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Media alert in an SIS epidemic model with logistic growth

Media alert in an SIS epidemic model with logistic growth JOURNAL OF BIOLOGICAL DYNAMICS, 2017 VOL. 11, NO. S1, 120–137 http://dx.doi.org/10.1080/17513758.2016.1181212 a b c c,d a Lianwen Wang , Da Zhou ,ZhijunLiu ,DashunXu and Xinan Zhang School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China; b c School of Mathematical Sciences, Xiamen University, Xiamen, People’s Republic of China; Department of Mathematics, Hubei University for Nationalities, Hubei, People’s Republic of China; Department of Mathematics, Southern Illinois University, Carbondale, IL, USA ABSTRACT ARTICLE HISTORY Received 10 August 2015 In general, media coverage would not be implemented unless the Accepted 18 April 2016 number of infected cases reaches some critical number. To reflect this feature, we incorporate the media effect and a critical number KEYWORDS of infected cases into the disease transmission rate and consider SIS epidemic model; media an susceptible-infected-susceptible epidemic model with logistic alert; logistic growth; growth. Our model analysis shows that early media alert and strong multiple endemic equilibria; media effects are preferable to decrease the numbers of infected bi-stability cases at endemic equilibria. Furthermore, we noticed that the model AMS SUBJECT may have up to three endemic equilibria and bi-stability can occur CLASSIFICATION in a threshold interval for the critical number. Note that the interval 34K18; 34K20; 92D30 depends on parameters for the focal disease and the media effect. It is possible to roughly estimate the interval for re-emerging diseases in a given region. Therefore, the result could be useful to health pol- icymakers. Global stability is also obtained when the model admits a unique endemic equilibrium. 1. Introduction Media has been utilized as a disease control measure, especially for epidemics associated with emerging and re-emerging infectious diseases [19] such as HIV/AIDS, SARS, H1N1, Ebola virus disease (EVD), Middle East Respiratory Syndrome. During the outbreak of the influenza A (H1N1) in 2009, mass media was extensively used by the Centers for Disease Control and Prevention of United States and WHO to keep the public aware of information relatedtothe pandemic [6]. It is believed that media use contributed to the control of the pandemic. WHO also indicated that media played an important role in controlling the spread of H7N9 in China in 2013 [31].Mediadoesnot onlyalertthegeneralpubliconthe hazard from the infectious diseases but also informs the public of the requisite preventive measures like wearing protective masks [25], vaccination, voluntary quarantine, avoidance of congregated places, etc. Therefore, the extensive use of media may bring in changes in public behaviour and reduce the frequency and probability of contacts with infected individuals so that the severity of a disease outbreak would be diminished [4, 9, 10, 13, 14, 21, 24]. CONTACT Zhijun Liu zhijun_liu47@hotmail.com; Lianwen Wang wanglianwen5.17@163.com;DaZhou zhouda@xmu.edu.cn; Dashun Xu dashunxu@siu.edu; Xinan Zhang zhangxinan@hotmail.com © 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. JOURNAL OF BIOLOGICAL DYNAMICS 121 In ordertostudytheimpactofmedia-likecontrolmeasures ondiseasetransmis- sion dynamics, several types of media function forms have been proposed to describe reduced disease transmission rates due to media use and compartmental models with these rates have been analysed (e.g. [9, 10, 13–17, 23, 24]). The deduction in the transmis- −mI sion rate was described by the form of β(1 − e ) with the parameter m > 0 reflecting how strongly media coverage can aeff ct contact infection [ 9]. With the rate, the analysis of a susceptible–exposed–infected model (SEI) shows that the model may exhibit peri- odicoscillationsforweakmediaeeff cts whileitmay havethreeendemic equilibriafor strong media effects [ 9]. The form of β − β I/(ν + I) was also used as the transmission rate with the deduction β I/(ν + I) due to media use [10, 13, 24]. A threshold dynamics was obtained for an SIS epidemic model. It is also shown that media coverage can lower infection and delay the arrival of the infection peak [10, 13]. However, the study of a sus- ceptible–vaccinated–infected–recovered epidemic model with a vaccinated class indicates that media eeff cts could be complicated and simplified understandings may even make the disease worse due to possible public panic [24]. The third function type with psycholog- ical/media eect ff s is of the form β/(1 + αI ), identified by Collinson and Heeff rnan (see [8, 33] and the references therein). Using a simple susceptible-exposed-infected-recovered model, Collinson and Heeff rnan [ 8] found that important measurements of an epidemic outbreak (such as peak magnitude of infection, peak time of infection peak and end of the outbreak) depend on the chosen media function. Their sensitivity analysis also showed such dependence for the sensitivities of model parameters. This makes it dicffi ult to iden- tify eecti ff ve disease control strategy and calls for more study on the eect ff s of mass media on disease transmission dynamics. Theoretical studies usually assume that media coverage aeff cts disease transmission dur- ing the whole time period of the disease spread (e.g. [9, 10, 14, 24]). In the reality, media coverage generally does not occur in the beginning stage of a disease spread. For instance, the early suspected cases of EVD died in December 2013 while the first notification by WHO on the outbreak [32] was not issued until 21 March 2014. In general, media deliv- ers alerts and timely reports infected cases only when certain number of infected cases is reached [28, 34, 38]. To include such feature of media/psychological effect, Xiao et al. [34] introduced a critical number of infected cases I and proposed a piecewise and discon- tinuous control function [7, 26] for disease control strategy (sliding mode control). Wang and Xiao [28] further constructed a Filippov SIR epidemic model to describe media eeff cts using the following transmission rate β, I < I , β(I) = (1) β exp (−αI), I > I . Their analysis shows that the model system can stabilize at either one of the equilibria fortheresulted subsystems or thenewendemicstateinducedbytheon-offmedia eeff ct, depending on the critical level I . They also demonstrated that proper combinations of critical levels and control intensities can lead to the desired case number. The transmission rate (1) was then generalized in a time-dependent way to study an influenza outbreak in Shannxi, China [35]. It was demonstrated that compartmental models can exhibit distinct dynamics, depend- ing on the chosen incidence rate (e.g. [20, 33]). To explore the media eect ff on disease 122 L. WANG ET AL. transmission dynamics we here propose a new transmission rate. Following the idea of the critical number of infected cases I ,weconsiderthefollowingnon-smoothbut continuous transmission rate: ⎨ β,0 ≤ I ≤ I , β(I) = (2) β , I > I , where p ≥ 0 represents the intensity of the media eeff ct on contact infection. If p = 0, β(I) is equal to the background transmission rate β, implying that media coverage does not occur. With this rate, we shall consider an SIS endemic model. Classical compartmental models with media eeff cts assume either a constant size of the total population or constant recruitment rate for the susceptible class. The assumption of varying total populations may be more reasonable for a relatively long-lasting disease or for a disease with high mortality rates. In fact, varying total populations were discussed before (e.g. [1, 3, 5, 9, 11, 22, 29, 36]). Here, we assume that the population of a commu- nity follows the logistic growth. For the sake of mathematical simplicity, we assume that newborns directly enter into the susceptible class and infected persons do not contribute to births and deaths in the susceptible class. Following works in [2, 9, 28], our SIS model reads dS S = rS 1 − − β(I)IS + γ I, dt a (3) dI = β(I)IS − (d +  + γ)I, dt where r is the intrinsic growth rate of the susceptible population, a denotes the carrying capacity of the community in the absence of infection, d is natural death rate, γ represents the recovered rate, and  is the disease-induced death rate. The analysis of model (3) with thetransmissionrate(2) showsthatthereexistsathreshold interval  for the critical num- ber I in which the model may be stabilized at one of two stable equilibria with different levels of infected cases. This implies that the policymaker may have to choose the critical number I according to the focal disease in order to minimize infected cases and also avoid unnecessary public panic. The global stability was also obtained for I not in the threshold interval. In the following, the analysis of existence of equilibria is presented in Section 2 while the local and global stabilities are given in Sections 3 and 4, respectively. A discussion section comes to the end of the work. 2. Existence of equilibria For our convenience, denote b =: r/a. Model (3) with the transmission rate (2) can be decomposed into two sub-systems dS = bS(a − S) − βIS + γ I, dt 0 ≤ I ≤ I,(4) dI = βIS − (d +  + γ)I, dt JOURNAL OF BIOLOGICAL DYNAMICS 123 and dS I = bS(a − S) − β IS + γ I, dt I I > I.(5) dI I = β IS − (d +  + γ)I, dt I The origin O(0, 0) and the disease-free equilibrium E (a,0) always exist. The basic repro- ductive number can be easily calculated, given by βa R = , d +  + γ from which one can see that media coverage does not change the basic reproduction number (e.g. [9, 10, 13, 15–17, 23, 24]). To obtain the existence of endemic equilibria, denote a d +  a b a  , a  a + I , I  (R − 1), 0 2 0 c ∗ 0 R a b d + 0 0 and consider two cases: 0 ≤ I ≤ I and I > I ,separately. c c In the case of 0 ≤ I ≤ I , the sub-system (4) has a unique positive equilibrium E (S , I ) c ∗ ∗ ∗ if and only if 0 < I ≤ I ,thatis, a < a ≤ a ,where S = a . ∗ c 0 2 ∗ 0 In the case of I > I ,the I component of a positive equilibrium for the sub-system (5) satisefi s the following equation a a 0 0 2p−1 p−1 f (I) = bI − abI + (d + ) = 0. (6) p p I I c c p p 2 2p−1 p−1 From (a /I ) bI − (a /I )abI =−(d + ) < 0, positive solutions to Equation (6) 0 0 c c satisfy 1/p I < R I  I.(7) c n That is, the positive solutions of Equation (6) must be in the interval (I , I ).Also, f (I ) = c n d + > 0. Therefore, we must have a > a due to I > I . Next, we discuss the existence 0 n c of positive solutions to Equation (6) in (I , I ) in two cases, 0 < p ≤ 1and p > 1. c n Case I. 0 < p ≤ 1. In this case, f (I) is strictly increasing. In fact, the derivative of f (I) is given by a a 0 0 p−2 p f (I) = bI (2p − 1)I − a(p − 1) ,for I ∈ (I , I ),(8) c n p p I I c c and we can show that f (I)> 0for 0 < p ≤ 1. If ≤ p ≤ 1, one clearly derives f (I)> 0. If 0 < p < ,wecan calculatethezeropointof f (I) as 1/p p − 1 I  R I > 0. (9) e 0 c 2p − 1 124 L. WANG ET AL. which, obviously, I is the unique maximum point of f (I).Since p p p I − I = R I , (10) e n 0 c 1 − 2p we have I < I for 0 < p < , and hence f (I)> 0on (I , I ).Tosum up,wealwayshave n e c n f (I)> 0if0 < p ≤ 1. Meanwhile, note that a b(a − a) 0 2 f (I ) = (11) may be positive, negative or zero. If f (I ) ≥ 0, which is equivalent to a ≤ a ,then c 2 Equation (6) has no positive real root in (I , I ).If f (I )< 0, which is equivalent to a > a , c n c 2 then Equation (6) admits a unique positive root, denoted by I ,satisfying I < I < I .That 1 c 1 n is, in the case of 0 < p ≤ 1, the sub-system (5) has a unique positive equilibrium, denoted by E (S , I ),ifand onlyif a > a ,where S = a (I /I ) and I ∈ (I , I ). 1 1 1 2 1 0 1 c 1 c n The following lemma is needed to discuss the case of p > 1. Lemma 2.1: Let p > 1, then a < a always holds, where 1 2 1/(2p−1) 2p−1 p (2p − 1) (d + ) I a  . p p−1 p p (p − 1) a b Proof: Consider the concave function H(x) = ln x, x > 0. Assign 2p − 1 2p − 1 d + x = a , x = I , 1 0 2 c p − 1 p a b p − 1 p λ = , λ = . 1 2 2p − 1 2p − 1 Note that λ + λ = 1, andH(x) is a strictly concave function. This implies thatH(λ x + 1 2 1 1 λ x )>λ H(x ) + λ H(x ), and we have 2 2 1 1 2 2 d +  p − 1 2p − 1 p 2p − 1 d + ln a + I = ln a + I 0 c 0 c a b 2p − 1 p − 1 2p − 1 p a b 0 0 p − 1 2p − 1 p 2p − 1 d + > ln a + ln I 0 c 2p − 1 p − 1 2p − 1 p a b (p−1)/(2p−1) p/(2p−1) 2p − 1 2p − 1 d + = ln a · I 0 c p − 1 p a b 1/(2p−1) 2p−1 p (2p − 1) (d + ) I = ln . p p−1 p p (p − 1) a b The monotonicity of H(x) leads to a < a .Theproofiscompleted. 1 2 Case II. p > 1. By Lemma 2.1, a < a holds. It follow from the formula (10) that I < I . 1 2 e n Note that the derivative f (I) givenbyEquation(8)is positive. I is the unique minimum e JOURNAL OF BIOLOGICAL DYNAMICS 125 + − point of f (I).Clearly, f (0 ) = f (I ) = d + . Next, we want to compare the sizes of I and I and determine the signs of f (I ) and f (I ). The following six cases are considered. Let us e c e denote p a b 0 0 I  . p − 1 d + 0 0 Case 1. I ≥ I and a ≥ a . It follows from I ≥ I that c 2 c c c d +  d +  p a b 2p − 1 0 0 a ≥ a + I ≥ a + · = a . 2 0 0 0 a b a b p − 1 d +  p − 1 0 0 Since a ≥ a > a ,wehave f (I ) ≤ 0, and a ≥ ((2p − 1)/(p − 1))a . Furthermore, from 2 1 c 0 Equation (9) one can get I ≤ I .Thus f (I ) ≤ f (I ) ≤ 0. Direct calculation shows that a > c e e c a ⇔ f (I )< 0holds.If f (I ) = 0, we have I < I .Thus, Equation(6)hasauniquepos- 1 e c c e itive root I ,satisfying I < I < I .If f (I )< 0, the conclusion is still true. Accordingly, 1 e 1 n c the sub-system (5) has a unique positive equilibrium E (S , I ). 1 1 1 0 0 Case 2. I ≥ I and a < a < a . It follows from I ≥ I that c 1 2 c c c 1/(2p−1) 2p−1 p (2p − 1) (d + ) 0 p a ≥ (I ) p p−1 p p (p − 1) a b 1/(2p−1) 2p−1 p (2p − 1) (d + ) p a b 2p − 1 = = a , p p−1 p p (p − 1) a b p − 1 d +  p − 1 that is, a >((2p − 1)/(p − 1))a . Thus, we have I < I from Equation (9) and f (I )> 0 c e c 0because of a < a . Moreover, one can show that a > a ⇔ f (I )< 0. Therefore, when 2 1 e a < a < a , Equation (6) has two different positive roots in (I , I ),denotedby I , I , 1 2 c n 2 3 where I ∈ (I , I ), I ∈ (I , I ). That is, the sub-system (5) admits two different positive 2 c e 3 e n equilibria E (S , I ),where S = a (I /I ) , i = 2,3. i i i i 0 i c Case 3. I ≥ I and a = a .Similar to theargumentin Case 2, it follows that c 1 ((2p − 1)/(p − 1))a ≤ a < a , and hence we have I ≤ I and f (I )> 0. Note that 0 2 c e c the equivalent relationship a = a ⇔ f (I ) = 0. We must have I < I .Thissuggests 1 e c e that Equation (6) only has one positive solution I ∈ (I , I ). Accordingly, for the sub- e c n system (5), there exists exactly one positive equilibrium E (S , I ),where S = a (I /I ) . e e e e 0 e c Case 4. I ≥ I and a < a < a . Recall that a < a ⇔ f (I )> 0and I is the unique c 0 1 1 e e minimum point of f (I) in (I , I ). One immediately deduces that Equation (6) has no pos- c n itive root in (I , I ) no matter I ≤ I or I > I .Hence,thesub-system (5)has no positive c n c e c e equilibrium. Case 5. I < I and a > a .From a > a > a , one deduces that f (I )< 0and f (I )< 0. c 2 2 1 c e Hence, Equation (6) only has one positive solution I ∈ (I , I ) no matter I ≤ I or I > I , 1 c n c e c e where I < I < I .Thatis,thesub-system(5)hasonlyonepositiveequilibrium E (S , I ). e 1 n 1 1 1 Case 6. I < I and a < a ≤ a .Clearly,one canhave a <((2p − 1)/(p − 1))a from c 0 2 2 0 I < I and a ≤ a <((2p − 1)/(p − 1))a .Hence, I < I .Noticethat f (I) is an increas- c 2 0 e c ing function on the interval (I , I ),and f (I ) ≥ 0. Equation (6) has no positive root in the c n c interval (I , I ). Namely, the sub-system (5) has no positive equilibrium. c n 126 L. WANG ET AL. Now, one summarizes the existence of the equilibria of model (3) as follows: Theorem 2.2: Model (3) always admits an equilibrium O(0, 0) andadisease-freeequilib- rium E (a,0).Ifa ≤ a (i.e.R ≤ 1), then model (3) has no endemic equilibrium. If a > a 0 0 0 0 (i.e. R > 1), we have the following conclusions. (1) Assume that 0 < p ≤ 1. (i) If a < a ≤ a , then E isauniqueendemicequilibrium; 0 2 ∗ (ii) If a > a , then E isauniqueendemicequilibrium. 2 1 (2) Assume that p >1and I ≥ I . (i) If a < a < a , E isauniqueendemicequilibrium; 0 1 ∗ (ii) If a = a , there are two endemic equilibrium, E and E ; 1 ∗ e (iii) If a < a < a , there exist three endemic equilibria, E , E and E ; 1 2 ∗ 2 3 (iv) If a = a , both E and E exist; 2 ∗ 1 (v) If a > a , E isauniqueendemicequilibrium. 2 1 (3) Assume that p >1and I < I . (i) If a < a ≤ a , E isauniqueendemicequilibrium; 0 2 ∗ (ii) If a > a , E isauniqueendemicequilibrium. 2 1 Remark 1: From (ii) of (1), (v) of (2) and (iii) of (3) in Theorem 2.2 one can deduce that E isauniqueendemic equilibriumif a > a . 1 2 By Theorem 2.2, if 0 < p ≤ 1, model (3) only has one endemic equilibrium, either E (in the case of a < a ≤ a ), or E (in the case of a > a ). The existence of positive equilibria 0 2 1 2 for the model in the case of p > 1 is illustrated in Figure 1. Here, we define the follow- ing different curves and regions for parameters a and I as follows: L ={(a, I ) | a = a }, c 0 c 0 0 0 L ={(a, I ) | a = a and I ≥ I }, L ={(a, I ) | a = a and I ≥ I }, Q ={(a, I ) | 1 c 1 c 2 c 2 c 1 c c c Figure 1. The existence of the endemic equilibria of model (3) in the case of p > 1. There are equilibria E and E on the curve L , E and E on L , three equilibria E , E and E in the region Q , a unique ∗ e 1 ∗ 1 2 ∗ 2 3 2 equilibrium E in the region Q , a unique equilibrium E in the region Q , and no endemic equilibrium 1 1 ∗ 3 in Q . See the content for the definitions of these regions. 4 JOURNAL OF BIOLOGICAL DYNAMICS 127 a > a }, Q ={(a, I ) | a < a < a and I ≥ I }, Q = C (Q ∪ L ∪ L ) (where U = 2 2 c 1 2 c 3 U 2 1 2 {(a, I ) | a < a ≤ a }), and Q ={(a, I ) | 0 < a ≤ a }. c 0 2 4 c 1 0 1 Note that the condition a ≤ a ≤ a with I > I in Theorem 2.2 is equivalent to I ≤ 1 2 c c c I ≤ I ,where 1/p p p−1 2p−2 p (p − 1) a b 1 0 2 I = max{I , I }, I = . c c c 2p−1 (2p − 1) R d + In the case of R > 1and p > 1, multiple endemic equilibria exist if the critical number I 0 c is in the threshold interval 1 2 := [I , I ]. (12) c c The existence region in Figure 1 is given by the union of L , L and Q . 1 2 2 3. Local stability of equilibria This section focuses on the local stability of equilibria. Corresponding to the equilibria O(0, 0), E and E , the Jacobian matrix of the sub-system (4) reads 0 ∗ ab − 2bS − βI −βS + γ J = . (13) βI βS − (d +  + γ) At O(0, 0) the determinant det(J(O)) =−ab(d +  + γ) < 0. Thus O(0, 0) is a saddle point. At E (a,0) it can be shown that det(J(E )) =−βab(a − a ), 0 0 (14) tr(J(E )) = β(a − a ) − ab. 0 0 If a > a ,thendet(J(E )) < 0, which means that E isasaddlepoint.If a < a ,then 0 0 0 0 det(J(E )) > 0and tr(J(E )) < 0. Hence E is locally asymptotically stable. Since 0 0 0 2 2 [−tr(J(E ))] − 4det(J(E )) = [β(a − a ) + ab] ≥ 0, (15) 0 0 0 E is a stable node or critical node or degenerate node. If a = a , it follows from 0 0 det(J(E )) = 0, tr(J(E )) < 0that E is a saddle-node (see Theorem 7.1 in [37]or 0 0 0 Theorem 2.11.1 in [18]). The Jacobian matrix at E (S , I ) can be written as ∗ ∗ ∗ ⎛ ⎞ γ b(a − a ) − − a −(d + ) ⎜ ⎟ d + J(E ) = . (16) ⎝ ⎠ b(d +  + γ)(a − a ) d + And hence, det(J(E )) = b(d +  + γ)(a − a ), ∗ 0 (17) γ b(a − a ) tr(J(E )) =− − a . ∗ 0 d + Recall that a > a holds when E exists. We have det(J(E )) > 0, tr(J(E )) < 0. Thus, E 0 ∗ ∗ ∗ ∗ is asymptotically stable. 128 L. WANG ET AL. In the following, we discuss the stability of the equilibria E , i = e,1,2,3. The Jacobian matrix of the sub-system (5) at E is given by ⎛   ⎞ i p 1−p ab − 2a b − βI I p(d +  + γ) − (d + ) ⎜ i ⎟ M = c . (18) ⎝ ⎠ p 1−p βI I −p(d +  + γ) By Equation (6), we have a I 1−p 0 i I = a − a . (19) i p I (d + ) c Hence, i p 1−p tr(M(E )) = a − 2a − βI I − p(d +  + γ) i 0 c a [γ − (d + )] I γ a + p(d +  + γ)(d + ) 0 i = − . (20) d +  I d + 1/p If γ ≤ d + ,thentr(M(E )) < 0alwaysholds.If γ> d + ,from I < I = R I ,it i i n c follows that a[γ − (d + )] γ a + p(d +  + γ)(d + ) tr(M(E )) < − d +  d + =−[a + p(d +  + γ)] < 0. (21) Therefore, we always have tr(M(E )) < 0. Fromformula(19),one canhave i p 1−p det(M(E )) = 2a (d +  + γ) − βI (d + )I − pa(d +  + γ) i 0 c = a (d +  + γ) (2p − 1) − (p − 1)R . (22) 0 0 Letusfirstdeterminethesignofdet (M(E )) at the equilibrium E .If ≤ p ≤ 1, both terms i 1 in thebracket of formula(22) arepositiveandhencedet(M(E )) > 0. If 0 < p < ,from Case I in Section 2 we know I < I < I . Therefore, 1 n e det(M(E )) > a (d +  + γ) (2p − 1) − (p − 1)R = 0. (23) 1 0 0 If p > 1, from the discussion of Cases 1and 5inSection 2 it follows that I > I ,implying 1 e thatinequality(23) stillholds.Insummary,wehavedet(M(E )) > 0and tr(M(E )) < 0. 1 1 Therefore, E is asymptotically stable. The stability of the equilibria E , E and E can be 1 e 2 3 determined similarly. At E ,one canhavedet(M(E )) = 0, implying that E is a degenerate e e e node. At E , it follows from I < I that det(M(E )) < 0. Thus, E isasaddlepoint.At E , 2 2 e 2 2 3 we have det(M(E )) > 0(since I > I ). Hence, E is stable. 3 3 e 3 JOURNAL OF BIOLOGICAL DYNAMICS 129 Tosumup, oneobtainsthefollowing result. Theorem 3.1: For model (3), the local stability of the equilibria is stated as follows: (i) The origin O(0, 0) isasaddlepoint; (ii) If a < a (i.e. R < 1), then the disease-free equilibrium E is locally stable. If a = a , 0 0 0 0 then E is a saddle-node. If a > a (i.e. R > 1), then E isasaddlepoint; 0 0 0 0 (iii) E , E , E are locally asymptotically stable whenever they exist. E is a saddle point, and ∗ 1 3 2 E is a degenerate node. To illustrate the existence and stabilities of multiple equilibria for model (3), we uti- lize some parameter values estimated from the influenza A (H1N1). Set the recovered rate −1 −1 γ = 0.0196 year [12], the disease-induced death rate  = 2.7397 year [25], and xfi the 1 −1 −3 natural death rate d = year ,and b = 1.9237 × 10 .Thebasicreproductivenumber of influenza A was estimated as 1.5 −3.1 [30]. For ease of demonstration, we naively set p = 3, β = 0.0139 and a = 600, and hence the basic reproductive number R is equal to 3.0. We shall consider two examples. Both examples show the occurrence of bi-stability, in which solutions may converge to one of two stable equilibria, depending on initial conditions. 0 0 Example 3.1: Set I = 45. We can calculate I = 21 and a = 600. Therefore, I > I ,and c 2 c c c a = a . In this case (see Theorem 2.2(2)), model (3) admits two stable endemic equilibria E and E , that is, bi-stability occurs (see the line L in Figure 1). Some solutions to the ∗ 1 2 model are illustrated in Figure 2.Notethat a = a is equivalent to I = I . The endemic 2 c ∗ equilibrium E lies on the horizontal line I = I . ∗ c Example 3.2: Set I = 50. In this case, I > I = 21, a = 597 and a = 651. The con- c c 1 2 ditions of (iii) of (2) in Theorem 2.2 are satisfied. Hence, model (3) has three positive equilibria E , E and E .Figure 3 illustrates some numerical solutions to the model. The ∗ 2 3 I=I 100 150 200 250 300 350 400 Figure 2. Phase plot of I verses S showing that two stable endemic equilibria E , E coexist. Here, we fix ∗ 1 −3 1 0 (a, b, β, d, γ , ) = (600, 1.9237 × 10 , 0.0139, , 0.0196, 2.7397),and set I = 45 > I = 21. I 130 L. WANG ET AL. 60 2 E I=I 100 150 200 250 300 350 Figure 3. Phase plot of I verses S showing that bi-stability occurs for I > I and a < a < a .Here, E c 1 2 ∗ and E are locally stable while E is a saddle point. I = 50 > I = 21 and other parameters take the 3 2 c same values as in Figure 2. stable manifolds of the saddle E split the phase plane into two regions. In the lower region, solutions approach to E while in the upper region solutions approach to E (see Figure 3). ∗ 3 4. Global stability analysis In this section, we study the global stability of model (3). It can be shown that the state variables of model (3) remain non-negative for non-negative initial conditions. Consider the biologically feasible region (ab + d + ) = (S, I) ∈ R : N = S + I ≤ K = + 1 . 4b (d + γ) Choosing the straight line S+I−Ab =0,similartotheproofofCorollaryin[36], we have the following two results. Lemma 4.1: Theclosedset is a positively invariant set for model (3). Theorem 4.2: E is globally asymptotically stable if 0 < a ≤ a and unstable if a > a . 0 0 0 Tothebestofour knowledge,byprecludingtheexistence ofalimitcycleweareableto proveglobalstabilityoftheuniqueendemicequilibriumofmodel (3). Theorem 4.3: There exist no limit cycles for model (3). Proof: Thefollowing twostepsareconsideredtoachieve ourconclusion. Step 1. We shall provethattherearenolimitcyclesintheregionbelowtheline I = I in the feasible region and in the region above the line I = I . Denote these two regions by and . 1 2 I JOURNAL OF BIOLOGICAL DYNAMICS 131 −1 Take the Dulac function D(S, I) = S .Inthe case of0 < I ≤ I , by the transformation x = S, y = ln I, (24) one can transfer sub-system (4) into dx = bx(a − x) − (βx − γ) e , dt (25) dy = βx − (d +  + γ). dt −1 and D(x, y) = x .Let N , N be the right-hand side functions of (25). Then 1 2 ∂(DN ) ∂(DN ) γ e 1 2 + =− − b < 0. (26) ∂x ∂y x Therefore, there are no limit cycles in the region below the line I = I . In the case of I > I , we can extend subsystem (5) to the case of I ≥ I because of the c c continuity of the transmission function β(I).Thatis, subsystem(5) canberewritten into dS I = bS(a − S) − β IS + γ I, dt I I ≥ I . (27) dI I = β IS − (d +  + γ)I, dt I Consider two cases p =1and p = 1. If p = 1, one can set x = S and y = I.With N and N 1 2 being the right sides of system (27), direct calculations show that ∂(DN ) ∂(DN ) γ y d +  + γ 1 2 + =− − − b < 0. (28) ∂x ∂y x x If p = 1, using the transformation 1−p x = S, y = ln I , (29) one obtains dx y y/(1−p) = bx(a − x) − βI x e + γ e , dt (30) dy −py/(1−p) = (1 − p)[βI x e − (d +  + γ)]. dt Therefore, ∂(DN ) ∂(DN ) γ 1 2 p y/(1−p) −py/(1−p) + =− e − pβI e − b < 0. (31) ∂x ∂y x Consequently, for all p > 0, one can have ∂(DN ) ∂(DN ) 1 2 + < 0. (32) ∂x ∂y Hence, there is no limit cycle in , the region above the line I = I .Weshouldpointout 2 c that inequalities (26) and (32) hold for y ∈ (−∞, +∞). 132 L. WANG ET AL. ΓΓ I=I C C 2 1 30 40 50 60 70 80 90 100 Figure 4. A limit cycle intersects with the line I = I at C and C . c 1 2 Step 2. We are now ready to show that model (3) has no limit cycle crossing the line I = I .Theidea is similartothatin[27, 28, 34]. Assume that  isalimit cycleacrosstheline I = I .Let  bethepartofthecycle c 1 below I = I of the cycle ,and  be the part above I = I , with the direction designated c 2 c in Figure 4.Letboth  and  include two intersection points C , C of  with the line 1 2 1 2 I = I .Theregion enclosed by  and the segment C C is denoted by G ,and theregion c 1 1 2 1 enclosed by  and the segment C C is denoted by G . 2 1 2 2 −−→ −−→ Let us choose two the directed-paths L : C C and L : C C ,asshownin Figure 4.It 1 1 2 2 2 1 can be seen that D(N dI − N dS) = + D(N dI − N dS) 1 2 1 2 1 2 1 2 = + + + − − D(N dI − N dS) 1 2 L L L L 1 2 1 2 1 2 = + − − D(N dI − N dS). (33) 1 2 ∪L  ∪L L L 1 1 2 2 1 2 Meanwhile, it is easy to obtain that + D(N dI − N dS) = 0. (34) 1 2 L L 1 2 Hence, from Green’s Theorem it follows that D(N dI − N dS) = + D(N dI − N dS) 1 2 1 2 ∪  ∪L  ∪L 1 2 1 1 2 2 ∂(DN ) ∂(DN ) 1 2 = + + dS dI. (35) ∂S ∂I G G 1 2 I JOURNAL OF BIOLOGICAL DYNAMICS 133 Obviously, one can have D(N dI − N dS) = 0. (36) 1 2 1 2 From Step 1, however, we know that there is no limit cycle in the regions Pi or Pi ,and 1 2 ∂(DN ) ∂(DN ) ∂(DN ) ∂(DN ) 1 2 1 2 + dS dI < 0, + dS dI < 0. ∂S ∂I ∂S ∂I G G 1 2 (37) A contradiction to Equation (35). Therefore, there are no limit cycles crossing the line I = I . To sum up, model (3) has no limit cycles. From Theorems 2.2 and 4.3, we immediately have Corollary 4.4: If model (3) admits a unique endemic equilibrium (either of E or E ), then ∗ 1 it is globally asymptotically stable. We can illustrate the global stability by numerical simulations. Set I = 40 and the remaining parameters take the same values as in Figure 2.Then a > a = 549. From Remark 1 and Corollary 4.4, the unique endemic equilibrium E (see the region Q in 1 1 Figure 1) of model (3) is globally asymptotically stable. Figure 5 shows that the number I of infections is stabilized at the level I . In the figure, it is also shown that the number I of infected cases is stabilized at decreased levels as either the intensity p of the media eeff ct increases or the critical number I decreases. That is, stronger media eect ff s and/or lower critical numbers lead to a decreased number of infections at the endemic equilibrium E . In fact, such decreasing eeff cts are also true for all endemic equilibria except E . I =40 I =35 p=3 p=6 45 p=9 I =30 0 0.2 0.4 0.6 0.8 1 Time (year) Figure 5. The number of infected cases is stabilized at decreased levels as either p increases or I decreases. I 134 L. WANG ET AL. 5. Discussion During a disease spread, only when the number of infected cases and/or the severity of infection are high enough to draw the attention of media and public health organiza- tions, alerts are issued and all related information is brought to the public through media coverage. The information gradually changes public behaviour, which reduces the chance of potential contact infection and eventually helps to curb the disease spread. Fast and dramatic changes in public behaviour can also occur subject to some intensive control mea- sures such as closing school and distancing certain groups of persons. Previous studies have introduced a critical number I to be the level for media and health organizations to take action. Functions with jump-discontinuity at I were used to describe the changing trans- missionrateduetothemedia eeff ct[ 28, 34, 35]. With the rate functions, media eect ff s on disease outbreaks were studied through compartmental epidemic models. It turns out that proper use of media coverage can curb disease outbreaks [28, 34]. In this paper, following the idea of the critical number I , we proposed the non-smooth function (2) to describe the transmission rate which is continuous at I . With this description, change in public behaviour is continuous. Using a susceptible–infected–susceptible model with a logistic growth in the susceptible class, we studied the media eeff ct on the transmission dynamics of an infectious disease in a given region. Ourmodel analysis showsthatwithoutthemediaeeff ctorwithrelativelyweakeeff ct (0 < p ≤ 1), the endemic equilibrium E or E (depending on the chosen I ) is globally ∗ 1 c asymptotically stable. With relatively strong media eect ff p >1,themodelmayhaveup to three endemic equilibria for the chosen critical number I in the threshold interval 1 2 = [I , I ]. Otherwise, the model admits a unique endemic equilibrium (which is glob- c c ally asymptotically stable, see Figure 1). In the former case, solutions to the model can converge to either one of two stable endemic equilibria (see Figures 2 and 3), depending on initial conditions. That is, bi-stability can occur. To avoid such uncertainty in practice, it is necessary to choose the critical number I below the value I . It is worthwhile to point 1 2 out that the critical values I , I depend only on the basic reproduction number R and c c the intensity p of the media eect ff and hence they are prescribed by the focal disease and the population in the given region. Therefore, it is possible for policymaker to roughly esti- mate the threshold interval for re-emerged diseases and then choose a reasonable critical number I to initiate media coverage. From the point of view of disease control, early media alerts (i.e. setting a small critical number for I )andstrongmediaeeff cts(i.e. p > 1) are definitely preferable. Our anal- ysis shows that if I < I ,equivalently a > a ,theendemicequilibrium E is globally c ∗ 2 1 asymptotically stable. However, if I > I , model solutions approach the unique equilib- rium E , causing that media coverage loses its impact on disease transmission. This should be avoided by health policymakers. Early media alerts and strong eect ff can decrease the numbers of infected cases at endemic equilibria I , i = e,1,2,3. For example, as the critical number I decreases, or the i c intensity p of the media eect ff increases, the number of infected cases can be stabilized at E with decreased numbers of cases (see Figure 5). Therefore, properly choosing the critical number and strengthening the media effect can reduce disease prevalence. The analysis of an SEI model also implies that media coverage can reduce the number of infected cases at endemic equilibria [9]. JOURNAL OF BIOLOGICAL DYNAMICS 135 The existence of multiple endemic equilibria was obtained in some previous studies −mI (see e.g. [9, 20]). Using the transmission rate β e with media parameter m,Cui et al. [9] found that the model may exhibit periodic oscillations for sucffi iently small media effects (small m) and may have multiple endemic equilibria for strong media eect ff s (large m). Unfortunately, the stabilities of the equilibria are not available and hence the solution behaviour is unknown for strong media eect ff s. Media/psychological eect ff s may be also 2 2 included in the incidence rate in a nonlinear and saturation way such as kI S/(1 + αI ) and simple compartmental models can exhibit complex dynamics. For instance, saddle- node bifurcation, Hopf bifurcation and homocyclic bifurcation can occur in a simple SIR epidemic model with the rate [20]. Therefore, the impact of media coverage on a dis- ease transmission dynamics could be complicated and simplified understandings may even make the disease worse (e.g. [24]). It needs to be further studied from distinct aspects [8]. Inordertoavoidthecomplexityofmathematicalanalysisweassumedthatthe logistic growth in the susceptible class depends only on the number of the susceptible, instead of thetotalpopulation size.Thisisasignicfi antsimplicfi ation.Asanendof thepaper,we would like to point out that this simplicfi ation is unlikely to change our major qualitative results about the media eeff ct, such as the existence of the threshold interval  and bi- stability. Acknowledgments The authors thank the editor and the anonymous reviewers for their constructive comments that help to improve the early version of this paper. Disclosure statement No potential conflict of interest was reported by the authors. Funding The work was supported by National Natural Science Foundation of China [No. 11261017, 11371161]. References [1] R.M. Anderson, H.C. Jackson, R.M. May, and A.D.M. Smith, Population dynamics of fox rabies in Europe, Nature 289 (1981), pp. 765–771. [2] F. Brauer, Models for the spread of universally fatal diseases,J.Math.Biol.28(1990), pp. 451–462. [3] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiol- ogy,Springer,NewYork, 2001. [4] S.BroderandR.C.Gallo, A pathogenic retrovirus (HTLV-III) linked to AIDS,NewEngl.J.Med. 311 (1984), pp. 1292–1297. [5] S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size,J.Math. Biol.28(1990), pp. 257–270. [6] CDC, H1N1 Flu, Center for Disease Control and Prevention Website. Available at http://www.cdc.gov/h1n1flu/. [7] N.S. Chong and R.J. Smith, Modeling avian inuenza fl using Filippov systems to determine culling of infected birds and quarantine, Nonlinear Anal. Real World Appl. 24 (2015), pp. 196–218. [8] S.Collinson andJ.M.Heeff rnan, Modelling the effects of media during an inuenz fl a epidemic , BMC Public Health 14 (2014), pp. 1–10. 136 L. WANG ET AL. [9] J.Cui,Y.Sun,and H. Zhu, The impact of media on the control of infectious diseases,J.Dynam. Differential Equations 20 ( 2008), pp. 31–53. [10] J-A. Cui, X. Tao, and H. Zhu, An SIS infection model incorporating media coverage,Rocky Mountain J. Math. 38 (2008), pp. 1323–1334. [11] H.W. Hethcote, Athousandandoneepidemicmodels,in Frontiers in Mathematical Biology,S.A. Levin, ed., Lecture Notes in Biomathematics, Vol. 100, Springer, Berlin, 1994, pp. 504–515. [12] S. Leekha, N.L. Zitterkopf, M.J. Espy, T.F. Smith,R.L. Thompson, and P. Sampathkumar, Dura- tion of influenza: A virus shedding in hospitalized patients and implications for infection control , Infect. Control Hosp. Epidemiol. 28 (2007), pp. 1071–1076. [13] Y. Liu and J-A. Cui, The impact of media coverage on the dynamics of infectious disease,Int.J. Biomath.1(2008), pp. 65–74. [14] R. Liu, J. Wu, and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases,Comput. Math.Methods Med.8(2007), pp. 153–164. [15] Y. 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Difference Equ. 2015 ( 2015), pp. 1–10. doi:10.1186/s13662-015-0477-8. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Biological Dynamics Taylor & Francis

Media alert in an SIS epidemic model with logistic growth

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JOURNAL OF BIOLOGICAL DYNAMICS, 2017 VOL. 11, NO. S1, 120–137 http://dx.doi.org/10.1080/17513758.2016.1181212 a b c c,d a Lianwen Wang , Da Zhou ,ZhijunLiu ,DashunXu and Xinan Zhang School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China; b c School of Mathematical Sciences, Xiamen University, Xiamen, People’s Republic of China; Department of Mathematics, Hubei University for Nationalities, Hubei, People’s Republic of China; Department of Mathematics, Southern Illinois University, Carbondale, IL, USA ABSTRACT ARTICLE HISTORY Received 10 August 2015 In general, media coverage would not be implemented unless the Accepted 18 April 2016 number of infected cases reaches some critical number. To reflect this feature, we incorporate the media effect and a critical number KEYWORDS of infected cases into the disease transmission rate and consider SIS epidemic model; media an susceptible-infected-susceptible epidemic model with logistic alert; logistic growth; growth. Our model analysis shows that early media alert and strong multiple endemic equilibria; media effects are preferable to decrease the numbers of infected bi-stability cases at endemic equilibria. Furthermore, we noticed that the model AMS SUBJECT may have up to three endemic equilibria and bi-stability can occur CLASSIFICATION in a threshold interval for the critical number. Note that the interval 34K18; 34K20; 92D30 depends on parameters for the focal disease and the media effect. It is possible to roughly estimate the interval for re-emerging diseases in a given region. Therefore, the result could be useful to health pol- icymakers. Global stability is also obtained when the model admits a unique endemic equilibrium. 1. Introduction Media has been utilized as a disease control measure, especially for epidemics associated with emerging and re-emerging infectious diseases [19] such as HIV/AIDS, SARS, H1N1, Ebola virus disease (EVD), Middle East Respiratory Syndrome. During the outbreak of the influenza A (H1N1) in 2009, mass media was extensively used by the Centers for Disease Control and Prevention of United States and WHO to keep the public aware of information relatedtothe pandemic [6]. It is believed that media use contributed to the control of the pandemic. WHO also indicated that media played an important role in controlling the spread of H7N9 in China in 2013 [31].Mediadoesnot onlyalertthegeneralpubliconthe hazard from the infectious diseases but also informs the public of the requisite preventive measures like wearing protective masks [25], vaccination, voluntary quarantine, avoidance of congregated places, etc. Therefore, the extensive use of media may bring in changes in public behaviour and reduce the frequency and probability of contacts with infected individuals so that the severity of a disease outbreak would be diminished [4, 9, 10, 13, 14, 21, 24]. CONTACT Zhijun Liu zhijun_liu47@hotmail.com; Lianwen Wang wanglianwen5.17@163.com;DaZhou zhouda@xmu.edu.cn; Dashun Xu dashunxu@siu.edu; Xinan Zhang zhangxinan@hotmail.com © 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. JOURNAL OF BIOLOGICAL DYNAMICS 121 In ordertostudytheimpactofmedia-likecontrolmeasures ondiseasetransmis- sion dynamics, several types of media function forms have been proposed to describe reduced disease transmission rates due to media use and compartmental models with these rates have been analysed (e.g. [9, 10, 13–17, 23, 24]). The deduction in the transmis- −mI sion rate was described by the form of β(1 − e ) with the parameter m > 0 reflecting how strongly media coverage can aeff ct contact infection [ 9]. With the rate, the analysis of a susceptible–exposed–infected model (SEI) shows that the model may exhibit peri- odicoscillationsforweakmediaeeff cts whileitmay havethreeendemic equilibriafor strong media effects [ 9]. The form of β − β I/(ν + I) was also used as the transmission rate with the deduction β I/(ν + I) due to media use [10, 13, 24]. A threshold dynamics was obtained for an SIS epidemic model. It is also shown that media coverage can lower infection and delay the arrival of the infection peak [10, 13]. However, the study of a sus- ceptible–vaccinated–infected–recovered epidemic model with a vaccinated class indicates that media eeff cts could be complicated and simplified understandings may even make the disease worse due to possible public panic [24]. The third function type with psycholog- ical/media eect ff s is of the form β/(1 + αI ), identified by Collinson and Heeff rnan (see [8, 33] and the references therein). Using a simple susceptible-exposed-infected-recovered model, Collinson and Heeff rnan [ 8] found that important measurements of an epidemic outbreak (such as peak magnitude of infection, peak time of infection peak and end of the outbreak) depend on the chosen media function. Their sensitivity analysis also showed such dependence for the sensitivities of model parameters. This makes it dicffi ult to iden- tify eecti ff ve disease control strategy and calls for more study on the eect ff s of mass media on disease transmission dynamics. Theoretical studies usually assume that media coverage aeff cts disease transmission dur- ing the whole time period of the disease spread (e.g. [9, 10, 14, 24]). In the reality, media coverage generally does not occur in the beginning stage of a disease spread. For instance, the early suspected cases of EVD died in December 2013 while the first notification by WHO on the outbreak [32] was not issued until 21 March 2014. In general, media deliv- ers alerts and timely reports infected cases only when certain number of infected cases is reached [28, 34, 38]. To include such feature of media/psychological effect, Xiao et al. [34] introduced a critical number of infected cases I and proposed a piecewise and discon- tinuous control function [7, 26] for disease control strategy (sliding mode control). Wang and Xiao [28] further constructed a Filippov SIR epidemic model to describe media eeff cts using the following transmission rate β, I < I , β(I) = (1) β exp (−αI), I > I . Their analysis shows that the model system can stabilize at either one of the equilibria fortheresulted subsystems or thenewendemicstateinducedbytheon-offmedia eeff ct, depending on the critical level I . They also demonstrated that proper combinations of critical levels and control intensities can lead to the desired case number. The transmission rate (1) was then generalized in a time-dependent way to study an influenza outbreak in Shannxi, China [35]. It was demonstrated that compartmental models can exhibit distinct dynamics, depend- ing on the chosen incidence rate (e.g. [20, 33]). To explore the media eect ff on disease 122 L. WANG ET AL. transmission dynamics we here propose a new transmission rate. Following the idea of the critical number of infected cases I ,weconsiderthefollowingnon-smoothbut continuous transmission rate: ⎨ β,0 ≤ I ≤ I , β(I) = (2) β , I > I , where p ≥ 0 represents the intensity of the media eeff ct on contact infection. If p = 0, β(I) is equal to the background transmission rate β, implying that media coverage does not occur. With this rate, we shall consider an SIS endemic model. Classical compartmental models with media eeff cts assume either a constant size of the total population or constant recruitment rate for the susceptible class. The assumption of varying total populations may be more reasonable for a relatively long-lasting disease or for a disease with high mortality rates. In fact, varying total populations were discussed before (e.g. [1, 3, 5, 9, 11, 22, 29, 36]). Here, we assume that the population of a commu- nity follows the logistic growth. For the sake of mathematical simplicity, we assume that newborns directly enter into the susceptible class and infected persons do not contribute to births and deaths in the susceptible class. Following works in [2, 9, 28], our SIS model reads dS S = rS 1 − − β(I)IS + γ I, dt a (3) dI = β(I)IS − (d +  + γ)I, dt where r is the intrinsic growth rate of the susceptible population, a denotes the carrying capacity of the community in the absence of infection, d is natural death rate, γ represents the recovered rate, and  is the disease-induced death rate. The analysis of model (3) with thetransmissionrate(2) showsthatthereexistsathreshold interval  for the critical num- ber I in which the model may be stabilized at one of two stable equilibria with different levels of infected cases. This implies that the policymaker may have to choose the critical number I according to the focal disease in order to minimize infected cases and also avoid unnecessary public panic. The global stability was also obtained for I not in the threshold interval. In the following, the analysis of existence of equilibria is presented in Section 2 while the local and global stabilities are given in Sections 3 and 4, respectively. A discussion section comes to the end of the work. 2. Existence of equilibria For our convenience, denote b =: r/a. Model (3) with the transmission rate (2) can be decomposed into two sub-systems dS = bS(a − S) − βIS + γ I, dt 0 ≤ I ≤ I,(4) dI = βIS − (d +  + γ)I, dt JOURNAL OF BIOLOGICAL DYNAMICS 123 and dS I = bS(a − S) − β IS + γ I, dt I I > I.(5) dI I = β IS − (d +  + γ)I, dt I The origin O(0, 0) and the disease-free equilibrium E (a,0) always exist. The basic repro- ductive number can be easily calculated, given by βa R = , d +  + γ from which one can see that media coverage does not change the basic reproduction number (e.g. [9, 10, 13, 15–17, 23, 24]). To obtain the existence of endemic equilibria, denote a d +  a b a  , a  a + I , I  (R − 1), 0 2 0 c ∗ 0 R a b d + 0 0 and consider two cases: 0 ≤ I ≤ I and I > I ,separately. c c In the case of 0 ≤ I ≤ I , the sub-system (4) has a unique positive equilibrium E (S , I ) c ∗ ∗ ∗ if and only if 0 < I ≤ I ,thatis, a < a ≤ a ,where S = a . ∗ c 0 2 ∗ 0 In the case of I > I ,the I component of a positive equilibrium for the sub-system (5) satisefi s the following equation a a 0 0 2p−1 p−1 f (I) = bI − abI + (d + ) = 0. (6) p p I I c c p p 2 2p−1 p−1 From (a /I ) bI − (a /I )abI =−(d + ) < 0, positive solutions to Equation (6) 0 0 c c satisfy 1/p I < R I  I.(7) c n That is, the positive solutions of Equation (6) must be in the interval (I , I ).Also, f (I ) = c n d + > 0. Therefore, we must have a > a due to I > I . Next, we discuss the existence 0 n c of positive solutions to Equation (6) in (I , I ) in two cases, 0 < p ≤ 1and p > 1. c n Case I. 0 < p ≤ 1. In this case, f (I) is strictly increasing. In fact, the derivative of f (I) is given by a a 0 0 p−2 p f (I) = bI (2p − 1)I − a(p − 1) ,for I ∈ (I , I ),(8) c n p p I I c c and we can show that f (I)> 0for 0 < p ≤ 1. If ≤ p ≤ 1, one clearly derives f (I)> 0. If 0 < p < ,wecan calculatethezeropointof f (I) as 1/p p − 1 I  R I > 0. (9) e 0 c 2p − 1 124 L. WANG ET AL. which, obviously, I is the unique maximum point of f (I).Since p p p I − I = R I , (10) e n 0 c 1 − 2p we have I < I for 0 < p < , and hence f (I)> 0on (I , I ).Tosum up,wealwayshave n e c n f (I)> 0if0 < p ≤ 1. Meanwhile, note that a b(a − a) 0 2 f (I ) = (11) may be positive, negative or zero. If f (I ) ≥ 0, which is equivalent to a ≤ a ,then c 2 Equation (6) has no positive real root in (I , I ).If f (I )< 0, which is equivalent to a > a , c n c 2 then Equation (6) admits a unique positive root, denoted by I ,satisfying I < I < I .That 1 c 1 n is, in the case of 0 < p ≤ 1, the sub-system (5) has a unique positive equilibrium, denoted by E (S , I ),ifand onlyif a > a ,where S = a (I /I ) and I ∈ (I , I ). 1 1 1 2 1 0 1 c 1 c n The following lemma is needed to discuss the case of p > 1. Lemma 2.1: Let p > 1, then a < a always holds, where 1 2 1/(2p−1) 2p−1 p (2p − 1) (d + ) I a  . p p−1 p p (p − 1) a b Proof: Consider the concave function H(x) = ln x, x > 0. Assign 2p − 1 2p − 1 d + x = a , x = I , 1 0 2 c p − 1 p a b p − 1 p λ = , λ = . 1 2 2p − 1 2p − 1 Note that λ + λ = 1, andH(x) is a strictly concave function. This implies thatH(λ x + 1 2 1 1 λ x )>λ H(x ) + λ H(x ), and we have 2 2 1 1 2 2 d +  p − 1 2p − 1 p 2p − 1 d + ln a + I = ln a + I 0 c 0 c a b 2p − 1 p − 1 2p − 1 p a b 0 0 p − 1 2p − 1 p 2p − 1 d + > ln a + ln I 0 c 2p − 1 p − 1 2p − 1 p a b (p−1)/(2p−1) p/(2p−1) 2p − 1 2p − 1 d + = ln a · I 0 c p − 1 p a b 1/(2p−1) 2p−1 p (2p − 1) (d + ) I = ln . p p−1 p p (p − 1) a b The monotonicity of H(x) leads to a < a .Theproofiscompleted. 1 2 Case II. p > 1. By Lemma 2.1, a < a holds. It follow from the formula (10) that I < I . 1 2 e n Note that the derivative f (I) givenbyEquation(8)is positive. I is the unique minimum e JOURNAL OF BIOLOGICAL DYNAMICS 125 + − point of f (I).Clearly, f (0 ) = f (I ) = d + . Next, we want to compare the sizes of I and I and determine the signs of f (I ) and f (I ). The following six cases are considered. Let us e c e denote p a b 0 0 I  . p − 1 d + 0 0 Case 1. I ≥ I and a ≥ a . It follows from I ≥ I that c 2 c c c d +  d +  p a b 2p − 1 0 0 a ≥ a + I ≥ a + · = a . 2 0 0 0 a b a b p − 1 d +  p − 1 0 0 Since a ≥ a > a ,wehave f (I ) ≤ 0, and a ≥ ((2p − 1)/(p − 1))a . Furthermore, from 2 1 c 0 Equation (9) one can get I ≤ I .Thus f (I ) ≤ f (I ) ≤ 0. Direct calculation shows that a > c e e c a ⇔ f (I )< 0holds.If f (I ) = 0, we have I < I .Thus, Equation(6)hasauniquepos- 1 e c c e itive root I ,satisfying I < I < I .If f (I )< 0, the conclusion is still true. Accordingly, 1 e 1 n c the sub-system (5) has a unique positive equilibrium E (S , I ). 1 1 1 0 0 Case 2. I ≥ I and a < a < a . It follows from I ≥ I that c 1 2 c c c 1/(2p−1) 2p−1 p (2p − 1) (d + ) 0 p a ≥ (I ) p p−1 p p (p − 1) a b 1/(2p−1) 2p−1 p (2p − 1) (d + ) p a b 2p − 1 = = a , p p−1 p p (p − 1) a b p − 1 d +  p − 1 that is, a >((2p − 1)/(p − 1))a . Thus, we have I < I from Equation (9) and f (I )> 0 c e c 0because of a < a . Moreover, one can show that a > a ⇔ f (I )< 0. Therefore, when 2 1 e a < a < a , Equation (6) has two different positive roots in (I , I ),denotedby I , I , 1 2 c n 2 3 where I ∈ (I , I ), I ∈ (I , I ). That is, the sub-system (5) admits two different positive 2 c e 3 e n equilibria E (S , I ),where S = a (I /I ) , i = 2,3. i i i i 0 i c Case 3. I ≥ I and a = a .Similar to theargumentin Case 2, it follows that c 1 ((2p − 1)/(p − 1))a ≤ a < a , and hence we have I ≤ I and f (I )> 0. Note that 0 2 c e c the equivalent relationship a = a ⇔ f (I ) = 0. We must have I < I .Thissuggests 1 e c e that Equation (6) only has one positive solution I ∈ (I , I ). Accordingly, for the sub- e c n system (5), there exists exactly one positive equilibrium E (S , I ),where S = a (I /I ) . e e e e 0 e c Case 4. I ≥ I and a < a < a . Recall that a < a ⇔ f (I )> 0and I is the unique c 0 1 1 e e minimum point of f (I) in (I , I ). One immediately deduces that Equation (6) has no pos- c n itive root in (I , I ) no matter I ≤ I or I > I .Hence,thesub-system (5)has no positive c n c e c e equilibrium. Case 5. I < I and a > a .From a > a > a , one deduces that f (I )< 0and f (I )< 0. c 2 2 1 c e Hence, Equation (6) only has one positive solution I ∈ (I , I ) no matter I ≤ I or I > I , 1 c n c e c e where I < I < I .Thatis,thesub-system(5)hasonlyonepositiveequilibrium E (S , I ). e 1 n 1 1 1 Case 6. I < I and a < a ≤ a .Clearly,one canhave a <((2p − 1)/(p − 1))a from c 0 2 2 0 I < I and a ≤ a <((2p − 1)/(p − 1))a .Hence, I < I .Noticethat f (I) is an increas- c 2 0 e c ing function on the interval (I , I ),and f (I ) ≥ 0. Equation (6) has no positive root in the c n c interval (I , I ). Namely, the sub-system (5) has no positive equilibrium. c n 126 L. WANG ET AL. Now, one summarizes the existence of the equilibria of model (3) as follows: Theorem 2.2: Model (3) always admits an equilibrium O(0, 0) andadisease-freeequilib- rium E (a,0).Ifa ≤ a (i.e.R ≤ 1), then model (3) has no endemic equilibrium. If a > a 0 0 0 0 (i.e. R > 1), we have the following conclusions. (1) Assume that 0 < p ≤ 1. (i) If a < a ≤ a , then E isauniqueendemicequilibrium; 0 2 ∗ (ii) If a > a , then E isauniqueendemicequilibrium. 2 1 (2) Assume that p >1and I ≥ I . (i) If a < a < a , E isauniqueendemicequilibrium; 0 1 ∗ (ii) If a = a , there are two endemic equilibrium, E and E ; 1 ∗ e (iii) If a < a < a , there exist three endemic equilibria, E , E and E ; 1 2 ∗ 2 3 (iv) If a = a , both E and E exist; 2 ∗ 1 (v) If a > a , E isauniqueendemicequilibrium. 2 1 (3) Assume that p >1and I < I . (i) If a < a ≤ a , E isauniqueendemicequilibrium; 0 2 ∗ (ii) If a > a , E isauniqueendemicequilibrium. 2 1 Remark 1: From (ii) of (1), (v) of (2) and (iii) of (3) in Theorem 2.2 one can deduce that E isauniqueendemic equilibriumif a > a . 1 2 By Theorem 2.2, if 0 < p ≤ 1, model (3) only has one endemic equilibrium, either E (in the case of a < a ≤ a ), or E (in the case of a > a ). The existence of positive equilibria 0 2 1 2 for the model in the case of p > 1 is illustrated in Figure 1. Here, we define the follow- ing different curves and regions for parameters a and I as follows: L ={(a, I ) | a = a }, c 0 c 0 0 0 L ={(a, I ) | a = a and I ≥ I }, L ={(a, I ) | a = a and I ≥ I }, Q ={(a, I ) | 1 c 1 c 2 c 2 c 1 c c c Figure 1. The existence of the endemic equilibria of model (3) in the case of p > 1. There are equilibria E and E on the curve L , E and E on L , three equilibria E , E and E in the region Q , a unique ∗ e 1 ∗ 1 2 ∗ 2 3 2 equilibrium E in the region Q , a unique equilibrium E in the region Q , and no endemic equilibrium 1 1 ∗ 3 in Q . See the content for the definitions of these regions. 4 JOURNAL OF BIOLOGICAL DYNAMICS 127 a > a }, Q ={(a, I ) | a < a < a and I ≥ I }, Q = C (Q ∪ L ∪ L ) (where U = 2 2 c 1 2 c 3 U 2 1 2 {(a, I ) | a < a ≤ a }), and Q ={(a, I ) | 0 < a ≤ a }. c 0 2 4 c 1 0 1 Note that the condition a ≤ a ≤ a with I > I in Theorem 2.2 is equivalent to I ≤ 1 2 c c c I ≤ I ,where 1/p p p−1 2p−2 p (p − 1) a b 1 0 2 I = max{I , I }, I = . c c c 2p−1 (2p − 1) R d + In the case of R > 1and p > 1, multiple endemic equilibria exist if the critical number I 0 c is in the threshold interval 1 2 := [I , I ]. (12) c c The existence region in Figure 1 is given by the union of L , L and Q . 1 2 2 3. Local stability of equilibria This section focuses on the local stability of equilibria. Corresponding to the equilibria O(0, 0), E and E , the Jacobian matrix of the sub-system (4) reads 0 ∗ ab − 2bS − βI −βS + γ J = . (13) βI βS − (d +  + γ) At O(0, 0) the determinant det(J(O)) =−ab(d +  + γ) < 0. Thus O(0, 0) is a saddle point. At E (a,0) it can be shown that det(J(E )) =−βab(a − a ), 0 0 (14) tr(J(E )) = β(a − a ) − ab. 0 0 If a > a ,thendet(J(E )) < 0, which means that E isasaddlepoint.If a < a ,then 0 0 0 0 det(J(E )) > 0and tr(J(E )) < 0. Hence E is locally asymptotically stable. Since 0 0 0 2 2 [−tr(J(E ))] − 4det(J(E )) = [β(a − a ) + ab] ≥ 0, (15) 0 0 0 E is a stable node or critical node or degenerate node. If a = a , it follows from 0 0 det(J(E )) = 0, tr(J(E )) < 0that E is a saddle-node (see Theorem 7.1 in [37]or 0 0 0 Theorem 2.11.1 in [18]). The Jacobian matrix at E (S , I ) can be written as ∗ ∗ ∗ ⎛ ⎞ γ b(a − a ) − − a −(d + ) ⎜ ⎟ d + J(E ) = . (16) ⎝ ⎠ b(d +  + γ)(a − a ) d + And hence, det(J(E )) = b(d +  + γ)(a − a ), ∗ 0 (17) γ b(a − a ) tr(J(E )) =− − a . ∗ 0 d + Recall that a > a holds when E exists. We have det(J(E )) > 0, tr(J(E )) < 0. Thus, E 0 ∗ ∗ ∗ ∗ is asymptotically stable. 128 L. WANG ET AL. In the following, we discuss the stability of the equilibria E , i = e,1,2,3. The Jacobian matrix of the sub-system (5) at E is given by ⎛   ⎞ i p 1−p ab − 2a b − βI I p(d +  + γ) − (d + ) ⎜ i ⎟ M = c . (18) ⎝ ⎠ p 1−p βI I −p(d +  + γ) By Equation (6), we have a I 1−p 0 i I = a − a . (19) i p I (d + ) c Hence, i p 1−p tr(M(E )) = a − 2a − βI I − p(d +  + γ) i 0 c a [γ − (d + )] I γ a + p(d +  + γ)(d + ) 0 i = − . (20) d +  I d + 1/p If γ ≤ d + ,thentr(M(E )) < 0alwaysholds.If γ> d + ,from I < I = R I ,it i i n c follows that a[γ − (d + )] γ a + p(d +  + γ)(d + ) tr(M(E )) < − d +  d + =−[a + p(d +  + γ)] < 0. (21) Therefore, we always have tr(M(E )) < 0. Fromformula(19),one canhave i p 1−p det(M(E )) = 2a (d +  + γ) − βI (d + )I − pa(d +  + γ) i 0 c = a (d +  + γ) (2p − 1) − (p − 1)R . (22) 0 0 Letusfirstdeterminethesignofdet (M(E )) at the equilibrium E .If ≤ p ≤ 1, both terms i 1 in thebracket of formula(22) arepositiveandhencedet(M(E )) > 0. If 0 < p < ,from Case I in Section 2 we know I < I < I . Therefore, 1 n e det(M(E )) > a (d +  + γ) (2p − 1) − (p − 1)R = 0. (23) 1 0 0 If p > 1, from the discussion of Cases 1and 5inSection 2 it follows that I > I ,implying 1 e thatinequality(23) stillholds.Insummary,wehavedet(M(E )) > 0and tr(M(E )) < 0. 1 1 Therefore, E is asymptotically stable. The stability of the equilibria E , E and E can be 1 e 2 3 determined similarly. At E ,one canhavedet(M(E )) = 0, implying that E is a degenerate e e e node. At E , it follows from I < I that det(M(E )) < 0. Thus, E isasaddlepoint.At E , 2 2 e 2 2 3 we have det(M(E )) > 0(since I > I ). Hence, E is stable. 3 3 e 3 JOURNAL OF BIOLOGICAL DYNAMICS 129 Tosumup, oneobtainsthefollowing result. Theorem 3.1: For model (3), the local stability of the equilibria is stated as follows: (i) The origin O(0, 0) isasaddlepoint; (ii) If a < a (i.e. R < 1), then the disease-free equilibrium E is locally stable. If a = a , 0 0 0 0 then E is a saddle-node. If a > a (i.e. R > 1), then E isasaddlepoint; 0 0 0 0 (iii) E , E , E are locally asymptotically stable whenever they exist. E is a saddle point, and ∗ 1 3 2 E is a degenerate node. To illustrate the existence and stabilities of multiple equilibria for model (3), we uti- lize some parameter values estimated from the influenza A (H1N1). Set the recovered rate −1 −1 γ = 0.0196 year [12], the disease-induced death rate  = 2.7397 year [25], and xfi the 1 −1 −3 natural death rate d = year ,and b = 1.9237 × 10 .Thebasicreproductivenumber of influenza A was estimated as 1.5 −3.1 [30]. For ease of demonstration, we naively set p = 3, β = 0.0139 and a = 600, and hence the basic reproductive number R is equal to 3.0. We shall consider two examples. Both examples show the occurrence of bi-stability, in which solutions may converge to one of two stable equilibria, depending on initial conditions. 0 0 Example 3.1: Set I = 45. We can calculate I = 21 and a = 600. Therefore, I > I ,and c 2 c c c a = a . In this case (see Theorem 2.2(2)), model (3) admits two stable endemic equilibria E and E , that is, bi-stability occurs (see the line L in Figure 1). Some solutions to the ∗ 1 2 model are illustrated in Figure 2.Notethat a = a is equivalent to I = I . The endemic 2 c ∗ equilibrium E lies on the horizontal line I = I . ∗ c Example 3.2: Set I = 50. In this case, I > I = 21, a = 597 and a = 651. The con- c c 1 2 ditions of (iii) of (2) in Theorem 2.2 are satisfied. Hence, model (3) has three positive equilibria E , E and E .Figure 3 illustrates some numerical solutions to the model. The ∗ 2 3 I=I 100 150 200 250 300 350 400 Figure 2. Phase plot of I verses S showing that two stable endemic equilibria E , E coexist. Here, we fix ∗ 1 −3 1 0 (a, b, β, d, γ , ) = (600, 1.9237 × 10 , 0.0139, , 0.0196, 2.7397),and set I = 45 > I = 21. I 130 L. WANG ET AL. 60 2 E I=I 100 150 200 250 300 350 Figure 3. Phase plot of I verses S showing that bi-stability occurs for I > I and a < a < a .Here, E c 1 2 ∗ and E are locally stable while E is a saddle point. I = 50 > I = 21 and other parameters take the 3 2 c same values as in Figure 2. stable manifolds of the saddle E split the phase plane into two regions. In the lower region, solutions approach to E while in the upper region solutions approach to E (see Figure 3). ∗ 3 4. Global stability analysis In this section, we study the global stability of model (3). It can be shown that the state variables of model (3) remain non-negative for non-negative initial conditions. Consider the biologically feasible region (ab + d + ) = (S, I) ∈ R : N = S + I ≤ K = + 1 . 4b (d + γ) Choosing the straight line S+I−Ab =0,similartotheproofofCorollaryin[36], we have the following two results. Lemma 4.1: Theclosedset is a positively invariant set for model (3). Theorem 4.2: E is globally asymptotically stable if 0 < a ≤ a and unstable if a > a . 0 0 0 Tothebestofour knowledge,byprecludingtheexistence ofalimitcycleweareableto proveglobalstabilityoftheuniqueendemicequilibriumofmodel (3). Theorem 4.3: There exist no limit cycles for model (3). Proof: Thefollowing twostepsareconsideredtoachieve ourconclusion. Step 1. We shall provethattherearenolimitcyclesintheregionbelowtheline I = I in the feasible region and in the region above the line I = I . Denote these two regions by and . 1 2 I JOURNAL OF BIOLOGICAL DYNAMICS 131 −1 Take the Dulac function D(S, I) = S .Inthe case of0 < I ≤ I , by the transformation x = S, y = ln I, (24) one can transfer sub-system (4) into dx = bx(a − x) − (βx − γ) e , dt (25) dy = βx − (d +  + γ). dt −1 and D(x, y) = x .Let N , N be the right-hand side functions of (25). Then 1 2 ∂(DN ) ∂(DN ) γ e 1 2 + =− − b < 0. (26) ∂x ∂y x Therefore, there are no limit cycles in the region below the line I = I . In the case of I > I , we can extend subsystem (5) to the case of I ≥ I because of the c c continuity of the transmission function β(I).Thatis, subsystem(5) canberewritten into dS I = bS(a − S) − β IS + γ I, dt I I ≥ I . (27) dI I = β IS − (d +  + γ)I, dt I Consider two cases p =1and p = 1. If p = 1, one can set x = S and y = I.With N and N 1 2 being the right sides of system (27), direct calculations show that ∂(DN ) ∂(DN ) γ y d +  + γ 1 2 + =− − − b < 0. (28) ∂x ∂y x x If p = 1, using the transformation 1−p x = S, y = ln I , (29) one obtains dx y y/(1−p) = bx(a − x) − βI x e + γ e , dt (30) dy −py/(1−p) = (1 − p)[βI x e − (d +  + γ)]. dt Therefore, ∂(DN ) ∂(DN ) γ 1 2 p y/(1−p) −py/(1−p) + =− e − pβI e − b < 0. (31) ∂x ∂y x Consequently, for all p > 0, one can have ∂(DN ) ∂(DN ) 1 2 + < 0. (32) ∂x ∂y Hence, there is no limit cycle in , the region above the line I = I .Weshouldpointout 2 c that inequalities (26) and (32) hold for y ∈ (−∞, +∞). 132 L. WANG ET AL. ΓΓ I=I C C 2 1 30 40 50 60 70 80 90 100 Figure 4. A limit cycle intersects with the line I = I at C and C . c 1 2 Step 2. We are now ready to show that model (3) has no limit cycle crossing the line I = I .Theidea is similartothatin[27, 28, 34]. Assume that  isalimit cycleacrosstheline I = I .Let  bethepartofthecycle c 1 below I = I of the cycle ,and  be the part above I = I , with the direction designated c 2 c in Figure 4.Letboth  and  include two intersection points C , C of  with the line 1 2 1 2 I = I .Theregion enclosed by  and the segment C C is denoted by G ,and theregion c 1 1 2 1 enclosed by  and the segment C C is denoted by G . 2 1 2 2 −−→ −−→ Let us choose two the directed-paths L : C C and L : C C ,asshownin Figure 4.It 1 1 2 2 2 1 can be seen that D(N dI − N dS) = + D(N dI − N dS) 1 2 1 2 1 2 1 2 = + + + − − D(N dI − N dS) 1 2 L L L L 1 2 1 2 1 2 = + − − D(N dI − N dS). (33) 1 2 ∪L  ∪L L L 1 1 2 2 1 2 Meanwhile, it is easy to obtain that + D(N dI − N dS) = 0. (34) 1 2 L L 1 2 Hence, from Green’s Theorem it follows that D(N dI − N dS) = + D(N dI − N dS) 1 2 1 2 ∪  ∪L  ∪L 1 2 1 1 2 2 ∂(DN ) ∂(DN ) 1 2 = + + dS dI. (35) ∂S ∂I G G 1 2 I JOURNAL OF BIOLOGICAL DYNAMICS 133 Obviously, one can have D(N dI − N dS) = 0. (36) 1 2 1 2 From Step 1, however, we know that there is no limit cycle in the regions Pi or Pi ,and 1 2 ∂(DN ) ∂(DN ) ∂(DN ) ∂(DN ) 1 2 1 2 + dS dI < 0, + dS dI < 0. ∂S ∂I ∂S ∂I G G 1 2 (37) A contradiction to Equation (35). Therefore, there are no limit cycles crossing the line I = I . To sum up, model (3) has no limit cycles. From Theorems 2.2 and 4.3, we immediately have Corollary 4.4: If model (3) admits a unique endemic equilibrium (either of E or E ), then ∗ 1 it is globally asymptotically stable. We can illustrate the global stability by numerical simulations. Set I = 40 and the remaining parameters take the same values as in Figure 2.Then a > a = 549. From Remark 1 and Corollary 4.4, the unique endemic equilibrium E (see the region Q in 1 1 Figure 1) of model (3) is globally asymptotically stable. Figure 5 shows that the number I of infections is stabilized at the level I . In the figure, it is also shown that the number I of infected cases is stabilized at decreased levels as either the intensity p of the media eeff ct increases or the critical number I decreases. That is, stronger media eect ff s and/or lower critical numbers lead to a decreased number of infections at the endemic equilibrium E . In fact, such decreasing eeff cts are also true for all endemic equilibria except E . I =40 I =35 p=3 p=6 45 p=9 I =30 0 0.2 0.4 0.6 0.8 1 Time (year) Figure 5. The number of infected cases is stabilized at decreased levels as either p increases or I decreases. I 134 L. WANG ET AL. 5. Discussion During a disease spread, only when the number of infected cases and/or the severity of infection are high enough to draw the attention of media and public health organiza- tions, alerts are issued and all related information is brought to the public through media coverage. The information gradually changes public behaviour, which reduces the chance of potential contact infection and eventually helps to curb the disease spread. Fast and dramatic changes in public behaviour can also occur subject to some intensive control mea- sures such as closing school and distancing certain groups of persons. Previous studies have introduced a critical number I to be the level for media and health organizations to take action. Functions with jump-discontinuity at I were used to describe the changing trans- missionrateduetothemedia eeff ct[ 28, 34, 35]. With the rate functions, media eect ff s on disease outbreaks were studied through compartmental epidemic models. It turns out that proper use of media coverage can curb disease outbreaks [28, 34]. In this paper, following the idea of the critical number I , we proposed the non-smooth function (2) to describe the transmission rate which is continuous at I . With this description, change in public behaviour is continuous. Using a susceptible–infected–susceptible model with a logistic growth in the susceptible class, we studied the media eeff ct on the transmission dynamics of an infectious disease in a given region. Ourmodel analysis showsthatwithoutthemediaeeff ctorwithrelativelyweakeeff ct (0 < p ≤ 1), the endemic equilibrium E or E (depending on the chosen I ) is globally ∗ 1 c asymptotically stable. With relatively strong media eect ff p >1,themodelmayhaveup to three endemic equilibria for the chosen critical number I in the threshold interval 1 2 = [I , I ]. Otherwise, the model admits a unique endemic equilibrium (which is glob- c c ally asymptotically stable, see Figure 1). In the former case, solutions to the model can converge to either one of two stable endemic equilibria (see Figures 2 and 3), depending on initial conditions. That is, bi-stability can occur. To avoid such uncertainty in practice, it is necessary to choose the critical number I below the value I . It is worthwhile to point 1 2 out that the critical values I , I depend only on the basic reproduction number R and c c the intensity p of the media eect ff and hence they are prescribed by the focal disease and the population in the given region. Therefore, it is possible for policymaker to roughly esti- mate the threshold interval for re-emerged diseases and then choose a reasonable critical number I to initiate media coverage. From the point of view of disease control, early media alerts (i.e. setting a small critical number for I )andstrongmediaeeff cts(i.e. p > 1) are definitely preferable. Our anal- ysis shows that if I < I ,equivalently a > a ,theendemicequilibrium E is globally c ∗ 2 1 asymptotically stable. However, if I > I , model solutions approach the unique equilib- rium E , causing that media coverage loses its impact on disease transmission. This should be avoided by health policymakers. Early media alerts and strong eect ff can decrease the numbers of infected cases at endemic equilibria I , i = e,1,2,3. For example, as the critical number I decreases, or the i c intensity p of the media eect ff increases, the number of infected cases can be stabilized at E with decreased numbers of cases (see Figure 5). Therefore, properly choosing the critical number and strengthening the media effect can reduce disease prevalence. The analysis of an SEI model also implies that media coverage can reduce the number of infected cases at endemic equilibria [9]. JOURNAL OF BIOLOGICAL DYNAMICS 135 The existence of multiple endemic equilibria was obtained in some previous studies −mI (see e.g. [9, 20]). Using the transmission rate β e with media parameter m,Cui et al. [9] found that the model may exhibit periodic oscillations for sucffi iently small media effects (small m) and may have multiple endemic equilibria for strong media eect ff s (large m). Unfortunately, the stabilities of the equilibria are not available and hence the solution behaviour is unknown for strong media eect ff s. Media/psychological eect ff s may be also 2 2 included in the incidence rate in a nonlinear and saturation way such as kI S/(1 + αI ) and simple compartmental models can exhibit complex dynamics. For instance, saddle- node bifurcation, Hopf bifurcation and homocyclic bifurcation can occur in a simple SIR epidemic model with the rate [20]. Therefore, the impact of media coverage on a dis- ease transmission dynamics could be complicated and simplified understandings may even make the disease worse (e.g. [24]). It needs to be further studied from distinct aspects [8]. 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Journal

Journal of Biological DynamicsTaylor & Francis

Published: Mar 17, 2017

Keywords: SIS epidemic model; media alert; logistic growth; multiple endemic equilibria; bi-stability; 34K18; 34K20; 92D30

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