Periodic solutions for a seasonally forced SIR model with impact of media coverage
 Jian Zu^{1} and
 Lin Wang^{2, 3}Email author
https://doi.org/10.1186/s1366201504778
© Zu and Wang; licensee Springer. 2015
Received: 8 July 2014
Accepted: 17 April 2015
Published: 1 May 2015
Abstract
In this paper, we study periodic solutions for a seasonally forced SIR model with impact of media coverage. Usually, media reports, information processing, and individuals’ alerted responses to the information can only arise as the number of infected individuals reaches and exceeds a certain level. The piecewise smooth righthand side is introduced to describe the impact of this kind of media coverage. Using LeraySchauder degree theory, we establish new results on the existence of at least one positive periodic solution for a seasonally forced SIR model with impact of media coverage. Some numerical simulations are presented to illustrate the effectiveness of such media coverage.
Keywords
MSC
1 Introduction
Many infectious diseases, such as measles, chickenpox, mumps, rubella, pertussis and influenza, show seasonal patterns of incidence [1–3]. The cause of seasonal patterns may vary from the periodic contact rates [3, 4], periodic fluctuation in birth and death rates [5–7], and periodic vaccination program [8]. Thus, it is natural to model these diseases by seasonally forced epidemiological models.
The media coverage is an important factor responsible for the transmission of an infectious disease. When a type of contagious disease appears and starts to spread, people’s response to the threat of disease is dependent on their perception of risk, which is affected by public and private information disseminated widely by the media. Massive news coverage and fast information flow have played an important role in affecting the outcome of infectious disease outbreak, such as the 2003 severe acute respiratory syndrome (SARS) and the 2009 H1N1 influenza epidemic [9–14].

S, I, R are the fractions of the susceptible, infective and recovered population,

μ and γ denote the birth (death) rate and recovery rate respectively, which are positive constant,

\(\beta(t)\) is the seasonallydependent transmission rate, which is a positive continuous Tperiodic function,

\(f(I)\) is a decreasing piecewise smooth factor which can describe the impact of media coverage on the transmission coefficient, given bywhere α is the factor of influences, σ is a small parameter and \(I_{c}\) is a critical level.$$ f(I)=\left \{ \begin{array}{@{}l@{\quad}l} 1,& I\leq I_{c},\\ 1+\sigma^{1}(e^{\alpha(I_{c}+\sigma)}1)(II_{c}),& I_{c}< I<I_{c}+\sigma,\\ e^{\alpha I},& I\geq I_{c}+\sigma, \end{array} \right . $$(1.2)
We think that \(f(I)\) in (1.2) is a good approximation to the discontinuous factor in [15] provided σ is small enough. Denote the basic reproduction number \(\mathcal{R}_{0} = \frac{\bar{\beta}}{\gamma+\mu}\) with \(\bar{\beta}=\frac{1}{T}\int_{0}^{T}\beta(t)\,dt\). When \(f(I)\equiv1\) in (1.1), Katriel [16] got the existence of periodic positive solutions for the periodically forced SIR model by LeraySchauder degree theory provided \(\mathcal{R}_{0}>1\). By GainesMawhin’s continuation theorem, Jódar et al. [17] obtained that a Tperiodic solution exists for a more general system whenever the condition \(\min_{t\in\mathbb{R}} \beta(t)> \gamma+ \mu\) holds; Bai and Zhou [18], Bai et al. [19], and Liu [20] studied the existence of periodic solutions for a periodically forced SIR model with saturated incidence rates. By persistence theory, Zhang and Zhao [21] studied a periodic epidemic model in a patchy environment; Sun et al. [22] studied the SEI model with seasonality comprehensively; Rebelo et al. [23] extended these results to some delay differential equations and partial differential equations.
When \(f(I)\) in (1.1) is a nonsmooth function, to the best of our knowledge, there are no results on the existence of periodic solutions. The methods we mentioned above cannot deal with the nonsmooth righthand sides directly.
In this paper, we use an integral version of LeraySchauder degree theory under Katriel’s frame to prove the existence of periodic solutions for our SIR model. Some numerical simulations are presented to illustrate the effectiveness of such media coverage. Our main results are as follows.
Theorem 1.1
Whenever \(\mathcal{R}_{0}>e^{\alpha}\), there exists at least one Tperiodic solution \((S(t), I(t), R(t))\) of (1.1)(1.2), all of whose components are positive.
Theorem 1.2
Whenever \(\mathcal{R}_{0}>e^{m I_{c}}\), there exists at least one Tperiodic solution \((S(t), I(t), R(t))\) of (1.1) and (1.3), all of whose components are positive.
This paper is organized as follows. In Section 2, we study the properties of the homotopy equation from the classical autonomous SIR model to our SIR model. In Section 3, we construct an equivalent integral equation and define a completely continuous operator. In Section 4, we prove the main theorem by LeraySchauder degree theory. In Section 5, some numerical simulations are presented to illustrate the effectiveness of such media coverage.
2 Homotopy equation and suitable domain
Lemma 2.1
\(\overline{D}\) is an invariant region with respect to (2.2). The diseasefree equilibrium \((S_{0}, I_{0}) = (1, 0)\) is the unique periodic solution of (2.2) satisfying \((S, I)\in\partial D\) for any \(\lambda\in[0,1]\).
Proof
 (i)
There exists \(t_{0}\in[0,T]\) such that \(I(t_{0})=0\).
 (ii)
There exists \(t_{0}\in[0,T]\) such that \(S(t_{0})=0\).
 (iii)
There exists \(t_{0}\in[0,T]\) such that \(S(t_{0})+I(t_{0})=1\).
In the case of (i), we have \(I(t_{0})=0\) and \(I'(t_{0})=0\), which implies \(I\equiv0\). Thus, the only possible periodic solution of \(S' = \mu(1S)\) is \(S\equiv1\).
In the case of (ii), we have \(S(t_{0})=0\) and \(S'(t_{0})=\mu>0\). Thus, it is easy to obtain that \(S(t)<0\) for \(t< t_{0}\) sufficiently close to \(t_{0}\), which contradicts the fact that \(\overline{D}\) is an invariant region.
Remark 2.2
For fixed \(t_{0}\), U is an open set in \(\mathbb{R}^{2}\). Furthermore, for any \(t\in[0,T]\), U with norm \(\(S,I)\=\max_{t\in [0,T]}(S(t)+I(t))\) is an open set in \(C[0,T]\times C[0,T]\).
Lemma 2.3
Let \(\mathcal{R}_{0}>e^{\alpha}\). If we choose \(\delta\in(\frac{e^{\alpha }}{\mathcal{R}_{0}}, 1)\), then there is no solution \((S, I)\) of (2.2) with \((S, I) \in\partial U\) for any \(\lambda\in[0,1]\).
Proof
In the first case, Lemma 2.1 and the fact that \((S_{0}, I_{0})\notin\partial U\) imply that \((S, I)\) is not a solution of (2.2).
3 Existence of periodic solutions
3.1 Equivalent integral equation and completely continuous operator
Lemma 3.1
\(P_{\lambda}\) in (3.6) is a completely continuous operator.
Proof
Since \(f(I)\) is a nonsmooth but continuous function, the operator \(P_{\lambda}\) is continuous with respect to S and I. Since \(\beta(t)\), \(\Phi(t)\) and \(\Phi^{1}(t)\) are all bounded in \([0,T]\), S and I are bounded on U, \(e^{\alpha}\leq f(I)\leq1\) on U, it is easy to see that the operator \(P_{\lambda}\) in (3.6) is uniformly bounded and equicontinuous, which implies \(P_{\lambda}\) in (3.6) is a completely continuous operator. □
3.2 Main results
 (1)
\((\operatorname{Id}P_{\lambda})(S,I)\neq0\) for all \((S,I)\in\partial U\), \(\lambda\in[0,1]\),
 (2)
\(\operatorname{deg}(\operatorname{Id}P_{0}, U, 0)\neq0\).
By Lemma 2.3, there are no solutions \((S, I)\) of (2.2) with \((S, I) \in\partial U\), \(\lambda\in[0,1]\). Now we prove that \(\operatorname{deg}(\operatorname{Id}P_{0}, U, 0)\neq0\).
Lemma 3.2
For \(\lambda=0\), (2.2) has only one periodic solution in U, which is endemic equilibrium: \((S^{\ast},I^{\ast}) = (\frac{\gamma+\mu}{\bar{\beta}}, \mu(\frac {1}{\gamma+\mu}\frac{1}{\bar{\beta}}))\).
Proof
Since \(\frac{e^{\alpha}}{\mathcal{R}_{0}}<\delta<1\), γ, μ and \(\bar{\beta}\) are positive constant, we have \(0< S^{\ast}<\frac{\delta }{e^{\alpha}}<\delta\) and \(0< I^{\ast}<\frac{\mu}{\gamma+\mu}<1\). For \(\lambda=0\), it is easy to calculate that \((S^{\ast}, I^{\ast})\) is the unique constant periodic solution in U, which implies that (2.2) has only one periodic solution in U. □
Lemma 3.3
[26]
Lemma 3.4
[26]
Lemma 3.5
1 is not the eigenvalue of \(DP_{0}(S^{\ast}, I^{\ast})\).
Proof
4 Simulation
In this section, we present some numerical examples to illustrate the effectiveness of such media coverage. Furthermore, we show how various parameters influence the solutions of our SIR model.
With the period \(T=2\pi\) of the forcing representing one year, we take \(\gamma= 14\frac{2\pi}{365}\) corresponding to a twoweek infectious period. We set \(\bar{\beta} = 4\gamma\), \(\mu=\frac{0.5}{2\pi}\), \(\beta (t)=\bar{\beta}(1+0.8\cos(t))\), \(I_{c}=0.15\) and \(\delta=0.05\). Let \([0,2\pi]\) be divided into \(k=200\) intervals equally. Given the initial point \((S^{\ast\ast}, I^{\ast\ast})=(\frac{\mu+\gamma}{\bar {\beta}}, \frac{\mu}{\mu+\gamma}\frac{\mu}{\bar{\beta}})\), which is the endemic equilibrium of SIR model without periodic transmission rate and impact of media coverage. The periodic solutions of system (1.1) can be solved by the Newton iteration method.
In a word, it is effective to reduce the infective population by media coverage.
5 Conclusion
In this paper, we study the existence of positive periodic solutions for a seasonally forced SIR model with impact of media coverage. This paper can be divided into two parts. In the first part, we construct a homotopy equation from an autonomous system to our SIR model. Using LeraySchauder degree theory, we establish a new result on the existence of at least one positive periodic solution for our SIR model. In the final part, some numerical simulations are presented to illustrate the effect of media coverage.
Declarations
Acknowledgements
The authors would like to express their sincere gratitude to Professor Yong Li for his enthusiastic guidance and constant encouragement. Jian Zu was supported by NSFC (Grant No. 11401089).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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