An SEIRS epidemic model with stochastic transmission
 Peter J Witbooi^{1}Email author
https://doi.org/10.1186/s1366201711666
© The Author(s) 2017
Received: 9 January 2016
Accepted: 31 March 2017
Published: 11 April 2017
Abstract
For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the diseasefree equilibrium whenever a certain invariant \(\mathcal{R}_{\sigma}\) is below unity. Here \(\mathcal{R}_{\sigma}< \mathcal{R}\), the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the diseasefree state if both components of the perturbation are nontrivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem.
Keywords
SEIRS model stochastic transmission almost sure exponential stability1 Introduction
In recent years, a number of articles have been published on stochastic differential equation models of population dynamics of infectious diseases. In comparison with models described by ordinary differential equations (ode), the stochastic differential equation (sde) models provide of course a means of accommodating randomness in the model. Two themes of special interest in the modeling of population dynamics of a disease are the stability of equilibrium points and the optimal control of interventions such as vaccination, quarantine, public health education and others. For sde models, optimal control problems and solutions are presented in [1] of Cai and Luo, [2] of Ishikawa and in [3]. In the stochastic setting, stability of equilibria and the long term persistence or extinction of a disease in a population have been studied in most of the sde models in the literature. Such studies use different versions of stability. Stochastic perturbation has also been studied in multigroup models, such as in [4–6] for example. In many cases it has been proved that the introduction of stochastic perturbations into an ode epidemic model system can possibly render an unstable diseasefree equilibrium of the ode system to become stable in the stochastic differential equation system. This phenomenon was highlighted in [7] by Chen et al., [8] by Gray et al. and in [9] for instance.
Since the basic models such as [10] by Li et al. on diseases of the SEIRS type, many variations have been presented in the literature, such as [11] of Melesse and Gumel. Starting with an ode model of SEIRS type, in this paper we study the effect of stochastic perturbations on the stability of the diseasefree equilibrium of the system. The models in [7] and in [8] have perturbations of the transmission rate from the Sclass to the Iclass. The latter models do not include a latent infection compartment such as the E compartment in SEIR type models. The current paper is among the first studies of a disease model with a latent infection compartment, having a perturbation of the disease transmission. In the literature there are stochastic models such as in [4] and [5] having latent infection compartments, but with the stochastic perturbations not directly aimed at transmission. We prove the existence of solutions which are almost surely global and positive. We also study stability of the diseasefree equilibrium. In particular, we introduce an invariant \(\mathcal{R}_{ \sigma}\) of the model that is related to the basic reproduction number \(\mathcal{R}\) of the underlying deterministic model, with \(\mathcal{R}_{ \sigma}< \mathcal{R}\). With the given type of randomness in the system, we prove that there is a greater chance of the disease vanishing from the population. The main results are illustrated with simulations.
2 Preliminaries
Notation 2.1
By \(\mathbb{R}^{n}_{+}\) (resp. \(\mathbb{R}^{n} _{++}\)) we denote the set of points in \(\mathbb{R}^{n}\) having only nonnegative (resp. strictly positive) coordinates.
We assume throughout the paper that we have a complete probability space \((\Omega, \mathcal{F}, \mathbb{P})\), equipped with a filtration, \(\{ {\mathcal{F}}_{t}\}_{t\geq0}\), that is right continuous and with \({\mathcal{F}}_{0}\) containing all the subsets having measure zero. We consider a onedimensional Wiener process \(W(t)\) on this filtered probability space.
By \(\mathcal {L}\) we denote the infinitesimal generator (see for instance [12]) associated with the function displayed in equation (2.1), defined for a function \(V(t,x)\in C^{1,2}(\mathbb{R} _{+} \times\mathbb{R}^{k})\).
Definition 2.2
See [13]
The following lemma was utilized in [9] and proved in [14]. For completeness we include the simple proof.
Lemma 2.3
For \(k\in\mathbb{N}\), let \(X(t)=(X_{1}(t), X _{2}(t), \ldots , X_{k}(t))\) be a bounded \(\mathbb{R}^{k}\)valued function and let \((t_{0,n})\) be any increasing unbounded sequence of positive real numbers. Then there is a family of sequences \((t_{l,n})\) such that, for each \(l\in\{ 1, 2, \ldots, k\}\), \((t_{l,n})\) is a subsequence of \((t_{l1,n})\) and the sequence \(X_{l}(t_{l,n})\) converges to a chosen limit point of the sequence \(X_{l}(t_{l1,n})\).
Proof
Let b be an upper bound for the functions \(X_{i}(t)\). In the compact set \([0,b]\), we can choose a limit point in the closure of the set \(\{ X_{1}(t_{0,n}) \vert n\in\mathbb{N}\}\) and select a convergent subsequence \((t_{1,n})\) of \((t_{0,n})\) for which the limit is the chosen limit point. In the same way we can start with the sequence \((t_{1,n})\) and pick a subsequence \((t_{2,n})\) for which \((X_{2}(t_{2,n}))\) is convergent, etc. □
Proposition 2.4
Let p, h and \(h_{*}\) be as above, and let \(h_{1}=h\vert _{(0,1]}\) be the (domain) restriction of h to \((0,1]\). Then \(h_{*}(p)\) is the absolute minimum of \(h_{1}\).
Proof
If \(p=\frac{1}{2}\), then \(h(x)=(4x)^{1}\) and the result follows easily. Thus, for the remainder of the proof we exclude the case \(p=\frac{1}{2}\). Then we observe that h tends to +∞ if \(x\to0^{+}\) and also, h tends to +∞ as \(x\to+\infty\). Using calculus we find that \(h'(x)\) is continuous on \(\mathbb{R}_{++}\) and has exactly one root \(x_{0}\), which is \(x_{0}=p\cdot \vert qp\vert ^{1}\). Therefore, the minimum of \(h_{1}\) is \(h(x_{0})\) whenever \(x_{0} \le1\) and is \(h(1)\) otherwise. Further, \(x_{0}\le1\) if and only if \(0\le p\le\frac{1}{3}\). The rest of the proof follows readily. □
3 The model
Melesse and Gumel [11] present a model for a disease of SEIRS type that may cause different stages of infectiousness in a patient. In a special case of the mentioned model, in this paper we study the effect of stochastic perturbations on the stability of the diseasefree equilibrium. The population, which at any time t consists of \(N(t)\) individuals, is regarded as being divided into four compartments or classes. These are called the susceptible, exposed, infectious and removed classes. Their sizes, at any time t, are denoted by \(S(t)\), \(E(t)\), \(I(t)\) and \(R(t)\), respectively. The equations of motion of the system are assumed to be given by the system (3.1) of stochastic differential equations. If \(\sigma=0\) then the system reduces to a system of ode, which can be called the underlying deterministic model or the underlying system of ode. For the system (3.1), the underlying system of ode coincides with a special case of the model in [11]. Inflow into the population is assumed to be all into the class of susceptibles, and it is at a rate \(\mu_{0}K\). Additionally there is flow from the recovered class into the class of susceptibles at a rate \(\alpha_{3}R\), due to loss of infectionacquired immunity. The mortality rates in the different classes are denoted by \(\mu_{i}\) (\(i=0,1,2,3\)) and this allows for higher mortality rates in classes which have been affected by the disease, such as also in [15] of Beretta et al. Hence the condition (3.2) below. The symbol β denotes the effective contact rate. The parameters \(\alpha_{1}\) and \(\alpha_{2}\) determine the rates at which individuals in the population pass from classes E to I and (respectively) from I to R.
Proposition 3.1
Proof
We now prove the existence of solutions which are almost surely global and positive.
Theorem 3.2
Given any initial value \(X_{0}=(S_{0}, E_{0}, I_{0}, R_{0})\in\Delta_{K}\), then the system (3.1) admits a unique solution \(X(t)=(S(t), E(t), I(t), R(t))\) on \(t\geq0\), and this solution remains in \(\Delta_{K}\) almost surely.
Proof
The coefficients of the system (3.1) are locally Lipschitz continuous. By [13], Theorem 3.5, for the given initial value \(X_{0}\in\Delta_{K}\) there is a unique local solution \(X(t)\) over the interval \(t\in[0,\tau_{\mathrm{en}} )\), where \(\tau_{ \mathrm{en}}\) is the explosion time.
Remark 3.3
4 Stability theorems
The concept of stability of a deterministic system of differential equations ramifies into different forms when dealing with stochastic differential equations. In this paper we shall focus on almost sure exponential stability, which is conceptually uncomplicated. We prove that when the basic reproduction number \(\mathcal{R}\) of the underlying deterministic model is below unity, then the diseasefree equilibrium is almost surely exponentially stable. We also prove stronger results on \(I(t)\) converging to zero, in terms of the analog \(\mathcal{R_{\sigma }}\) of \(\mathcal{R}\).
Theorem 4.1
If \(\mathcal{R} < 1\), then the diseasefree equilibrium of the system (3.1) is almost surely exponentially stable.
The proof of Theorem 4.1 will be presented following a discussion which is relevant to all the stability results that we derive in this paper.
Item 4.2
A construction and notation.
Item 4.3
A useful inequality for \(\mathcal{L}V(X(u))\).
Remark 4.4
Proof of Theorem 4.1
Theorem 4.1 implies that if a diseasefree equilibrium is locally asymptotically stable with respect to the underlying odesystem, then it is almost surely exponentially stable with respect to the stochastic model, in particular, the perturbations do not disrupt the stability of the diseasefree equilibrium.
Remark 4.5
We now present the main result of this paper, which proves that the stochastic perturbation improves the stability of the diseasefree equilibrium for small values of the perturbation parameter.
Theorem 4.6
 (1)
\(\mathcal{R}_{\sigma}< 1\),
 (2)
\(\sigma^{2}\le\frac{c\beta}{Kh_{*}}\) with \(c=\frac{ \alpha_{1}}{\mu_{1}+\alpha_{1}}\),
Proof
In Section 5 we shall further reflect on Theorem 4.6. Also, Theorem 4.8 below combines very well with Theorem 4.6. However, while our main result Theorem 4.6 focused on small perturbations, let us briefly address the case of larger perturbations. The following theorem asserts that, for sufficiently large values of the perturbation parameter σ, the disease will eventually vanish from the population.
Theorem 4.7
Proof
Theorem 4.8
Proof
Remark 4.9
(a) Theorem 4.6 is much more significant than Theorem 4.7 because in disease modeling, in practice one is more interested in smaller perturbations rather than the larger perturbations. Let us denote the bounds on σ specified in Theorems 4.6 and 4.7 by \(\theta_{1}\) and \(\theta_{2}\) respectively. If \(\theta_{1} > \theta _{2}\), then these theorems can be combined, guaranteeing the diseasefree equilibrium to be almost surely exponentially stable irrespective of the magnitude of σ.
(b) Of course, Theorem 4.8 serves to extend Theorems 4.6 and 4.7.
5 Simulations
Theorem 4.6 suggests that \(\mathcal{R}_{\sigma}\) is an approximation for a threshold that decides stability in a way similar to the basic reproduction number. Simulations show that it is a rather useful approximation. For a nonnegative stochastic process, almost sure convergence to 0 can be tested by computing the (approximation over finitely many paths, of the) mean of sample paths. If the mean of I is not asymptotically stable, then I is not almost surely exponentially stable. The simulations that were run produced trajectories of the mean of I which consistently appears to converge to a value which is smaller than, or at least not bigger than, in the deterministic case.
These simulations are obtained by considering an influenza infection of the type in [11] and [17]. The relevant parameter values for \(\alpha_{1}\), \(\alpha_{3}\), \(\mu_{0}\), \(\mu_{1}\) and \(\mu_{3}\) are taken directly from [11] and other parameters values are derived. Our value for \(1/\alpha_{2}\) is obtained by taking the sum of the average times spent in the \(I_{(\cdot)}\)compartments (of [11]). The value of \(\mu_{2}\) is taken as \(1.025\mu_{0}\). The parameter β is not kept fixed in these simulations. Here we note that infections that are aerially transmitted will spread faster when people are in high density locations with poor ventilation. So for instance the same disease has a higher value for the effective contact rate, when considered in a concentration camp as compared to an ordinary small village or rural area.
Initial values in millions are \(S(0)=6\), \(E(0)=2\), \(I(0)= 0.9\) and \(R(0)= 1.05\).

the mean of I, taken over 1,000 sample paths and indicated as ‘I (1000ave)’, and

the Iclass trajectory ‘I determ’ (broken line) of the underlying deterministic model,

\(K=10\) (in millions),
Numerical values of the fixed parameters
Parameter  Value 

\(\mu_{0}\)  \((60\times365)^{1}=4.566\times10^{5}\) 
\(\mu_{1}\)  4.566 × 10^{−5} 
\(\mu_{2}\)  \(1.025\mu_{0}=4.680\times10^{5}\) 
\(\mu_{3}\)  4.566 × 10^{−5} 
\(\alpha_{1}\)  1.9^{−1} = 0.5263 
\(\alpha_{2}\)  0.2 
\(\alpha_{3}\)  \((83.33)^{1}=0.012\) 
Parameters such as p and σ are difficult to compute. We choose \(p=0.333\) for simulation. The parameters β and σ are varied in order to obtain different values of \(\mathcal{R}_{\sigma}\).
6 Conclusion
In this paper we constructed an SEIRS model, with stochastic perturbations which can be viewed as linked to the transmission rate out of the class of susceptibles. We proved that the system of stochastic differential equations permits solutions that are almost surely global and positive. The model permits a diseasefree equilibrium which we showed to be almost surely exponentially stable whenever the basic reproduction number of the underlying deterministic model is below unity, and even slightly beyond under given conditions.
Biologically we observe, in particular, the following effect of a stochastic perturbation on the disease transmission in the case of a deterministic compartmental model which allows for a latently infectious class. Given a small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase, the stability of the diseasefree state will be improved if both components of the perturbation are nontrivial. If any one of the components of the perturbation is zero, then the stability will not be disturbed. The simulations confirm the proven results and also provide further insights, such as about the behavior of the mean of the Iclass trajectories.
Declarations
Acknowledgements
The author acknowledges financial support by the National Research Foundation of South Africa.
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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