Global dynamics of an SEIR epidemic model with discontinuous treatment
 Tailei Zhang^{1},
 Ruini Kang^{1},
 Kai Wang^{2}Email author and
 Junli Liu^{3}
https://doi.org/10.1186/s1366201506950
© Zhang et al. 2015
Received: 10 May 2015
Accepted: 11 November 2015
Published: 25 November 2015
Abstract
We consider a susceptibleexposedinfectedremoved (SEIR) epidemic model with discontinuous treatment strategies. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on nonsmooth analysis, the basic reproductive number \(\mathcal{R}_{0}\) is proved to be a sharp threshold value which completely determines the dynamics of the model. If \(\mathcal{R}_{0}\leq 1\), then there only exists a diseasefree equilibrium which is globally stable. If \(\mathcal {R}_{0}>1\), the diseasefree equilibrium becomes unstable and there exists a unique endemic equilibrium which is globally stable. The numerical simulations indicate that strengthening treatment measures after infective individuals reach some level is beneficial to disease control. Furthermore, we discuss that the disease will die out in a finite time, which is impossible for the corresponding SEIR model with continuous treatment.
Keywords
1 Introduction
Infectious diseases can arise if the host’s protective immune mechanisms are compromised and the organism inflicts damage on the host. It shows that the infectious disease can causes millions of deaths every year. Hence, how to prevent or slow down the transmission of infectious diseases is a very important problem. Many methods for control of infectious diseases are extensively applied, such as treatment, quarantine, isolation, immunity etc. To understand how to control and eradicate infectious disease is also one of the main goals of mathematical epidemiology. At the same time, well understanding for dynamic behaviors of infectious disease is a benefit for diseases controlling. Researchers have proposed many epidemic models to understand the mechanism of disease transmission (see [1–6] and the references therein).
Treatment plays a very important role in controlling the spread of diseases such as HIV/AIDS, tuberculosis, malaria, which are the top three single disease killers in the world. In recent years, some mathematical models incorporating treatment have been studied by many researchers (see [7–15] and the references therein). In [14], Wang and Ruan proposed an epidemic model to simulate the limited resources for the treatment of patients, which can occur because patients have to be hospitalized but there are limited beds in hospitals, or there is not enough medicine for treatments. In [15], Wang adopted a constant treatment, which simulates a limited capacity for treatment. Note that a constant treatment is suitable when the number of infectives is large. Li et al. [10] constructed an SIR epidemic model with nonlinear incidence and treatment. The results show that a backward bifurcation occurs if the capacity is small and there exist bistable endemic equilibria if the capacity is low. Recently, Guo et al. considered an SIR epidemic model with discontinuous treatment strategies. The results show that discontinuous treatment strategies would be superior to continuous ones [11]. On the other hand, there are a lot of infectious diseases (e.g. TB, HIV/AIDS, malaria, SARS etc.) which have latent periods. That is, a susceptible individual first goes through a latent period after infection before becoming infectious. When we use mathematical models to analyze the abovementioned diseases, omitting the latent period will lead to some inaccurate results on their transmission law. Under this circumstance, the epidemic models with latent periods play a very important role in epidemiology. Some models with latent periods can be called SEI, SEIS, SEIR or SEIRS type, respectively [16–18]. In [16], the authors studied the global dynamics of an SEIR model with this saturating contact rate.
This work was intended as an attempt to motivate [11] where infective individuals are removed from the infective class due to the treatment at a discontinuous rate. At the same time, the latent period is considered in the model. Due to this discontinuous treatment strategy, the resulting model is a discontinuous system. Some nonsmooth analysis techniques [19] are used for this system. The paper is organized as follows. In the next section, we will construct the model and introduce the main assumptions for a discontinuous treatment function. In Section 3, positivity and boundedness of the solution in the sense of Filippov for the model will be clearly discussed. We obtain the existence of possible equilibria, the basic reproductive number, and the stability of equilibria in Section 4. In Section 5, we summarize our main results and discuss the possibility of the extinction of the infectives in a finite time.
2 Model and preliminaries
 (H_{1}):

\(\phi: [0,\infty)\rightarrow[0,\infty)\) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. There is no loss of generality in assuming ϕ is continuous at \(I=0\), otherwise we define \(\phi(0)\) to be \(\phi(0^{+})\). Here \(\phi(0^{+})\) denotes the right limit of \(\phi(I)\) as \(I\to0^{+}\).
3 Positivity and boundedness
In this section, we will prove that the solutions exist on \([0,+\infty)\) and are nonnegative. The main result is as follows.
Theorem 3.1
Suppose that assumption (H_{1}) holds and let \((S(t),E(t), I(t))\) be the solution with initial condition (2.3) of model (2.2) on \([0,T)\). Then \((S(t),E(t), I(t))\) is nonnegative and bounded on \([0,T)\).
Proof
(i) \(E_{0}=I_{0}=0\).
From (3.2), we see that \(E(t)=I(t)=0\) for all \(t\in[0,T)\).
(ii) \(E_{0}>0\), \(I_{0}=0\).
(iii) \(E_{0}=0\), \(I_{0}>0\).
(iv) \(E_{0}>0\), \(I_{0}>0\).
Our next goal is to prove the boundedness of the solutions of model (2.4).
Remark 3.1
 (i)
The solution \((S(t), E(t), I(t))\) of (2.4) exists on \([0,+\infty)\) and \(S(t)>0\) (\(t>0\)), \(E(t)\geq0\) (\(t>0\)), \(I(t)\geq0\) (\(t>0\)).
 (ii)
If \(E(0)=0\) and \(I(0)=0\), then the solution \((S(t), E(t), I(t))\) of (2.4) exists on \([0,+\infty)\), \(S(t)>0\) (\(t>0\)), \(E(t)\equiv0\) (\(t\geq0\)), \(I(t)\equiv0\) (\(t\geq0\)).
 (iii)
If one of \(E(0)\) and \(I(0)\) is greater than zero, then the solution \((S(t), E(t), I(t))\) of (2.4) exists on \([0,+\infty)\) and \(S(t)>0\) (\(t>0\)), \(E(t)>0\) (\(t>0\)), \(I(t)>0\) (\(t>0\)).
4 Stability of equilibria
In this section, we show the stability of the equilibria for model (2.2). We first discuss the existence of the equilibria as follows.
We next claim that \(\mathcal{R}_{0}\) is the basic reproductive number for the model (2.2) which will determine the existence of an endemic equilibrium.
Theorem 4.1
Suppose that assumption (H_{1}) holds. If \(\mathcal{R}_{0}\leq1\), then there only exists a diseasefree equilibrium \(P_{0}(\frac{\Lambda}{\mu},0,0)\). If \(\mathcal{R}_{0}>1\), then there exists a unique positive endemic equilibrium \(P^{*}(S^{*}, E^{*}, I^{*})\) except \(P_{0}\).
Proof
As \(\mathcal{R}_{0}\leq1\), we have \(g(0)\leq\phi(0)\). Since \(g(I)\) is nonincreasing on I and \(\phi(I)\) is nondecreasing on I. For this reason, the inclusion (4.3) is only valid at \(I=0\). Hence, the model (2.2) has a unique diseasefree equilibrium as long as \(\mathcal{R}_{0}\leq1\).
In the next part, we show the global stability of the diseasefree equilibrium and the endemic equilibrium. We do this in several steps. We first prove their local stability as follows.
Theorem 4.2
Assume (H_{1}) holds. The diseasefree equilibrium \(P_{0}\) is locally asymptotically stable if \(\mathcal{R}_{0}<1\), and is unstable if \(\mathcal{R}_{0}>1\).
Proof
We have shown that there exists a positive endemic equilibrium if and only if \(\mathcal{R}_{0}>1\) in Theorem 4.1. Here, we will establish its local stability.
Theorem 4.3
Suppose that assumption (H_{1}) holds. If \(\mathcal{R}_{0}>1\), the endemic equilibrium \(P^{*}\) of the system (2.2) is locally asymptotically stable.
Proof
We next prove global stability of the diseasefree equilibrium and endemic equilibrium. We need to use the LaSalletype invariance principle for the differential inclusion (Theorem 3 in [21]) to prove their global stability.
Theorem 4.4
Suppose that assumption (H_{1}) holds. If \(\mathcal{R}_{0}\leq1\), then the diseasefree equilibrium \(P_{0}\) of (2.4) is globally asymptotically stable.
The following theorem states the global stability of the endemic equilibrium \(P^{*}\).
Theorem 4.5
Suppose that assumption (H_{1}) holds. If \(\mathcal{R}_{0}> 1\), then the endemic equilibrium \(P^{*}\) of (2.4) is globally asymptotically stable.
Proof
Remark 4.1
From Theorems 4.24.5, we can claim that the basic reproduction number \(\mathcal{R}_{0}\) is a sharp threshold value and that the global dynamical behaviors of the system (2.4) and the outcome of the disease are completely determined. In other words, when \(\mathcal {R}_{0}\leq1\), the diseasefree equilibrium \(P_{0}\) is globally stable so that the disease goes to extinction, while if \(\mathcal{R}_{0}>1\), the endemic equilibrium \(P^{*}\) is globally stable so that the disease remains endemic.
5 Discussion
We have considered an SEIR epidemic model that incorporates the discontinuous treatment strategies. Unlike previous SEIR epidemic models, we are interested in finding the impact of the adoption of a discontinuous treatment function.
The basic reproductive number \(\mathcal{R}_{0}\) is derived under some reasonable assumptions on the discontinuous treatment function. It is a sharp threshold parameter which completely determines the global dynamics of the model (2.4) and whether the disease goes to extinction or not. When \(\mathcal{R}_{0}\leq1\), the diseasefree equilibrium is globally stable so that the disease always dies out, and when \(\mathcal{R}_{0}>1\), the diseasefree equilibrium is unstable while the endemic equilibrium emerges as the unique positive equilibrium and it is globally stable.
It shows that the disease goes to extinction. This numerical verification supports Theorem 4.4. In addition, we find that different values of \(c_{2}\) can affect the peak values of the infective. Figure 2 reflects that larger values of \(c_{2}\) can reduce the peak values of the infective. Therefore, we can also prevent the spread of disease by increasing the treatment rate after the number infective individuals reach some high level. From the numerical simulations, strengthening the treatment rate after the infective individuals reach some level is also effective for disease control, even though we do not take any treatment measures at the initial time of the disease’s outbreak.
 (H_{2}):

\(h(I): [0,\infty)\rightarrow[0,\infty)\) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. Furthermore, \(h(0)=0\) and \(h(I)\) is discontinuous at \(I=0\).
From the form of \(t^{*}\), we find that \(t^{*}\) is increasing in the initial exposed and infective individuals but decreasing in the initial treatment rate \(h(0^{+})\). If we take more effective control measures for infectious diseases at the initial time of the diseases’ spread, then the diseases go to extinction more quickly. The number \(\frac{\varepsilon\beta\Lambda }{\mu(\mu+\varepsilon)(\mu+\alpha+\gamma)}\) is just the basic reproductive number of the SEIR model without treatment. The above analysis shows that the disease can go to extinction in a finite time under a discontinuous treatment strategy.
Declarations
Acknowledgements
This work was supported by the Fundamental Research Funds for the Central Universities (310812152002), the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ1018) and National Natural Science Foundation of P.R. China (11201399, 11461073).
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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