds and hence a Hopf bifur- cation occurs when R = R (m) and m is small. 0 0 5.3. Numerical Simulations For the purpose of simulations, here we ﬁx some of the parameters in Table I and shall consider the cases when γ and m are varied. First we consider the case when the disease transmission is mild with a lower reproduction number. In the case when γ = 0.05 and all other Table I. Part of the parameters for the simulations Parameters Value Carrying capacity K 5,000,000 Intrinsic growth rate of the population b 0.001 −8 Contact transmission rate µ 1.2 × 10 Time that an exposed becomes infected 10 Natural death rate of the population d 0.001 The Impact of Media on the Control of Infectious Diseases 49 ∗ ∗ ∗ Table II. Endemic equilibrium (S ,E ,I ) when m> 0 is varied. In the table, except for the parameters given in Table I, here we have γ = 0.05. In this case, R = 1.188 and R = 5.52 0 H ∗ ∗ ∗ ParametermS E I m = 0 4208333 6597 13194 −6 m = 1 × 10 4261135 6235 12468 −6 m = 6 × 10 4457313 4790 9580 parameters as in Table I, we have R = 1.188. As shown in Fig. 2(a), (b), the transmission of the disease experiences multiple peaks without the media alert, the thin curve represents the case when m = 0, the applica- tion of media was not considered. The other two thicker curves represent the cases when m = 0.000001 and m = 0.000006, respectively. As shown in Table II, if γ = 0.05, we have R = 1.188 and R = 5.52. For all the cases, 0 H the endemic equilibrium is a spiral sink which is local asymptotically sta- ble. The population in each compartment approaches its equilibrium value. From the simulation results in Fig. 2, one can see that the effective media coverage (larger values of m) stabilizes the oscillation, and less number of the individuals become infected in the course of transmission. The media impact to the transmission is also simulated in Fig. 3(a), (b) where γ is reduced to 0.02, with all other parameters are given in Table I. 6. DISCUSSION 6.1. Multiple Peaks of the Transmission and the Media Impact We knew that when m = 0, the Hopf bifurcation occurs and a periodic solution appears. When the media impact is not considered, if R > 1 and close to R , the disease will be endemic with multiple peaks. The time between between the two peaks can be approximated by 2π 2π T = = ! . (c + d + γ) But when the media coverage/alert is introduced, or when 0 <m<m is sufﬁciently small, if there are multiple peaks, the time between each of 50 Cui, Sun, and Zhu (a) 5e+06 4.8e+06 4.6e+06 4.4e+06 4.2e+06 4e+06 3.8e+06 0 1000 2000 3000 4000 5000 6000 The number of susceptibles S(t) (b) 1000 2000 3000 4000 5000 6000 The number of infected I(t) Figure 2. Simulations for the case when γ = 0.05. Here R = 1.188 and R = 5.52. The 0 H −6 −6 thickness of the curves increases when the parameter m = 0 changes from 0, 10 to 6 × 10 . the two peaks can be approximated by 2π T = . b(c + d + γ) bck 1 b (c + d + γ) 1 + m[ (1 − ) + (1 − )] + O(m ) R R R µR R 0 0 0 0 0 This shows that the media alert shortens the time of the secondary peak of the disease transmission. This effect is also veriﬁed by the simu- lations in Fig. 3 (a), (b). J(t) S(t) The Impact of Media on the Control of Infectious Diseases 51 (a) 5e+06 4e+06 3e+06 2e+06 1e+06 0 1000 2000 3000 4000 5000 6000 The number of susceptibles S(t) (b) 1.2e+06 1e+06 1000 2000 3000 4000 5000 6000 The number of infected I(t) Figure 3. Simulations for the case when γ = 0.02. Here R = 2.97 and R = 8.26. The thick- 0 H −6 −6 ness of the curves increases when the parameter m = 0 changes from 0, 10 to 6 × 10 . 6.2. The Media Coverage/Alert and the Endemic State Note that whenever R > 1, an endemic equilibrium appears and its ∗ ∗ ∗ coordinates (S ,E ,I ) aregivenby mc ∗ S K c ∗ ∗ ∗ ∗ ∗ bS 1 − − (c + d)E = 0,S = e ,I = E . K R γ J(t) S(t) 52 Cui, Sun, and Zhu ∗ ∗ ∗ If we consider S , E and I as functions of m> 0, then we have ∗ ∗ 2S dE ⎪ dS b 1 − − (c + d) = 0, dm K dm (6.1) ∗ ∗ mc ∗ mc ∗ ⎪ dS cK cmK dE E E γ γ = e + e . dm R γ R γ dm 0 0 Thus, we get ∗ 2 ⎪ cb(S ) 1 − ⎪ ∗ dS K = , 2S ⎪ dm ⎪ ∗ ⎪ γ(c + d) − mcbS 1 − ∗ ∗ S 2S ∗ 2 cb(S ) 1 − 1 − dE 1 K K = · , 2S ⎪ dm c + d ⎪ ∗ ⎪ γ(c + d) − mcbS 1 − ∗ ∗ dI c dE ⎩ = . dm γ dm Since the endemic equilibrium is locally-asymptotically stable, we have ∗ ∗ 2S dS ∗ ∗ γ(c + d) − mcbS (1 − )> 0. Hence > 0, thus S is always an K dm dE increasing function of m, and if 1 < R < 2 one can also verify that < dm dI ∗ ∗ 0 and < 0, therefore, E and I are decreasing functions of m. This is dm veriﬁed by the numerical simulationsin Table II and Fig. 3 (a), (b). 6.3. Other Comments and Further Improvement In this paper, we are trying to explore the impact of media coverage to the transmission of infection diseases. The model (2.1) is a toy model for the purpose of analyzing the impact of media on the spreading of the −mI disease. In the model, we used a contact transmission rate β(I ) = µe . For further study, it would be ideal to consider more realistic contact transmission rates to reﬂect the impact of media coverage and alertness. Yet, the analysis of such a new model can be mathematically more chal- lenge due to the high dimension of the models and nonlinearity of the incidence function. The Impact of Media on the Control of Infectious Diseases 53 REFERENCES 1. Brauer, F., and Castillo-Chavez, C.(2000). Mathematical Models in Population Biology and Epidemics. Springer-Verlag, New York. 2. Busenberg, S., and Cooke, K.(1993). Vertically Transmitted Diseases. Springer-Verlag, New York. 3. Capasso, V.(1993). Mathematical Structure of Epidemic System, Lecture Note in Biomath- ematics, Vol. 97. Springer, Berlin. 4. Capasso, V., and Serio, G.(1978). 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Journal of Dynamics and Differential Equations – Pubmed Central
Published: May 18, 2007
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