# Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease

Authors: Olusegun Michael Otunuga
Journal: International Journal of Stochastic Analysis
 (2017)

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## Abstract

We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (R0) and stochastic (R0) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0<1), epidemic can still grow initially (if R0>1) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.


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## Body

## 1. Introduction

Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics. Among these model is the SEI susceptible-exposed-infectious model. This model is used by some author in studying disease transmission of the Severe Acute Respiratory Syndrome (SARS) disease. Several authors [1–3] have studied other models like SEIR and SEIRS to analyze the spread of the disease. Guihua and Zhen [3] studied the deterministic SEI model by providing conditions for the global asymptotic stability of the infection-free and epidemic equilibrium using the higher dimensional Poincare-Bendixson property.In this paper, we are interested in studying the effect of stochastic fluctuations in the disease transmission rates in the susceptible-exposed-infected epidemic model. We assume that a susceptible individual first goes through latent period after infection before becoming infectious. We consider a case where the disease is infectious in the latent period and the infected period. We study the role of external noise in the transmission rate. We assume the external noise is a Gaussian white noise. According to Méndez et al. [4], Langevin equations that describe system with real noise should be interpreted as a Stratonovich equation, instead of Ito equation. Due to this reason, we develop a Stratonovich stochastic dynamic SEI model by introducing noise in the transmission rates.The paper is organized as follows.In Section2, we present a Stratonovich stochastic SEI model by allowing the transmission rates to fluctuate around a mean value. The Stratonovich model is now converted into its Ito version. In Section 3, we show that the solution of the stochastic SEI model discussed in Section 2 exists and it is positive. By linearizing the Ito version of the stochastic SEI model around the infection-free equilibrium, we give a closed form expectation of the susceptible, exposed and infected. In Section 4, the closed form value for the stochastic reproductive number R0 is given. This is used to discuss and analyze the stability of the infection-free equilibrium. In Section 5, a numerical simulation is presented to verify our claim. The conclusion of the work is given in Section 6.


## 2. Stochastic SEI Model

We consider the SEI model for description of the population dynamics for SARS and other similar diseases. The host population is partitioned into three compartments: the susceptible, exposed (latent), and infectious, with sizes denoted byS, E, and I, respectively. The total population N=S+E+I. The SEI model is described by the following system of differential equation:(1)dS=-β1SE-β2SI-μS+δdt,St0=S0,dE=β1SE+β2SI-γE-μ+α1Edt,Et0=E0,dI=γE-κI-μ+α2Idt,It0=I0,where δ is the recruitment constant, β1 and β2 are rates of efficient contact in the latent period and infected period, respectively, μ>0 is the natural death rate, γ>0 is the transfer rate from the exposed to the infectious compartment, α1≥0, α2≥0 are rates of disease-caused death, and κ is the rate coefficient of segregation after disease. From (1) and the fact that the total population N=S+E+I, we have N satisfying the equation(2)dN=δ-μN-α1E-α2+κIdt.It follows from (2) that the population size N may vary with time and limt→∞⁡sup⁡N(t)≤δ/μ. Hence, we consider model (1) in the feasible region:(3)T≔S,E,I∈R+3:0≤S+E+I=N≤δμ.Here, R+ denotes nonnegative real number. It can be shown that T is positively invariant with respect to (1).The system has two equilibriums: the infection-free equilibriumP0 and the endemic equilibrium P1. The infection-free equilibrium P0≔S∗=δ/μ,E∗=0,I∗=0 exists on the boundary, ∂T, of T while the endemic equilibrium P1=S,E,I exists in the interior of T with (4)S=α2+κ+μα1+μ+γβ1α2+κ+μ+β2γ,E=α2+κ+μβ1δ-μα1+γ+μ+β2δγα1+μ+γβ1α2+κ+μ+β2γ,I=γα2+κ+μE.By settingδ=μ, we make the sizes S, E, and I into percentages. This reduces the feasible region T to(5)T≔S,E,I∈R+3:0≤S+E+I=N≤1.Let(6)R0≔β1κ+μ+α2+β2γγ+μ+α1κ+μ+α2S∗.If R0≤1, P0 is the only equilibrium in T. If R0>1, the unique endemic equilibrium P1 exists in T. Note that model (1) is similar to the model considered by Guihua and Zhen in [3]. They showed using LaSalle’s invariance principle that the disease-free equilibrium, P0, is globally asymptotically stable in T if R0≤1 and unstable if R0>1. We define R0 in (6) as the deterministic basic reproductive number.By allowing the transmission ratesβ1 and β2 to fluctuate around a mean value, we introduce external fluctuations in the model as follows: (7)βit≡βi+σiCit,i=1,2,where Ci(t), i=1,2, are independent noise term with zero mean, and σi>0, i=1,2 are noise intensity, a measure of the amplitude of fluctuation with respect to βi, i=1,2, respectively. Substituting (7) into (1), the dynamic model (1) reduces to the Stratonovich stochastic model:(8)dS=-β1SE-β2SI-μS+δdt-σ1SE∘dW1t-σ2SI∘dW2t,St0=S0,dE=β1SE+β2SI-γ+μ+α1Edt+σ1SE∘dW1t+σ2SI∘dW2t,Et0=E0,dI=γE-κ+μ+α2Idt,It0=I0,where Wi(t), i=1,2, are independent standard Wiener processes defined on stochastic basis Ω,F,Ftt≥0,P; ∘ denotes the Stratonovich integral [4, 5].We convert (8) to its Ito’s equivalent using the Stratonovich-Ito conversion (Bernardi et al. [6]) given below.Theorem 1.
The Ito Stochastic differential equation(9)dX=at,Xdt+∑j=1Mbjt,XdWjthaving the same solution as the N-dimensional Stratonovich SDE with an M-dimensional Wiener process (10)dX=a_t,Xdt+∑j=1Mbjt,X∘dWjthas drift coefficient a(t,X) that is defined in terms of a_(t,X), componentwise, by (11)ait,X=a_it,X+12∑k=1N∑j=1Mbk,jt,X∂bi,j∂xkt,X.Using Theorem1, the Ito equivalent of (8) is given by(12)dS=-β1SE-β2SI-μS+δ+12σ12SEE-S+12σ22SI2dt-σ1SEdW1t-σ2SIdW2t,St0=S0,dE=β1SE+β2SI-γ+μ+α1E-12σ12SEE-S-12σ22SI2dt+σ1SEdW1t+σ2SIdW2t,Et0=E0,dI=γE-κ+μ+α2Idt,It0=I0.


## 3. Existence and Uniqueness of Positive Solution of (12)

In this section, we show that not only does the stochastic model (12) have a unique global solution but also the solution will remain within T whenever it starts from there.Following Theorem3.5 of Khasminskii [7], we use Theorem 4 below to show the existence and uniqueness of positive solution of (12).Definition 2.
LetC1,2(D) be the family of nonnegative functions V(t,u) defined on D such that they are continuously differentiable with respect to t and twice continuously differentiable with respect to u.Definition 3.
Define the domainUn by Un=x<n. One defines the differential operator L for a function V(t,u)∈C1,2 corresponding to a stochastic differential equation with drift and diffusion coefficients A(t,u) and B(t,u), respectively, by(13)LVt,u=∂Vt,u∂t+AT∂Vt,u∂u+12tr⁡BT∂2Vt,u∂u2B,where ∂V(t,u)/∂u=∂V(t,u)/∂u1,…,∂V(t,u)/∂un and ∂2V(t,u)/∂u2=∂2V(t,u)/∂ui∂uji,j. Define R++ to be set of positive real numbers. Using the substitution E=N-S-I, one reduces (12) to model governing only S and I and applies Theorem 4 to show existence and uniqueness of positive solution of the reduced model.Theorem 4.
Suppose that (12) satisfies the classical existence and uniqueness theorem in every cylinder [a,b]×UR and, moreover, that there exists a nonnegative function V∈C1,2([0,T]×R++2→R++) such that for some constant c>0(14)LV≤cV,VR=infx>RVt,x⟶∞,asR⟶∞.Then, for every random variable X(t0)=(S(t0),I(t0)) independent of the processes W(t)-W(t0), there exists a solution X(t)=(S(t),I(t)) of the reduced stochastic differential equation (12) (with E=N-S-I) which is an almost surely continuous stochastic process and is unique up to equivalence.Proof.
It is easy to show that (12) satisfies the classical existence and uniqueness theorem in every cylinder [a,b]×UR. Define V:[0,T]×R++2→R++ by(15)VS,I=S-ln⁡S+I-ln⁡I.Note that, for any positive real number u, we have u-ln⁡(u)≥1 and u≤2(u-ln⁡u)-(2-2ln⁡(2)). It then follows that V≥2. Also, using the fact that 0≤E=N-S-I≤1, N≤1, we have(16)LV=1-1S-β1SN-S-I-β2SI-μS+δ+12σ12SN-S-IN-2S-I+12σ22SI2+1-1IγN-S-I-κ+μ+α2I+12S2σ12S2N-S-I2+σ22S2I2≤cV,where c=δ+κ+2μ+α2+2max⁡2β1+2γ,σ12/2,σ22/2.
It can be easily shown thatVn=inf|X|>n⁡V(t,X)→∞ as n→∞. The result follows. The existence of E(t)=N-S(t)-I(t) follows immediately.The fact that the solution(S,E,I) of (12) remains in T follows from Corollary 3.1 of Khasminskii [7].


### 3.1. Closed Form Expectation of Susceptible, Exposed, and Infected Population NearP0

We study the condition under which system (12) evolves into an endemic state by analyzing the endemic behavior of the linearized version of (12) around the infection-free equilibrium P0.Using the transformation(17)u=SEI-S∗00, we rewrite (12) to get the nonlinear version: (18)du1=f1t,udt-S∗+u1u2σ1dW1t-S∗+u1u3σ2dW2t,du2=f2t,udt+S∗+u1u2σ1dW1t+S∗+u1u3σ2dW2t,du3=f3t,udt,where (19)f1t,u=-β1S∗+u1u2-β2S∗+u1u3-μu1-12S∗+u12u2σ12+12S∗+u1u22σ12+12S∗+u1u32σ22,f2t,u=β1S∗+u1u2+β2S∗+u1u3-γ+μ+α1u2+12S∗+u12u2σ12-12S∗+u1u22σ12-12S∗+u1u32σ22,f3t,u=γu2-κ+μ+α2u3.The linearization of (12) around the infection-free equilibrium P0 is equivalent to the linearization of (18) around its trivial solution (u=0), given by(20)du=Audt+B1udW1t+B2udW2t,ut0=u0,where (21)A=-μ-β1+σ12S∗2S∗-β2S∗0-νβ2S∗0γ-κ+μ+α2,B1=0-σ100σ10000S∗,B2=00-σ200σ2000S∗,ν=γ+μ+α1-β1+σ12S∗2S∗.It follows that the expected value,E[u(t)], of the solution of (20) is given by(22)Eut=m1e-z1t+c1e-z2t+c2e-z3tm¯2e-z2t+m¯3e-z3tm2e-z2t+m3e-z3t,where mi=E[ui(t0)], i=1,2,3, m¯i=mi/γ(κ+μ+α2-zi), c1=β2m2+(β1+σ12S∗/2)m¯2/z2-μS∗, c2=β2m3+(β1+σ12S∗/2)m¯3/z3-μS∗,   z1=μ, zi=1/2(α2+κ+μ+ν)+(-1)i(α2+κ+μ-ν)2+4γβ2S∗, i=2,3.


## 3.1. Closed Form Expectation of Susceptible, Exposed, and Infected Population NearP0

We study the condition under which system (12) evolves into an endemic state by analyzing the endemic behavior of the linearized version of (12) around the infection-free equilibrium P0.Using the transformation(17)u=SEI-S∗00, we rewrite (12) to get the nonlinear version: (18)du1=f1t,udt-S∗+u1u2σ1dW1t-S∗+u1u3σ2dW2t,du2=f2t,udt+S∗+u1u2σ1dW1t+S∗+u1u3σ2dW2t,du3=f3t,udt,where (19)f1t,u=-β1S∗+u1u2-β2S∗+u1u3-μu1-12S∗+u12u2σ12+12S∗+u1u22σ12+12S∗+u1u32σ22,f2t,u=β1S∗+u1u2+β2S∗+u1u3-γ+μ+α1u2+12S∗+u12u2σ12-12S∗+u1u22σ12-12S∗+u1u32σ22,f3t,u=γu2-κ+μ+α2u3.The linearization of (12) around the infection-free equilibrium P0 is equivalent to the linearization of (18) around its trivial solution (u=0), given by(20)du=Audt+B1udW1t+B2udW2t,ut0=u0,where (21)A=-μ-β1+σ12S∗2S∗-β2S∗0-νβ2S∗0γ-κ+μ+α2,B1=0-σ100σ10000S∗,B2=00-σ200σ2000S∗,ν=γ+μ+α1-β1+σ12S∗2S∗.It follows that the expected value,E[u(t)], of the solution of (20) is given by(22)Eut=m1e-z1t+c1e-z2t+c2e-z3tm¯2e-z2t+m¯3e-z3tm2e-z2t+m3e-z3t,where mi=E[ui(t0)], i=1,2,3, m¯i=mi/γ(κ+μ+α2-zi), c1=β2m2+(β1+σ12S∗/2)m¯2/z2-μS∗, c2=β2m3+(β1+σ12S∗/2)m¯3/z3-μS∗,   z1=μ, zi=1/2(α2+κ+μ+ν)+(-1)i(α2+κ+μ-ν)2+4γβ2S∗, i=2,3.


## 4. Stability Analysis of Infection-Free EquilibriumP0

Using TheoremA.2 of Tornatore et al. [2], we show the global stability of the nonlinear stochastic system (12). Notice that, in order to avoid epidemic invasion, we must have zi>0, i=1,2,3. This is equivalent to (23)β1+σ122S∗κ+μ+α2S∗+β2γS∗-γ+μ+α1κ+μ+α2<0.Also (23) implies that ν>0.Define(24)R0=β1+σ12/2S∗κ+μ+α2+β2γγ+μ+α1κ+μ+α2S∗.Remark 5.
It follows that (23) is equivalent to R0<1. We can rewrite R0 in terms of R0 by combining (6) and (24) as follows: (25)R0=R0+σ12S∗22γ+μ+α1.We call the constant R0 defined in (25) the stochastic basic reproductive number.Using the following theorem, we get conditions for stochastic asymptotic stability of the infection-free equilibrium.Definition 6.
The solutionu(t)=0 of system (20) is said to be(1)
p-stable (p>0) for t≥0, if(26)supu≤n,t≥s⁡Eutp⟶0,asn⟶0,s≥0;(2)
asymptoticallyp-stable, if it is p-stable and moreover E|u(t)|p→0 as t→∞;(3)
exponentiallyp-stable, if, for some positive constants A and c, (27)Eutp≤Aupexp⁡-ct-s;(4)
whenp=1, one says stability in the mean and for p=2, one says stability in mean square.Theorem 7.
If(28)R0<1holds, then the solution u(t)=0 of (20) is globally asymptotically stable. Furthermore, if R0>1, the solution u(t)=0 is unstable.Proof.
If (28) holds, it follows from (25) that R0<1 and (22) implies limt→∞⁡E[|u(t)|]=0. If R0>1, then R0>1 and E[|u(t)|]→∞ as t→∞.Remark 8.
Contrary to Remark3.1 of Kim and Lin [1], if (29)1-σ12S∗22γ+μ+α1<R0<1,then epidemic can grow initially, leading to transient epidemic advance. Note that condition (29) implies R0<1 and R0=R0+σ12(S∗)2/2(γ+μ+α1)>1. Hence, z1>0, z2>0, and z3<0. The transient epidemic advance is caused by the noise intensity, σ1, in the rate of efficient contact in the latent period, β1.Remark 9.
Note that the global stability of the trivial solutionu=0 of (20) is equivalent to the global stability of the infection-free equilibrium P0=(S∗,0,0).The following theorem shows that the expected valueE[u(t)] in (22) does not always converge to the trivial solution u=0 of (20) if R0=1. In fact, if R0=1, then E[u(t)]→0 as t→∞ if m3=E[u3(t0)]=E[I(t0)]=0.Theorem 10.
IfR0=1, then E[u(t)] in (22) converge to the trivial solution u=0 of (20) if E[u3(t0)]=0.Proof.
From (23) and (24), R0=1 is equivalent to β1+σ12/2S∗(κ+μ+α2)+β2γS∗=(γ+μ+α1)(κ+μ+α2). If R0=1, then ν=β2γ/κ+μ+α2S∗, z1=μ, z2=α2+κ+μ+ν, z3=0, and E[u(t)]→c2m¯3m3 as t→∞, where c2=-(γ+μ+α1)(κ+μ+α2)/γμm3S∗, m¯3=(κ+μ+α2)/γm3. If m3=0, then E[u(t)]→0 as t→∞.We show the global stability of the infection-free equilibriumP0 and trivial solution u=0 of the nonlinear stochastic systems (12) and (18), respectively.Theorem 11.
If the trivial solutionu(t)=0 for linear system (20) with drift and diffusion coefficients F(t,u) and G(t,u), respectively, is asymptotically stable and the drift and diffusion coefficients f(t,u) and g(t,u), respectively, of nonlinear system (18) satisfy the inequality(30)ft,u-Ft,u+gt,u-Gt,u<cuin a sufficiently small neighbourhood of u=0, with a sufficiently small constant c, then the trivial solution u(t)=0 of system (18) is globally asymptotically stable.Proof.
In a sufficiently small neighbourhood ofu=0, choose ϵ>0 sufficiently small so that |u|<ϵ. We have(31)ft,u-Ft,u+gt,u-Gt,u=-β1u1u2-β2u1u3-12S∗+u12u2σ12+12S∗+u1u22σ12+12S∗+u1u32σ22+σ12S∗2u222+β1u1u2+β2u1u3+12S∗+u12u2σ12-12S∗+u1u22σ12-12S∗+u1u32σ22-σ12S∗2u2221/2+2u1u2σ12+2u1u3σ221/2≤2β1+2β2+σ12+σ22+2σ1+σ2ϵu1+3S∗+1σ12ϵu2+σ22S∗ϵu3<cu, where c=2β1+2β2+(3S∗+2)σ12+(S∗+1)σ22+2(σ1+σ2)ϵ.


## 5. Numerical Simulations

Figure1 verifies the global stability criteria discussed in Theorems 7, 10, and 11 and Remark 8.Figure 1
Graph of deterministic and stochastic trajectories of SEI model for initial conditionsS0=0.9, E0=0.08, I0=0.02, σ1=0.15, σ2=0.8, R0=0.9346, and R0=0.9845; (R0<1, R0<1).
(a) (b) (c)Figure1 shows the graph of deterministic and stochastic trajectories of SEI model for initial conditions S0=0.9, E0=0.08, and I0=0.02. Here, we see that both R0 and R0 are both less than one and both trajectories converge to the infection-free equilibrium (S∗,0,0)=(1,0,0).The transient epidemic advance is evident in all Figures2(a), 2(b), and 2(c). The number of exposed (latent) individuals drastically increased for some time period. During the same period, the number of infected individuals increased and the number of susceptible individuals decreased.Figure 2
Graph of deterministic and stochastic trajectories of SEI model for the case1-σ1(S∗)2/2(γ+μ+α1)<R0<1 of Remark 8 with initial conditions S0=0.9, E0=0.08, I0=0.02, β1=0.1, β2=5.5, μ=0.3=δ (S∗=1), γ=0.103, κ=3.47, α1=0.01, α2=1, σ1=0.9, σ2=0.1, R0=0.5297, and R0=1.5103.
(a) (b) (c)


## 6. Conclusion

The purpose of this work is to shed more light on the effects of the presence of external perturbations (in the form of white noise) in the disease transmission rates in a SEI model. By extending a SEI system of deterministic differential equations into a system of stochastic differential equations, we show the existence of solution of the stochastic model and derive the stochastic basic reproductive number,R0, corresponding to the deterministic reproductive number, R0. The stochastic basic reproductive number R0 converges to the deterministic basic reproductive number R0 as the noise intensity σ1 tends to zero. We show that even if the infection-free equilibrium is stable, we might still have a transient epidemic advance. A condition in which this advance can happen is derived and analyzed. From our result, we conclude that the transient epidemic advance is induced by the noise intensity, σ1, of the white noise. It follows from (7) that as the noise intensity σ1 increases, the fluctuations around the mean number of infected individuals also increase thereby causing an initial epidemic growth provided that (29) is satisfied.



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