# Global Analysis of a Novel Nonlinear Stochastic SIVS Epidemic System with Vaccination Control

Authors: Xiaona Leng; Tao Feng; Xinzhu Meng
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper proposes a stochastic SIVS epidemic system with nonlinear saturated infection rate under vaccination and investigates the dynamics predicted by the model. By using Itô’s formula and Lyapunov methods, we first study the existence and uniqueness of global positive solution. Then we investigate the stochastic dynamics of the system and obtain the thresholds which govern the extinction and the spread of the epidemic disease. Results show that large stochastic noises can lead to the extinction of epidemic diseases; that is, stochastic disturbances can suppress the outbreak of epidemic diseases. Finally, we carry out a series of numerical simulations to demonstrate the performance of our theoretical findings.


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## Body

## 1. Introduction

Infectious diseases pose a serious threat to public health around the world. Therefore, the study of epidemic diseases has been part of the burning issues of scientists. Mathematically, after the pioneering work of Kermack and McKendrick on the epidemic regularity of the Black Plague in London by the famous SIR model [1], mathematical models have been used extensively by scientists to study the spread and control of diseases [2–7]. There are many ways to suppress the spread of disease, for instance, cutting off transmission routes, paying attention to food hygiene, and vaccination. Vaccination is an effective method of preventing infectious diseases and many scientists have explored the effect of vaccination on diseases [8–17]. For instance, Li and Ma [8] established an SIS epidemic model with vaccination. The model has the following form: (1)dSdt=1-qΛ-βSI-u+nS+rI+ηV,dIdt=βSI-u+r+aI,dVdt=qΛ+nSt-u+ηV.The incidence rate in model (1) is bilinear, which means that the number of people infected by a patient within a unit time is proportional to the total number of susceptible individuals in the environment. However, the number of susceptible individuals to contact with a patient within a unit time is limited, and many scholars have noted that the saturated incidence is more accurate to describe the spread of epidemic [18–21]. Besides, the WHO reports that licensed vaccines are currently available to prevent or contribute to the prevention and control of twenty-five preventable infections. But the effect of vaccination is not absolute. In other words, some people who are vaccinated still have the risk of being infected [22, 23]. Moreover, epidemic diseases may be subject to uncertain environmental disturbances, such as fluctuations of birth rate, death rate, and infection rate. These phenomena can be characterized by stochastic processes. Numerous scholars have introduced stochastic interferences into differential systems, and the stochastic dynamics of such systems were investigated [24–36].Motivated by the above works, in this paper, we suppose the following:(i)
β1, β2 (usually 0≤β2≤β1), respectively, stand for the average contact rate of an infective individual explored to a susceptible and a vaccinated individual per unit time at time t. β2=0 indicates the immune efficiency is 100%, β2=β1 indicates the immunization was completely ineffective, and 0<β2<β1 indicates the immunity is effective but not completely effective. Authors in [8] discussed the case β2=0.(ii)
The infection rates from infective individual to a susceptible and a vaccinated individual areβ1SI/(1+p1I) and β2VI/(1+p2I), respectively. Here p1,p2>0 are constants.(iii)
Environmental interference mainly affects the rate of infection, and we haveβi→βi+σiB˙i, i=1,2.Then model (1) becomes(2)dS=1-qΛ-β1SI1+p1I-u+nS+rI+ηVdt-σ1SI1+p1IdB1t,dI=β1SI1+p1I+β2VI1+p2I-u+r+aIdt+σ1SI1+p1IdB1t+σ2VI1+p2IdB2t,dV=qΛ+nS-u+ηV-β2VI1+p2Idt-σ2VI1+p2IdB2t,where S(t), I(t), and V(t), respectively, stand for the density of susceptible and infective and vaccinated individuals at time t; Λ is a constant input of new numbers into the population; q means a fraction of vaccinated newborn; u represents the natural death rate of S(t), I(t), and V(t); n is the proportional coefficient of vaccinated susceptible; r is the recovery rate of I(t); η stands for the rate of losing their immunity for vaccinated individuals; and a is the disease-caused death rate of infectious individuals I(t).Throughout this paper, let(Ω,F,{F}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). Function Bi(t)(i=1,2) is a Brownian motion defined on the complete probability space Ω, and the intensity of Bi(t) is σi(i=1,2). For an integrable function X(t) on 0,+∞, we define Xt=(1/t)∫0tX(s)ds.


## 2. The Existence and Uniqueness of Global Positive Solutions

Due to physical meaning, variablesS(t), I(t), and V(t) in model (2) should remain nonnegative for t≥0. We next prove that this is actually the case and, furthermore, the positive solution is unique.Lemma 1.
For any initial value(S(0),I(0),V(0))∈R+3, model (2) has a unique local positive solution (S(t),I(t),V(t)) in interval t∈0,τe, where τe is the explosion time.Proof.
For any initial value(S(0),I(0),V(0))∈R+3, the coefficients of system (2) are locally Lipschitz continuous. Thus model (2) has a unique local positive solution (S(t),I(t),V(t)) in interval t∈0,τe, and we complete the proof.Theorem 2.
For any initial value(S(0),I(0),V(0))∈R+3, model (2) has a unique positive solution (S(t),I(t),V(t))∈R+3 on t≥0 with probability  1.Proof.
By Lemma1, we only need to prove that τe=∞a.s. To this end, let k0>0 be a sufficiently large constant such that S(0), I(0), and V(0) all lie in [1/k0,k0]. For each k≥k0(k∈N+), define the stopping time (3)τk=inf⁡t∈0,τe:Xt∉1k0,k0,St∉1k0,k0orIt∉1k0,k0.Easily, τk is a monotonically increasing function when k→∞. Let τ∞=limk→∞⁡τk, and thus τ∞≤τea.s. Now we need to prove τ∞=∞a.s.; otherwise, there exist two constants T>0 and ϵ∈(0,1) such that P{τ∞≤T}>ϵ. Thus, there is an integer k1≥k0 such that(4)Pτ∞≤T>ϵ,k≥k1.Define a C3-function V:R+3→R+: (5)QS,I,V=S-1-ln⁡S+I-1-ln⁡I+V-1-ln⁡V.Applying Itô’s formula to stochastic differential system (2) yields (6)dQ=LQdt-σ1S-1I1+p1IdB1t+σ1I-1S1+p1IdB1t+σ2I-1V1+p2IdB2t-σ2V-1I1+p2IdB2t,where (7)LQ=1-1S1-qΛ-β1SI1+p1I-u+nS+rI+ηV+σ12I221+p1I2+1-1Iβ1SI1+p1I+β2VI1+p2I-u+r+aI+σ12S221+p1I2+σ22V221+p2I2+1-1VqΛ+nS-u+ηV-β2VI1+p2I+σ22I221+p2I2≤Λ+β1+β2I+3u+n+r+a+η+12σ12I2+S2+12σ22I2+V2.By (2), we have(8)dS+I+V=Λ-uS+I+V-aIdt≤Λ-uS+I+Vdt.Thus (9)St+It+Vt≤Λu+e-utS0+I0+V0-Λu.Therefore, for any t≤τk and each k, we have (10)St+It+Vt≤Λu,ifS0+I0+V0≤Λu,S0+I0+V0,ifS0+I0+V0>Λu.So(11)LQ≤Λ+β1+β2Λu+3u+n+r+a+η+σ12+σ22Λ2u2≤K0,where K0 is a positive constant.
Then(12)dQ≤K0dt-σ1S-1I1+p1IdB1t+σ1I-1S1+p1IdB1t+σ2I-1V1+p2IdB2t-σ2V-1I1+p2IdB2t.Integrating (12) from 0 to τk∧T and taking expectation on both sides, we have(13)EQSτk∧T,Iτk∧T,Vτk∧T≤QS0,I0,V0+K0T.Let Ωk={τk≤T}, and from inequality (4) we can see that P(Ωk)≥ϵ. Note that, for every ω∈Ωk, there exists at least one of S(τk,ω), I(τk,ω), and V(τk,ω) that equals either k or 1/k. As a result, we have(14)QSτk∧T,Iτk∧T,Vτk∧T≥k-1-ln⁡k∧1k-1-ln⁡1k.Applying (13) and (14), we get (15)QS0,I0,V0+K0T≥E1ΩkωQSτk∧T,Iτk∧T,Vτk∧T≥ϵk-1-ln⁡k∧1k-1-ln⁡1k,where 1Ωk is the indicator function of Ωk.
Whenk→∞, we have (16)∞>QS0,I0,V0+K0T=∞.This is a contradiction. So τ∞=∞ and we complete the proof.Without losing generality, in this paper, we always assume thatS(0)+I(0)+V(0)≤Λ/u.


## 3. Extinction

Theorem 3.
Let(S(t),I(t),V(t)) be the solution of model (1) with initial value (S(0),I(0),V(0))∈R+3. If one of the following conditions holds: (i)
σi≥uβi/Λ, i=1,2, and R11=β12/2σ12(u+r+a)+β22/2σ22(u+r+a)<1,(ii)
σi≤uβi/Λ, i=1,2, and R12=Λ(β1+β2)/u(u+r+a)-Λ2(σ12+σ22)/2u2(u+r+a)<1,(iii)
σ1≥uβ1/Λ, σ2≤uβ2/Λ, and R21=Λβ2/u(u+r+a)+β12/2σ12(u+r+a)-Λ2σ22/2u2(u+r+a)<1,(iv)
σ1≤uβ1/Λ, σ2≥uβ2/Λ, and R22=Λβ1/u(u+r+a)+β22/2σ22(u+r+a)-Λ2σ12/2u2(u+r+a)<1,then epidemic disease I(t) goes extinct; that is, limt→+∞⁡I(t)=0. Moreover, we have (17)limt→+∞⁡Vt=Λuuq+nu+n+η,limt→+∞⁡St=Λuu1-q+ηu+n+ηa.s.Proof.
Applying Itô’s formula toln⁡I(t), we have(18)dln⁡It=β1S1+p1I+β2V1+p2I-u+r+a-12σ12S21+p1I2-12σ22V21+p2I2dt+σ1S1+p1IdB1t+σ2V1+p2IdB2t.Integrating from 0 to t and dividing both sides of (18) by t, one has(19)ln⁡Itt=1t∫0tβ1Ss1+p1Is+β2Vs1+p2Is-u+r+a-12σ12S2s1+p1Is2-12σ22V2s1+p2Is2ds+M1tt+M2tt+ln⁡I0t=1t∫0t-σ122Ss1+p1Is-β1σ122+β122σ12-σ222Vs1+p2Is-β2σ222+β222σ22-u+r+ads+M1tt+M2tt+ln⁡I0t,where (20)M1t=∫0tσ1Ss1+p1IsdB1s,M2t=∫0tσ2Vs1+p2IsdB2sare real-valued continuous martingales.
Thus(21)limsupt→∞⁡M1,M1tt=limsupt→∞⁡1t∫0tσ12S2s1+p1Is2ds≤σ12Λ2u2<+∞,limsupt→∞⁡M2,M2tt=limsupt→∞⁡1t∫0tσ22V2s1+p2Is2ds≤σ22Λ2u2<+∞.By the strong law of large numbers [37], we have (22)limt→∞⁡M1tt=0,limt→∞⁡M2tt=0.Case  1. When σi≥uβi/Λ, i=1,2, and R11<1, applying (19) we have(23)ln⁡Itt≤β122σ12+β222σ22-u+r+a+M1tt+M2tt+ln⁡I0t=u+r+aβ122σ12u+r+a+β222σ22u+r+a-1+M1tt+M2tt+ln⁡I0t=u+r+aR11-1+M1tt+M2tt+ln⁡I0t.Taking the limit superior on both sides of (23), we have (24)limsupt→∞⁡ln⁡Itt≤u+r+aR11-1<0.
Thus(25)limt→∞⁡It=0.Case  2. When σi≤uβi/Λ, i=1,2, and R12<1, applying (19) we have(26)ln⁡Itt≤Λβ1+β2u-Λ2σ12+σ222u2-u+r+a+M1tt+M2tt+ln⁡I0t=u+r+aΛβ1+β2uu+r+a-Λ2σ12+σ222u2u+r+a-1+M1tt+M2tt+ln⁡I0t=u+r+aR12-1+M1tt+M2tt+ln⁡I0t.Taking the limit superior on both sides of (26), we have (27)limsupt→∞⁡ln⁡Itt≤u+r+aR12-1<0.Thus (28)limt→∞⁡It=0.Case  3. When σ1≥uβ1/Λ, σ2≤uβ2/Λ, and R21<1, applying (19) we have(29)ln⁡Itt≤β122σ12+Λβ2u-Λ2σ222u2-u+r+a+M1tt+M2tt+ln⁡I0t=u+r+aβ122σ12u+r+a+Λβ2uu+r+a-Λ2σ222u2u+r+a-1+M1tt+M2tt+ln⁡I0t=u+r+aR21-1+M1tt+M2tt+ln⁡I0t.Taking the limit superior on both sides of (29), we have (30)limsupt→∞⁡ln⁡Itt≤u+r+aR21-1<0.Thus (31)limt→∞⁡It=0.Case  4. When σ1≤uβ1/Λ, σ2≥uβ2/Λ, and R22<1, applying (19) we have(32)ln⁡Itt≤Λβ1u-Λ2σ122u2+β222σ22-u+r+a+M1tt+M2tt+ln⁡I0t=u+r+aΛβ1uu+r+a-Λ2σ122u2u+r+a+β222σ22u+r+a-1+M1tt+M2tt+ln⁡I0t=u+r+aR22-1+M1tt+M2tt+ln⁡I0t.Taking the limit superior on both sides of (32), we have (33)limsupt→∞⁡ln⁡Itt≤u+r+aR22-1<0.Thus (34)limt→∞It=0.Applying (8) we have (35)St+It+Vt=e-utS0+I0+V0+∫0tΛ-aIseusds.Applying L’Hospital rule, one has(36)limt→∞St+Vt=limt→∞S0+I0+V0+∫0tΛ-aIseusdseut-It=limt→∞⁡Λ-aItu=Λua.s.Applying (36) and limt→+∞I(t)=0, there exist two arbitrarily small constants ε1,ε2>0 such that when t>T, we have (37)It≤ε1,St+Vt≤Λu+ε2,and so we have (38)dV=qΛ+nS-u+ηV-β2VI1+p2Idt-σ2VI1+p2IdB2t≤qΛ+nS+V-n+u+ηVdt+ε1σ2V1+p2ε1dB2t≤qΛ+nΛu+ε2-n+u+ηVdt+ε1σ2V1+p2ε1dB2t.Letting t→+∞ and applying the arbitrariness of ε1 and ε2, we obtain (39)limt→+∞⁡Vt≤Λuuq+nu+n+ηa.s.Similarly, we have (40)limt→+∞⁡St≤Λuu1-q+ηu+n+ηa.s.Then, applying (36), we have (41)limt→+∞⁡Vt=Λuuq+nu+n+η,limt→+∞⁡St=Λuu1-q+ηu+n+ηa.s.This completes the proof.


## 4. Persistence in Mean

Theorem 4.
Let(S(t),I(t),V(t)) be a solution of model (2) with initial value (S(0),I(0),V(0))∈R+3. If condition (42)R2=Λβ2uu+r+a-Λ2σ12+σ222u2u+r+a>1holds, then epidemic disease I(t) is persistence in mean; that is, I∗≥limsupt→∞〈I(t)〉≥liminft→∞〈I(t)〉>I∗.Proof.
Integrating from0 to t and dividing both sides of (2) by t, we have(43)St+Vt=Λu-u+auIt-1utSt-S0+It-I0+Vt-V0.
Choosepˇ=max⁡{p1,p2}, and then, applying Itô’s formula to ln⁡I(t)+pˇI(t), we have(44)dln⁡It+pˇIt=β1S1+pˇI1+p1I+β2V1+pˇI1+p2I-u+r+a-pˇu+r+aI-12σ12S21+p1I2-12σ22V21+p2I2dt+σ1S1+pˇI1+p1IdB1t+σ2V1+pˇI1+p2IdB2t≥β1S+β2V-pˇu+r+aI-u+r+a-12σ12S2-12σ22V2dt+σ1S1+pˇI1+p1IdB1t+σ2V1+pˇI1+p2IdB2t.Integrating from 0 to t and dividing both sides of (44) by t, one has(45)ln⁡It+pˇItt≥β1St+β2Vt-u+r+a1+pˇIt-Λ22u2σ12+σ22+M3tt+M4tt+ln⁡I0+pˇI0t≥β2St+Vt-u+r+a1+pˇIt-Λ22u2σ12+σ22+M3t+M4t+ln⁡I0+pˇI0t=Λβ2u-β2u+au+pˇu+r+aIt-u+r+a-Λ22u2σ12+σ22+Φt,where (46)Φt=-β2utSt-S0+It-I0+Vt-V0+M3t+M4t+ln⁡I0+pˇI0t,M3t=∫0tσ1Ss1+pˇIs1+p1IsdB1s,M4t=∫0tσ2Vs1+pˇIs1+p2IsdB2sare local continuous martingales.
Thus(47)limsupt→∞⁡M3,M3tt=limsupt→∞⁡1t∫0tσ12S2s1+pˇIs21+p1Is2ds≤σ12Λ2u21+pˇΛu<+∞,limsupt→∞⁡M4,M4tt=limsupt→∞⁡1t∫0tσ22V2s1+pˇIs21+p2Is2ds≤σ22Λ2u21+pˇΛu<+∞.By the strong law of large numbers, we have (48)limt→∞⁡M3tt=0,limt→∞⁡M4tt=0.From (45), we obtain(49)It≥1Ω1Λβ2u-Λ22u2σ12+σ22-u+r+a+Φt-pˇItt,0<It<1;1Ω1Λβ2u-Λ22u2σ12+σ22-u+r+a+Φt-ln⁡It+pˇItt,1≤It,where (50)Ω1=β2u+au+pˇu+r+a.Hence, we have limt→∞((St+It+Vt)/t)=0 and limt→∞(ln⁡It/t)=0 as I(t)≥1 and limt→∞(Φt/t)=0.
Taking the limit inferior on both sides of (49), we have(51)liminft→∞⁡It≥u+r+aΩ1Λβ2uu+r+a-Λ2σ12+σ222u2u+r+a-1=u+r+aΩ1R2-1=I∗>0.On the other hand, choose p^=min⁡p1,p2, and, applying (44), we have(52)dln⁡It+p^It=β1S1+p^I1+p1I+β2V1+p^I1+p2I-u+r+a-p^u+r+aI-12σ12S21+p1I2-12σ22V21+p2I2dt+σ1S1+p^I1+p1IdB1t+σ2V1+p^I1+p2IdB2t≤β1S+β2V-p^u+r+aI-u+r+adt+σ1S1+p^I1+p1IdB1t+σ2V1+p^I1+p2IdB2t.Integrating from 0 to t and dividing both sides of (52) by t, one has(53)ln⁡It+p^Itt≤β1St+β2Vt-u+r+a1+p^It+M5tt+M6tt+ln⁡I0+p^I0t≤β1St+Vt-u+r+a1+p^It+M5t+M6t+ln⁡I0+p^I0t=Λβ1u-β1u+au+p^u+r+aIt-u+r+a+Ψt,where (54)Ψt=-β1utSt-S0+It-I0+Vt-V0+M5t+M6t+ln⁡I0+p^I0t,M5t=∫0tσ1Ss1+pIs1+p1IsdB1s,M6t=∫0tσ2Vs1+pIs1+p2IsdB2sare local continuous martingales.
Thus(55)limsupt→∞⁡M5,M5tt=limsupt→∞⁡1t∫0tσ12S2s1+p^Is21+p1Is2ds≤σ12Λ2u21+p^Λu<+∞,(56)limsupt→∞⁡M4,M4tt=limsupt→∞⁡1t∫0tσ22V2s1+p^Is21+p2Is2ds≤σ22Λ2u21+p^Λu<+∞.By the strong law of large numbers, we have (57)limt→∞⁡M5tt=0,limt→∞⁡M6tt=0.Applying limt→∞((St+It+Vt)/t)=0, we have limt→∞(Ψt/t)=0.
From (53), we obtain(58)It≤1Ω2Λβ1u-u+r+a+Ψt-ln⁡It+p^Itt,where (59)Ω2=β1u+au+p^u+r+a.Applying (51) yields limt→+∞⁡I(t)>0, and then limt→∞⁡(ln⁡It/t)=0.
Taking the limit superior on both sides of (58), we have(60)limsupt→∞⁡It≤u+r+aΩ2Λβ1uu+r+a-1=I∗>0.This completes the proof.


## 5. Conclusions and Numerical Simulations

This paper investigates a stochastic epidemic system with nonlinear incidence rate under vaccination. We first demonstrate the existence and uniqueness of global positive solution of model (2). Then we study the persistence in mean and extinction of the stochastic SIV system. Our result shows that large noises can lead to the extinction of epidemic diseases. In addition, notice that the basic reproduction number for epidemic disease to die out is negatively related to the intensity of interference, which indicates that stochastic impact can suppress the epidemic diseases. Compared with the existing work in [8, 10], the model constructed in this paper also considered the efficiency of vaccination. Authors in [8, 10] considered the case that the average contact rate of an infective individual explored to a vaccinated individual per unit time at time t is β2=0, and we study the more general case 0≤β2≤β1, which is more consistent with the actual situation. In addition, since the number of susceptible and vaccinated individuals to contact with a patient within a unit time is limited, in this paper, we employ a saturated incidence to describe the spread of an epidemic; that is, βiI/(1+piI), i=1,2.We next give some numerical simulations to support our results. Considering the following discrete equations:(61)Si+1=Si+1-qΛ-β1SiIi1+p1Ii-u+nSi+rIi+ηViΔt-σ1SiIi1+p1IiΔB1k,Ii+1=Ii+β1SiIi1+p1Ii+β2ViIi1+p2Ii-u+r+aIiΔt+σ1SiIi1+p1IiΔB1k+σ2ViIi1+p2IiΔB2k,Vi+1=ViqΛ+nSi-u+ηVi-β2ViIi1+p2IiΔt-σ2ViIi1+p2IiΔB2k,where Δt=0.01 and ΔBjk≜B(tk+1)-B(tk), (j=1,2,3) obeys the Gaussian distribution N(0,Δt). To this end, we set (62)q=0.2,Λ=0.3,β1=0.2,β2=0.2,u=0.1,n=0.1,r=0.2,a=0.2,η=0.1,p=1,q=1.Under the setting for plotting Figures1(b)–1(e), we have (63)limt→+∞⁡Vt=Λuuq+nu+n+η=1.2,limt→+∞⁡St=Λuu1-q+ηu+n+η=1.8.Figure 1
Time sequence diagram of model (2). (a) The deterministic model with σ1=0 and σ2=0; (b) extinction of model (2) with σ1=0.5 and σ2=0.5; (c) extinction of model (2) with σ1=0.25 and σ2=0.25; (d) extinction of model (2) with σ1=0.6 and σ2=0.2; (e) extinction of model (2) with σ1=0.2 and σ2=0.6; (f) persistence in mean of model (2) with σ1=0.1 and σ2=0.1.
(a)
(b)
(c)
(d)
(e)
(f)Figure1(a) is the time sequence diagram of model (2) with noises σ1=0 and σ2=0. In this case, the disease I(t) in model (2) is persistent. Figures 1(b)–1(e) are the time sequence diagram of model (2), satisfying conditions (i)–(iv) in Theorem 3, respectively. In Figure 1(b), δ1=δ2=0.5>0.258, R11=0.32<1, and this satisfies condition (i) of Theorem 3. In Figure 1(c), δ1=δ2=0.25<0.258, R12=0.98<1, and this satisfies condition (ii) of Theorem 3. In Figure 1(d), δ1=0.6>0.258, δ2=0.2<0.258, R21=0.95<1, and this satisfies condition (iii) of Theorem 3. In Figure 1(e), δ1=0.2<0.258, δ2=0.6>0.258, R22=0.95<1, and this satisfies condition (iv) of Theorem 3. From Figures 1(b)–1(e), the disease I(t) in model (2) is extinct. Comparing Figure 1(a) with Figures 1(b)–1(e), we find that random interference can lead to the extinct of disease. This is in line with our conclusion in Theorem 3.In Figure1(f), R2=1.02>1. In this case, 0.55=I∗≥limsupt→∞〈I(t)〉≥liminft→∞〈I(t)〉≥I∗=0.009, and the disease I(t) in model (2) is persistence in mean, and this is in line with our conclusion in Theorem 4.



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