# The Power of Delay on a Stochastic Epidemic Model in a Switching Environment

Authors: Amine El Koufi
Journal: Complexity
 (2022)

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## Abstract

In recent years, the world knew many challenges concerning the propagation of infectious diseases such as avian influenza, Ebola, SARS-CoV-2, etc. These epidemics caused a change in the healthy balance of humanity. Also, the epidemics disrupt the economies and social activities of countries around the world. Mathematical modeling is a vital means to represent and control the propagation of infectious diseases. In this paper, we consider a stochastic epidemic model with a Markov process and delay, which generalizes many models existing in the literature. In addition, we show the stochastic threshold for the extinction of the disease. Furthermore, numerical examples are discussed to confirm the theoretical result.


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## Body

## 1. Introduction and Preliminaries

For years, the development of more realistic mathematical models in epidemiology has attracted the attention of many researchers. Consequently, many deterministic models have been suggested to understand and control the propagation of infectious diseases.The first mathematical model that describes the spread of an epidemic in a human population divided into three categories (susceptible (S), infected (I), and recovered (R)) was initially proposed by Kermack and Mackendrick [1]. Then, several generalizations of the SIR model of Kermack and Mackendrick were proposed. For example, in [2], a mathematical model called SEIR was proposed, which describes the propagation of an epidemic in a human population divided into four classes: susceptible, exposed, infected, and recovered. In [3], Cheng et al. investigated a network-based SIQS infectious disease model with a particular incidence rate. They proved that the disease-free equilibrium of the system is globally asymptotically stable, and the unique endemic equilibrium is globally attractive. Zhang et al. [4] constructed an epidemic model with a saturated treatment function described by the following ordinary differential equation system:(1)dStdt=Λ−ρSt−βStIt1+kIt,dItdt=βStIt1+kIt−ρ+θ+μIt−r¯It1+ηIt,dRtdt=μIt+r¯It1+ηIt−ρRt.The parameters in system (1) have the following meaning: Λ represent the recruitment rate of the susceptible class, ρ specify the death rate of the population, and θ denotes the death rate due to the epidemic. Moreover, the parameter μ is the death rate due to the epidemic. βSI/1+kI is the saturated incidence rate of disease [5], β is the contact rate, and k is a parameter that measures the psychological or inhibitory effect of the population. There exist other types of incidence rates, and each one represents some advantages in modeling (for more detail, see Table 1). r¯I/1+ηI is saturated treatment function [4]; r¯ designates the cure rate and η measure the extent of the effect of the infected being delayed for treatment (see, [4]).Table 1
Some nonlinear incidence rates.
Incidence nameExpressionReferenceStandard incidence rateβSI/N[6]Saturated incidence rateβSI/1+kI[5]Beddington–DeAngelis functional responseβSI/1+k1S+k2I[7]Crowley–Martin functional responseβSI/1+k1S+k2I+k1k2SI[8]Incidence with media coverage effectβ1−β2I/I+m[9]General functional responseβSI/1+k1S+k2I+k3SI[10]However, the stochastic model represents some advantages compared with the deterministic one. In effect, the propagation of transmissible diseases is naturally randomized. Thus, it is necessary to introduce an environmental noise into the system to present the fluctuation of the environment. Generally, there exist many types of environmental noise: white noise [11], colored noise [12], Lévy noise [13], etc., and every one represents some advantages in modeling. Rajasekar et al. [14] studied a stochastic epidemic model with a saturated incidence rate and saturated treatment function. While El Koufi et al. [12] proposed an epidemic model incorporating with the Lévy process. Other works have used two types of noise in their models. For example, Li and Guo [15] presented a stochastic SIS epidemic model in which they mixed three types of noise, namely, white noise, colored noise, and Lévy noise. In the works [16, 17], the authors proposed models with delay and Markovian switching for ecological populations. We can use the telegraph (or colored) noise to describe the mutation of the system from one state to another. Indeed, the propagation of an epidemic is affected by external factors such as nutrition, pandemics, climate, etc. For example, the transmission rate of the influenza epidemic in the spring is not the same as in the summer. Thus, the system frequently transitions from one regime to another. Many scholars studied the effect of telegraphic noise on the transmission of an epidemic (see, for example, [18–23]).To develop the model (1) and the stochastic model proposed by Rajasekar et al. [14], we suppose that the transmission rate of disease β fluctuates randomly around an average value and that the system often switches between two or more environmental regimes.We assume that the stochastic perturbations are of the white and telegraphic noises types, that is,(2)β⇝βrt+σrtdWt,where Wt is a Brownian motion and σ is the intensity of noise, rt is a right-continuous Markov chain.Then, we present a new stochastic epidemic model for a human population with a saturated incidence rate and saturated treatment function in regime-switching described by the following stochastic hybrid differential equations:(3)dSt=Λrt−ρrt+mrtSt−βrtStIt1+krtIt+mrtSt−τ1e−ρrtτ1+μrtIt−τ2e−ρrtτ2dt−σrtStIt1+krtItdWt,dIt=βrtStIt1+krtIt−ρrt+θrt+μrtIt−r¯rtIt1+ηrtItdt+σrtStIt1+krtItdWt,dRt=μrtIt+r¯rtIt1+ηrtIt−ρrtRt−mrtSt−τ1e−ρrtτ1−μrtIt−τ2e−ρrtτ2dt,with the initial value(4)Sθ=φ1θ≥0,Iθ=φ2θ≥0,Rθ=φ3θ≥0,θ∈−τ,0,φiθ>0,i=1,2,3,where φ1θ,φ2θ,φ3θ∈C−τ,0;ℝ+33 with C−τ,0;ℝ+3 is the Banach space of continuous functions, mapping the interval −τ,0 into ℝ+3, and τ=maxτ1,τ2.The parameterm stands for the successful vaccination rate of susceptible individuals. The quantity μIt−τ2e−ρτ2 represents the case that an individual survives natural death before reverting to the susceptible class, τ2 is the length of immunity period of the recovered. The quantity St−τ1 indicates the period in which the vaccine has not yet given its effect, and τ1 is the validity period of the vaccination.Throughout this paper,Wt is a Brownian motion defined in a complete probability space Ω,ℱ,ℱtt≥0,ℙ with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right-continuous while ℱ0 contains all ℙ-null sets), σ is the intensity of noise. rt is a right-continuous Markov chain defined in the same space as the Brownian motion, and supposedly independent of the Brownian motion. The Markov chain rt takes values in a finite-state space V=1,2,…,N with the generator Ψ=φuv1≤u,v≤N defined for T>0 by(5)ℙrt+T=v|rt=u=ϕuvT+oTϕ,ifu≠v,1+ϕuuT+oT,ifu=v,where φuv is the transition rate from u to v and φuv≥0 if u≠v, while Tex translation failed. To ensure that the system goes switching from one regime to another regime. In the present paper, we suppose that the Markov chain is irreducible. Then, there exists a unique stationary distribution π=π1,…,πm of rt such that πΨ=0, with ∑i=1Nπi=1 and πi>0, for each i in V. Therefore, for any sequence g=g1,…,gNT, let g^=mini∈Vgi and gˇ=maxi∈Vgi.The stochastic differential equations with Markovian switching [24] have the following form:(6)dxt=fxt,rtdt+hxt,rtdWt,t≥0,withx0=x0∈ℝn,where f:ℝn×V⟶ℝn, and h:ℝn×V⟶ℝn×m. We present the operator ℒ associated with equation (6) and for any function Mx,e twice continuously differentiable for x by(7)ℒMx,e=Mtx,e+Mxx,efx,e+12tracehTx,eMxxx,ehx,eij+∑i∈VφeiMx,e,where(8)Mt=∂M∂t,Mx=∂M∂x1,…,∂M∂xn,Mxx=∂2M∂xi∂xjn×n.Then the generalized Itô’s formula [25] is displayed by(9)dMxt,e=ℒMxt,edt+Mxxt,ehxt,edWt.Theorem 1.
For any initial value defined in (4), the model (2) has a unique solution St,It,Rt for all t≥−τ and the solution will remain in ℝ+3 with probability one.With the same step in [11], we can prove the above theorem, and the following region is almost surely positively invariant [12].(10)Y=St,It,Rt∈ℝ+3:K≤St+It+Rt≤Λˇρ^.Our model represents many effects compatible with the natural word as(i)
The memory effect represents an essential component of the natural population [26–28].(ii)
The regime-switching. In fact, the epidemic disease often switches between two or more different regimes of the environment [18, 19]. So, integrating the colored noise into the epidemic system allows the latter to switch between different environmental regimes. Often, the switching between environmental regimes is usually memoryless. In addition, the waiting time for the ensuing switch follows the exponential distribution [25]. Then, we model the regime-switching by a Markov chain rtt≥0 taking value in a finite-state space V=1,2,…,N.(iii)
The vaccination represents an intervention medical that plays a significant role in the reduction of the infected class [29].This paper aims to study the extinction of disease. By proposing a threshold that includes the noise term of the stochastic system, we establish sufficient conditions for the extinction of disease. Moreover, our model (2) can be used to represent the switching between two or more regimes of environment, which differ by factors such as nutrition or socio-cultural factors. In addition, the model (2) represents a generalized form of many SIR epidemic models existing in the literature. The rest of this paper is as follows: in Section2, we prove the sufficient conditions for the extinction of the disease. In Section 3, we make simulations to confirm our theoretical results.


## 2. Extinction

The main question in mathematical epidemiology is the determination of the conditions that ensure the disappearance of an epidemic in a population. For the deterministic system, the answer to this question is made when the value ofℛ0 (the basic reproduction number [30]) is less than or equal to one. In general, the value of ℛ0 depends on the parameters of the deterministic system. So, for our stochastic delayed system, we have defined a threshold value that depends on the delay and the random effects defined by(11)ℛswh=λˇΛˇρ^+m^1−e−ρ^τ2χ,where χ=∑i=1Nπiρi+θi+μi+r¯iκ1+σ2i/2κ2. The following theorem presents a condition for the extinction of the disease in model (2) in the function of the threshold value ℛswh.Theorem 2.
LetSt,It,Rt be a positive solution of system (2) with the initial value given in (4), if ℛswh<1, then(12)limsupt⟶∞lnItt≤∑i=1Nπiρi+θi+μi+r¯iκ1+σ2i2κ2ℛswh−1,a.s.
In other words,It will go to zero exponentially, namely, the disease dies out with probability one.Proof.
Applying Itô’s formula tolnIt yields(13)dlnIt=λrtSt1+krtIt−ρrt+θrt+μrt−r¯rt1+ηrtIt−σ2rtS2t1+krtIt2dt+σrtSt1+krtItdWt.
By (7), we get(14)dlnIt≤λrtSt−ρrt+θrt+μrt+r¯rtκ1+σ2rt2κ2dt+σrtSt1+krtItdWt.
Integrating both sides of the above inequality from 0 tot and dividing by t, we have(15)lnItt≤λˇ1t∫0tSsds−1t∫0tρrs+θrs,(16)+μrs+r¯rsκ1+σ2rs2κ2dt−lnI0t+ℳtt,where ℳt=∫0tσrsSs/1+krsIsdWs is a local martingale with quadratic variation expressed as(17)ℳt,ℳt=∫0tσrsSs1+krsI2ds≤σ22ct.
By the strong law of large numbers for local martingales (see, [31]), we result(18)limt⟶∞ℳtt=0a.s.
On the other hand, in view of the system (2), we derive(19)dSt+It+μrte−ρrtτ2∫t−τ2tIsds+mrte−ρrtτ1∫t−τ1tSsds=Λ−ρrt+θrt+μrt1−e−ρrtτ1+r¯rt1+ηrtItIt−ρrt+mrt1−e−ρrtτ2Stdt.
Thus, we have(20)St+It+μrte−ρrtτ2∫t−τ2tIsdst+mrte−ρrtτ1∫t−τ1tSsdst−S0+I0+μr0e−ρr0τ2∫−τ20Isdst−mr0e−ρr0τ1∫−τ10Ssdst=Λrt−ρrt+θrt+μrt1−e−ρrtτ1+r¯rt1+ηrtIt1t∫0tIsds−ρrt+mrt1−e−ρrtτ21t∫0tSsds≤Λˇ−ρ^+θ^+μ^1−e−ρ^τ1+r^κ11t∫0tIsds−ρ^+m^1−e−ρ^τ21t∫0tSsd.
Consequently, we have(21)1t∫0tSsds≤Λˇρ^+m^1−e−ρ^τ2−ρ^+θ^+μ^1−e−ρ^τ1+r¯^κ1ρ^+m^1−e−ρ^τ21t∫0tIsds.
By substituting the inequality (10) in (16), we find(22)lnItt≤λˇΛˇρ^+m^1−e−ρ^τ2−λˇρ^+θ^+μ^1−e−ρ^τ1+r¯^κ1ρ^+m^1−e−ρ^τ21t∫0tIsds−1t∫0tρrs+θrs+μrs+r¯rsκ1+σ2rs2κ2dt−lnI0t+ℳtt≤λˇΛˇρ^+m^1−e−ρ^τ2−1t∫0tρrs+θrs+μrs+r¯rsκ1+σ2rs2κ2dt−lnI0t+ℳtt.
Taking the limit superior on both sides of (22) and combining with (18), using the ergodic theory of the Markov chain, we obtain(23)limsupt⟶∞lnItt≤λˇΛˇρ^+m^1−e−ρ^τ2−∑i=1Nπiρi+θi+μi+r¯iκ1+σ2i2κ2≜∑i=1Nπiρi+θi+μi+r¯iκ1+σ2i2κ2ℛswh−1a.s.
Ifℛswh<1, we obtain that(24)limsupt⟶∞lnItt≤0,a.s,which implies that(25)limt⟶∞It=0a.s.
However, the diseases in the system (2) die out exponentially with probability one.Remark 1.
Through Theorem2, we find an impressive result, namely, when ℛswh is less than one infective It go to extinction. Thus, the evanishment of the disease depends on the delay and the noise value.


## 3. Numerical Simulations

In this part, we present some numerical simulations in order to support our analytical results. For this, we use the approximate numerical resolution method for the stochastic differential equations of Euler and Maruyama. Many authors have chosen this method to give a numerical approximation of solutions for some stochastic differential equations. Then, we consider the Markov chainrtt≥0 taking values on the state space V=1,2, with a generator defined by(26)Φ=−112−2.and the stationary distribution π=2/3,1/3. Figure 1 demonstrates the path of the Markov chain rtt≥0.Figure 1
The trajectory of the Markov chainrt.Therefore, to simulate, we take the parameter values presented in Table2. Next, a simple computation gives(27)ℛswh=λˇΛˇρ^+m^1−e−ρ^τ2χ=0.3515<1.Table 2
Parameters value used in simulation of model (2).
ParameterState 1State 2Λ0.250.3ρ0.50.6δ0.60.5η0.60.4k0.50.4m0.40.2θ0.3 70.4μ0.20.12τ10.10.2τ20.50.6Then by Theorem2, for any initial (3), the solution of (3) satisfies(28)limsupt⟶∞lnItt≤0,a.s.Thus, the disease in model (2) is extinct, Figure 2 supports this result.Figure 2
The stochastic trajectory ofIt.By choosing different noise values as Case 1:σ1=0.1, σ1=0.1, Case 2: σ1=0.2, σ1=0.3, and Case 3: σ1=0.4, σ1=0.6, according to Figure 3, we see that when the value of noises increases, It tends more rapidly to zero, then we conclude that a large value of noise leads to the disappearance of the epidemic from the population.Figure 3
The stochastic trajectory ofIt with different noise values.In order to examine the effect of vaccination immunity on the dynamics of an epidemic in the model (2), we keep the same parameter values in Table 1, and we vary the value of the period immunity τ1. So, we see that when the period of validity of the vaccination is sufficiently large, the epidemic disease can be decreased in the population (see, Figure 4).Figure 4
The effects of the validity periodτ1 to the system (2).


## 4. Conclusion

In this paper, we have proposed a stochastic epidemic model with Markov switching. Our model represents a generalization of many models existing in the literature. For example, the stochastic model proposed by Rajasekar et al. in [14], when m=0, the regime-switching and delay are not considered. Also, using the stochastic Lyapunov approach, sufficient conditions are established to guarantee the extinction of epidemic disease by introducing a suitable stochastic threshold value ℛswh, which depends on delay terms and stochastic noise. We have obtained exactly the following results:(i)
LetSt,It,Rt be a positive solution of system (2) with the initial value given in (4), if(29)ℛswh=λˇΛˇρ^+m^1−e−ρ^τ2χ,<1,then epidemic infection dies out with probability one.Then, from the expression ofℛswh, we remark that a large value of white noise is helpful to control the propagation of infectious diseases (see Figure 3). In addition, the long period of vaccination validation can reduce the number of infected people in the population (see Figure 4). In our future works, we can study the memory effect on the dynamics of model (2) by using the new generalized fractional derivative presented by Hattaf in [32].



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