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Periodic solutions for a seasonally forced SIR model with impact of media coverage
Advances in Difference Equations volume 2015, Article number: 136 (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.
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].
Recently, Wang and Xiao have used piecewise continuous transmission rate to describe that the media coverage exhibits its effect once the number of infected individuals exceeds a certain critical level [15]. In this paper, we study the following periodic forced SIR model:
in which

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 by
$$ f(I)=\left \{ \begin{array}{@{}l@{\quad}l} 1,& I\leq I_{c},\\ 1+\sigma^{1}(e^{\alpha(I_{c}+\sigma)}1)(II_{c}),& I_{c}< I<I_{c}+\sigma,\\ e^{\alpha I},& I\geq I_{c}+\sigma, \end{array} \right . $$(1.2)where α is the factor of influences, σ is a small parameter and \(I_{c}\) is a critical level.
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.
In [24], Liu and Xiao consider the nonperiodic coefficient SIR model with
where \(m>0\) is a factor of influence and \(I_{c}\) is a critical level. Using the same method as in Theorem 1.1, we can present the following theorem without a proof.
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.
Homotopy equation and suitable domain
In the rest of this paper, we assume that \(f(I)\) has the expression (1.2). Observing system (1.1), we have \(\frac{dS}{dt}+\frac{dI}{dt}+\frac {dR}{dt}\equiv0\). Since \(S(t)\), \(I(t)\), \(R(t)\) are fractions of the population, we have \(S(t)+I(t)+R(t)=1\) for all t. Because R does not appear in the first two equations in (1.1), it is sufficient to consider the existence of periodic solutions of following systems:
with
In order to prove the existence of periodic solutions of (2.1), we consider the following homotopy system:
where \(\lambda\in[0,1]\). Let
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
First, we will prove that \(\overline{D}\) is an invariant region. In fact, it follows from model (2.2) that
Second, we will prove that the diseasefree equilibrium \((S_{0}, I_{0}) = (1, 0)\) is the unique periodic solution of (2.2) satisfying \((S, I)\in\partial D\). We assume that \((S, I)\in\partial D\) is a solution of (2.2), which means that at least one of the following conditions holds:

(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 \(S(t_{0})+I(t_{0})=1\).
We now consider each of these three cases:
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.
In the case of (iii), we get
Because the case of \(I(t_{0})=0\) has been discussed, we only discuss the case \(S(t_{0})+I(t_{0})=1\), \((S+I)'(t_{0})<0\), which contradicts the fact that \(\overline{D}\) is an invariant region. □
To use the continuity method, we need to choose a bounded open set \(U\subseteq D\) such that there is no solution \((S,I)\) of (2.2) satisfying \((S, I)\in\partial U\) for any \(\lambda\in[0,1]\). Since \((S_{0}, I_{0})\in\partial D\) is a solution of (2.2), we need to exclude \((S_{0}, I_{0})\) from the boundary of U we chose. Following the idea of Katriel [16], we take U to be the open subset of D given by
where \(\delta\in(0,1)\) is to be fixed.
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
Suppose \((S,I)\in\partial U\). Then either \((S,I) \in\partial D\) or \((S, I)\in D\) and
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).
In the second case, we can infer that \(I(t)>0\) and \(S(t)\geq\delta\), \(\forall t\in[0,T]\). If \((S,I)\) is a solution of (2.2), we can divide the second equation of (2.2) by I. Integrating over \([0,T]\), we obtain that
Thus
By the assumption \(\delta> \frac{e^{\alpha}}{\mathcal{R}_{0}}\), we have
which is a contradiction. □
Existence of periodic solutions
Equivalent integral equation and completely continuous operator
We rewrite (2.2) as
where \(F_{1}(S,I,\lambda)=\mu\bar{\beta}S I  \lambda(\beta (t)f(I)SI\bar{\beta}SI)\), and \(F_{2}(S,I,\lambda)=\bar{\beta} S I + \lambda(\beta(t)f(I)SI\bar{\beta}SI)\). If \(\Phi(t)\) is the fundamental solution matrix of
satisfying \(\Phi(0)=\operatorname{Id}\), we have
Then (3.1) can be transformed into the equivalent integral equation
If S, I is a Tperiodic solution of (3.1), then
Since \((I\Phi(T))\) is invertible, we have
Substituting (3.4) into (3.2), we have
Define an operator \(P_{\lambda}: C[0,T]\times C[0,T]\rightarrow C[0,T]\times C[0,T]\) such that
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. □
Main results
We recall that the existence of a periodic solution \((S, I)\) of (3.5) can be assured by LeraySchauder degree theory [25] if the following conditions hold:

(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
When \(\lambda=0\), (2.2) is an autonomous system. \(\mu(1S)\bar{\beta}SI\) and \(\bar {\beta}SI  (\mu+\gamma)I\) are both \(C^{1}\) in D. If we set the Dulac function to be \(h=\frac{1}{SI}\) in D, we have
so there is no closed orbit in D.
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]
Let Ω be a bounded open set in the Banach space X. Assume that a completely continuous field \(f=\operatorname{Id}F:\bar{\Omega}\rightarrow X\) has no zero points on ∂Ω, and there are only finite zero points \(x_{1}, x_{2}, \ldots, x_{n}\) in Ω. Then we have the index formula
Lemma 3.4
[26]
Let Ω be a bounded open set in the Banach space X. \(x_{0}\) is the zero point of a completely continuous field \(f=\operatorname{Id}F:\bar {\Omega}\rightarrow X\) in Ω. We assume that f is Frechet differentiable at \(x_{0}\) and 1 is not the eigenvalue of \(A=F'(x_{0})\). Then \(x_{0}\) is an isolated zero point of f, and
where \(\beta=\sum_{\lambda_{j}>1\lambda_{j}\in\sigma(A)}\beta_{j}\) and \(\beta _{j} = \operatorname{dim}\bigcup_{k=1}^{\infty} \operatorname{Ker}(\lambda_{j}IA)^{k}\).
Define the operator
Obviously,
Thus, \(DP_{0}[(S^{\ast}, I^{\ast})]\) is the Frechet derivative of \(P_{0}[(S,I)]\).
Lemma 3.5
1 is not the eigenvalue of \(DP_{0}(S^{\ast}, I^{\ast})\).
Proof
Let λ be the eigenvalue of operator \(DP_{0}(S^{\ast}, I^{\ast})\):
Multiplying \(\Phi^{1}(t)\) by both sides of (3.11) and taking the derivative with respect to t, we have
which is equal to
If 1 is an eigenvalue of \(DP_{0}(S^{\ast}, I^{\ast})\), we have
The characteristic polynomial of the above matrix is
It is easy to calculate that \(p(0)>0\), and \(p(\omega i)\neq0\) for \(\omega\in\mathbb{R}\), which means that the matrix has no imaginary or 0 eigenvalues, so that (3.14) has no periodic solutions except \((V, W)=(0,0)\), which is a contradiction to (3.11). □
Now we will finish to prove that \(\mathrm{\deg}(\operatorname{Id}  P_{0}, U, 0)\neq0\). \(\operatorname{Id}  P_{0}\) is a completely continuous field, 1 is not the eigenvalue of \(DP_{0}(S^{\ast}, I^{\ast})\). By Lemma 3.4, we have \(\operatorname{index}(\operatorname{Id}  P_{0}, (S^{\ast},I^{\ast }))=(1)^{\beta}\neq0\). Since we have proved that \((\operatorname{Id}P_{0})(S,I)\neq0\) for all \((S,I) \in\partial U\), and \((S^{\ast}, I^{\ast})\) is the unique zero point in U, by Lemma 3.3 we have
Finally, by LeraySchauder degree theory, we obtain
Thus, there exists at least one positive periodic solutions for a seasonally forced SIR model with impact of media coverage.
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 Figure 1, we make ten steps of Newton iteration to get the approximate infective population and susceptible population of system (1.1) with different α. Obviously, the infective population of system (1.1) with impact of media coverage (\(\alpha >0\)) is lower than the infective population of system (1.1) without impact of media coverage (\(\alpha=0\)). The effectiveness of impact of media coverage grows as α grows. The solutions in both cases are locally stable and the error is about 10^{−6}.
In Figure 2, we fix \(\alpha=5\) and make ten steps of Newton iteration to get the approximate infective population and susceptible population of system (1.1) with different \(I_{c}\). Obviously, the infective population of system (1.1) becomes smaller as \(I_{c}\) decreases. The solutions in both cases are locally stable and the error is about 10^{−6}.
In a word, it is effective to reduce the infective population by media coverage.
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.
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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).
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Zu, J., Wang, L. Periodic solutions for a seasonally forced SIR model with impact of media coverage. Adv Differ Equ 2015, 136 (2015). https://doi.org/10.1186/s1366201504778
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DOI: https://doi.org/10.1186/s1366201504778
MSC
 34C25
 37J45
 92B05
Keywords
 periodic solution
 SIR model
 media effects
 nonsmooth righthand sides