# Analysis of a Stochastic SIR Model with Vaccination and Nonlinear Incidence Rate

Authors: Amine El Koufi; Jihad Adnani; Abdelkrim Bennar; Noura Yousfi
Journal: International Journal of Differential Equations
 (2019)

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## Abstract

We expand an SIR epidemic model with vertical and nonlinear incidence rates from a deterministic frame to a stochastic one. The existence of a positive global analytical solution of the proposed stochastic model is shown, and conditions for the extinction and persistence of the disease are established. The presented results are demonstrated by numerical simulations.


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## Body

## 1. Introduction

Mathematical models play an important role in the analysis and control of infectious diseases, and thus effective measures can be taken to reduce its transmission as much as possible. The study of mathematical models in epidemiology has received much attention from many scientists, and some novel results are obtained [1–4]. Recently, stochastic analysis has widely been applied in mathematical modeling in biology [5–9].For compartmental mathematical models, the total population is divided into three classes, namely, susceptible populationS(t), infected population I(t), and recover population R(t). For more details, see [10]. For some diseases, such as AIDS, rubella, varicella, hepatitis B, hepatitis, syphilis, and mumps, it is one of the main transmission modes that infected mothers infect their unborn or newborn offsprings, called vertical transmission [11]. Meng and Chen [12] proposed an SIR epidemic model with vaccination and vertical transmission mode as follows:(1)dSdt=−βSI−bS+1−mpdI+b1−mS+R,dIdt=βSI−pd+rI,dRdt=rI−bR+dmpI+mbS+R,where b is the mortality rate in the susceptible and the recovered individuals and d is the mortality rate in the infective individuals. The constants p and q (p+q=1) are vertical transmission rates, namely, p and q are, respectively, the proportion of the offspring of infective parents that are susceptible individuals and the rest are born infected. The arrival of newborns constitutes a recruitment rate of bS+R into the susceptible individuals and qdI into the infectious individuals. m0≤m≤1 is the successful vaccination proportion to the newborn from S and R, r is the recover rate in the infective individuals into recovered individuals, and β is the contact rate. System (1) has a basic reproduction number R0 defined by(2)R0=1−mβpd+r.On the contrary, environmental fluctuations have great influence on all aspects of real life. The aim of this work is to study the effect of these environmental fluctuations on the transmission rateβ. We assume that the stochastic perturbations are of white noise type, that is, β⟶β+σB˙t, where Bt is a Brownian motion and σ is the intensity. Then, the stochastic version corresponding to the deterministic model (1) with general incidence rate is as follows:(3)dS=−βSI1+α1S+α2I+α3SI−bS+1−mpdI+b1−mS+Rdt−σSI1+α1S+α2I+α3SIdBt,dI=βSI1+α1S+α2I+α3SI−pd+rIdt+σSI1+α1S+α2I+α3SIdBt,dR=rI−bR+dmpI+mbS+Rdt,where Bt is the independent standard Brownian motions defined on 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).g
S
,
I
=
β
S
I
/
1
+
α
1
S
+
α
2
I
+
α
3
S
I is the incidence rate, where α1,α2,α3≥0. In addition, the incidence rate is the number of new infected situations by population in a determined time period. To model the disease transmission process, several authors employ the following bilinear incidence rate βSI, where β is a positive constant [13]. Yet, there exist many forms of nonlinear incidence rate and every form presents some advantages [14–16]. It is very important to note that gS,I=βSI/1+α1S+α2I+α3SI is a general form which represents mutual interference between S and I. in particular cases:(1)
Ifα1=α2=α3=0, gS,I becomes a bilinear incidence rate(2)
Ifα1=α2=0 or α1=α3=0, gS,I becomes a saturated incidence rate [17](3)
Ifα3=0, gS,I becomes a Beddington–DeAngelis functional response [18](4)
Ifα3=α1α2, gS,I becomes the Crowley–Martin functional response presented in [19]Next, we consider thed-dimensional stochastic system:(4)dxt=fxt,tdt+gxt,tdBt,where fx,t is a function in IRd defined in t0,+∞ and gx,t is a d×m matrix, and f and g are locally Lipschitz functions in x. Btt≥0 is a d-dimensional standard Wiener process defined on the above probability space.Denote byC2,1IRd×t0,+∞;IR+ the family of all nonnegative functions Ux,t defined on IRd × t0,+∞ such that they are continuously twice differentiable in x and once in t. The differential operator ℒ [16] associated with (4) is defined by(5)ℒ=∂∂t+∑i=1dfix,t.∂∂xi+12∑i,j=1dgTx,tgx,tij.∂∂xi∂xj.If the differential operatorℒ acts on a function U∈C2,1IRd×t0,+∞;IR+, then(6)ℒU=Utx,t+Uxx,tfx,t+12Tracgx,tTUxxx,tgx,t,where Utx,t=∂U/∂t, Uxx,t=∂U/∂x1,…,∂U/∂xd, and Uxxx,t=∂2U/∂xi∂xj.The organization of this paper is as follows. In Section2, we show the existence of unique positive global solution to the given SDE system. Extinction and persistence in mean results are explored in Section 3 and Section 4, respectively. In Section 5, the analytical results are illustrated with the support of numerical examples. Finally, we close the paper with conclusion and future directions.


## 2. Existence and Uniqueness of the Nonnegative Solution

As we are dealing with the population model, the positive solution of the model is of our interest. The coefficients of (3) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (3) may explode at a finite time. The following theorem shows that the solution is positive and will not explode at a finite time.We define a subsetΔ of IR3 as follows:(7)Δ=x,y,z∈IR+3:x+y+z=1.Theorem 1.
For any given initial valueX0=S0,I0,R0∈Δ, there is a unique positive solution Xt=St,It,Rt of (3) on t≥0, and the solution will remain in Δ with probability 1, namely, St,It,Rt∈Δ for all t≥0 almost surely (briefly a.s.).Proof 1.
Since the coefficients of system (3) are locally Lipschitz continuous, for any initial value S0,I0,R0∈Δ, there is a unique local solution on 0,τe, where τe is the explosion time. To show this solution is global, we need to show that τe=∞a.s. For this, we define the stopping time τ by(8)τ=inft∈0,τe:St≤0,orIt≤0,orRt≤0,with the traditional setting inf∅=∞ (as usual ∅ denotes the empty set). We have τ≤τe. If we can show that τ=∞ a.s., then τe=∞ a.s. and St,It,Rt∈Δ for all t≥0. In other words, to complete the proof, all we only need to show is that τ=∞. Assume that this statement is false, then there exists a T>0 such that Pτ<T>0. Define a C2 function U, by the expression(9)US,I,R=−lnSIR.
Using Itô’s formula, we get(10)dUXt=−1S−βSIfS,I−bS+1−mpdI+b1−mS+Rdt−1IβSIfS,I−pd+rIdt−1RrI−bR+dmpI+mbS+Rdt+12σIfS,I2+12σSfS,I2dt+σIfS,I−σSfS,IdBt.
SincefS,I≥1, and S≤1, I≤1, we get,(11)dUXt≤Hdt+σI−SfS,IdBt,where H=β+2b+pd+r+σ2. Then, we have(12)UXt≤UX0+∫0tHds+∫0tσIs−SsfSs,IsdBs.
Note that some components ofXτ equal 0; thus, limt⟶τUXt=+∞.
Lettingt⟶τ in (12), we obtain(13)+∞≤UX0+∫0τHds+∫0τσIs−SsfSs,IsdBs<+∞,which contradicts our assumption. Then, τ=∞ a.s. This completes the proof of theorem.


## 3. Extinction

In the following, we give a condition for the extinction of the disease. Let(14)Rs=R01−m−σ221+α11−m2pd+r.Theorem 2.
LetSt,It,Rt be the solution of system (3) with initial value S0,I0,R0∈Δ. Assume that(i)
σ2>β2/2pd+r or(ii)
Rs<1andσ2<β
Then,(15)limsupt⟶+∞lnItt≤β22σ2−pd+r<0a.s.ifiholds,limsupt⟶+∞lnItt≤R˜s−1pd+r<0a.s.ifiiholds.Namely,It tends to zero exponentially a.s., i.e., the disease dies out with probability 1.Proof 2.
Applying Itô’s formula to system (3) leads to(16)dlnIt=βSfS,I−pd+r−σ2S22f2S,Idt+σSfS,IdBt.
Integrating both sides of (16) from 0 to t, we get(17)lnIt=−σ22∫0tSfS,I−βσ22du+β22σ2t−pd+rt+Mt+lnI0≤β22σ2−pd+rt+Mt+lnI0,where Mt=∫0tσSu/fSu,IudBu, which is a local continuous martingale, and M0=0. Moreover, its quadratic variation is(18)M,Mt=∫0tσSufSu,Iu2du≤σ2ta.s.
By the large number theorem for martingales [21], we obtain(19)limt⟶+∞Mtt=0a.s.
If condition (i) is satisfied, dividing byt and taking the limit superior of both sides of (17), we get(20)limsupt⟶+∞lnItt≤β22σ2−pd+r<0a.s.
If the condition (ii) is satisfied, note that(21)βS1+α1S+α2I+α3SI=β1+α1−β1−S1+α11+α1S+α2I+α3SI−βα2I1+α11+α1S+α2I+α3SI−βα3SI1+α11+α1S+α2I+α3SI≤β1+α1≤β1+α11−m.
It follows that(22)lnItt=∫0tβSfS,I−pd+r−σ2S22f2S,Idu+Mtt+lnI0t≤β1+α11−m−pd+r−σ221+α11−m2+Mtt+lnI0t=pd+rβ1+α11−mpd+r−σ221+α11−m2pd+r−1+Mtt+lnI0t=pd+rRs−1+Mtt+lnI0t.
Taking the superior limit of both sides of (22), we obtain(23)limsupt⟶+∞lnItt≤pd+rRs−1<0a.s.,which implies that limt⟶+∞It=0.Remark 1.
From Theorem 1, we can get that the disease will die out ifRs<1, and the white noise is not large such that σ2<β, while if the white noise is large enough such that condition (i) is satisfied, then the infectious disease of system (3) goes to extinction almost surely.


## 4. Persistence

Here, we investigate the condition for the persistence of the disease. The basic reproduction numberR0(see [20]), is the threshold between disease extinction and persistence, with extinction for R0≤1 and persistence for R0>1 in the deterministic model. In the stochastic model, we define the threshold of persistence for disease as(24)Rs∗=1−mR0−σ221+α11−m21−mpd+r.Definition 1.
System (3) is said to be persistent in the mean if(25)liminft⟶∞1t∫0tIudu>0a.s.
We define(26)xt=1t∫0txudu.Theorem 3.
IfRs∗>1, then the solution St,It,Rt of system (3) with initial initial value S0,I0,R0∈Δ is persistent in mean. Moreover, we have(27)liminft⟶∞It≥1+α11−mpd+rχRs∗−1a.s.,where χ=β1−mr+dmp/b1−m+α2+α3pd+r/β1−m.Proof 3.
We have(28)St+It+Rt=1.
Integrating from 0 tot and dividing by t>0 the third equation of system (3), we get(29)Rt−R0t=mbSt+r+dmpIt−b1−mRt.
From (28), one can get(30)St=−r+dmpb+1−mIt+1−m+φtb,where φt=Rt−R0/t. From Itô’s formula and (3), we obtain(31)d1+α11−mlnIt+α2+α3I=1+α11−mβS1+α1S+α2I+α3SI−1+α11−mpd+r−σ21+α11−mS221+α1S+α2I+α3SI2dt+α2+α3βSI1+α1S+α2I+α3SI−pd+rI+1+α11−mσS1+α1S+α2I+α3SIdB+α2+α3σSI1+α1S+α2I+α3SIdB≥1+α11−mβS1+α1+α2I+α3I−1+α11−mpd+r−σ21+α11−mS221+α1S+α2I+α3SI2dt,α2+α3βSI1+α1+α2I+α3I−pd+rI+1+α11−mσS1+α1+α2I+α3IdB+α2+α3σSI1+α1+α2I+α3IdB≥β1−mS−1+α11−mpd+r−α2+α3pd+rI−σ221+α11−mdt+σ1−mSdB.
Integrating from 0 tot and dividing by t>0 on both sides of yields(32)1+α11−mlnIt−1+α11−mlnI0t+α2+α3I−α2+α3I0t≥β1−mS−1+α11−mpd+r−α2+α3pd+rI−σ221+α11−mdt+Mtt≥β1−m−r+dmpb+1−mIt+1−m+φtb−α2+α3pd+rI−σ221+α11−m+Mtt−1+α11−mpd+r−α2+α3pd+r=−β1−mr+dmpb+1−m+α2+α3pd+rβ1−mIt+β1−m2+β1−mφtb−1+α11−mpd+r−σ221+α11−m+Mtt.
We can rewrite the inequality (32) as(33)It≥1χβ1−m2−1+1−mα1pd+r−σ221+α11−m+1χβ1−mφtb+Mtt−1+α11−mlnIt−1+α11−mlnI0t−α2+α3It−α2+α3I0t,where χ=β1−mr+dmp/b−1+m+α2+α3pd+r/β1−m. We can see that Rt≤1 and It≤1. Thus, one has limt⟶+∞Rt/t=0, limt⟶+∞It/t=0, and limt⟶+∞φt=0. Taking the inferior limit of both sides of (33) yields(34)liminft⟶+∞It≥1χβ1−m2−1+α11−mpd+r−σ221+α11−m≥1+α11−mpd+rχβ1−m21+α11−mpd+r−σ221+α11−m2pd+r−1=1+α11−mpd+rχRs∗−1.
This completes the proof of theorem.


## 5. Numerical Simulations

In order to illustrate our theoretical results, we give some numerical simulations. The values ofm, σ, and β will be varied over the different examples.Example 1.
We choose the parameters in system (3) as follows:(35)m=0.9,β=0.6,p=0.1,b=0.2,d=0.4,r=0.2,σ=0,α1=0.6,α2=0.1,α3=0.1.
By calculation, we haveR0=0.4902; in this case, the disease dies out as shown in Figure 1(a). By choosing m=0.3, we obtain R0=1.635, and we deduce that the disease persists in the population. Figure 1(b) illustrates this result.Figure 1
Simulations of the pathS(t), I(t), and R(t) for the corresponding deterministic system (1).
(a)
(b)Example 2.
We choose the parameters in system (3) as follows:(36)m=0.2,β=0.8,p=0.5,b=0.2,d=0.4,r=0.2,σ=0.9,α1=0.6,α2=0.1,α3=0.1.
A simple computation shows that(37)Rs=0.9965<1,R0=1.0811,σ2=0.6400>0.1280=β22pd+r.
It follows that the condition of Theorem 2 is satisfied. We conclude that the disease dies out; Figure2 illustrates this result.Figure 2
Simulations of the path (a)S(t), (b) I(t), and (c) R(t) for the deterministic system (1) and the corresponding stochastic system (3).
(a)
(b)
(c)Example 3.
On the contrary, we choosem=0.05 and σ = 0.1 by simple calculation, and it can be found that Rs∗=1.2102>1 which implies that the disease persists (Figure 3).Figure 3
Simulations of the path (a)S(t), (b) I(t), and (c) R(t) for the deterministic system (1) and the corresponding stochastic system (3).
(a)
(b)
(c)


## 6. Conclusion

This article discusses a stochastic SIR epidemic model with vertical transmission and vaccination and nonlinear incidence rate. We have shown that when the noise is so small such thatσ2<β, the extinction of the disease can be determined by the value of Rs, i.e., if Rs<1, the disease dies out. Moreover, the disease dies out when the white noise is large enough such that σ2>β2/2pd+r.The persistence of the disease is determined byRs∗, i.e., if Rs∗>1, the disease persists. We presented some numerical simulations to illustrate the obtained analytical results. To go further in this study, we can give a new dimension to the stochastic SIR epidemic model (3) by introducing a different type of noise which is the random telegraph noise (RTN), modeled by the Markov chain. We will investigate this case in our future works.



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