# The Complete Parameters Analysis of the Asymptotic Behaviour of a Logistic Epidemic Model with Two Stochastic Perturbations

Authors: Dejun Fan; Ke Wang; Ling Hong
Journal: Mathematical Problems in Engineering
 (2009)

---

## Abstract

A simple model of the dynamics of an infectious disease, taking into account
environmental variability in the form of Gaussian white noise in the disease transmission rate
and the increase in mortality rate due to disease, has been investigated. The probability
distribution for the proportion of infected animals, plus its mean, mode, and variance, is found
explicitly.


---

## Body

## 1. Introduction

In order to describe the dynamics of bovine tuberculosis in possum populations in New Zealand, Roberts and Saha [1] introduced an epidemic model:
(1.1)Ż(t)=(p-1)BZ+(βC-α)(1-Z)Z,
where Z(t) is the proportion of the population infected with disease against the total population; p(∈[0,1]) is the vertical distribution probability; B is the birth rate independent on the total population; β is the constant transmission rate; C is the contact rate between individuals; α is the increase of the death rate suffering from infectious disease. It turns out that this model can be used to study other epidemics as well. It is easy to see that (1.1) is a well-known logistic equation with intrinsic growth rate:
(1.2)r=(p-1)B+βC-α,
and carrying capacity K=1+(p-1)B/(βC-α).Taking into account random perturbation to the disease transmission coefficientβ, (1.1) becomes
(1.3)dZ=[(p-1)BZ+(βC-α)(1-Z)Z]dt+ρC(1-Z)ZdB(t).By using the Fokker-Planck equation method, Roberts and Saha in [1] investigated the asymptotic behavior of solutions of (1.3) when R0=(pB+βC)/(α+β)>1 (i.e., intrinsic growth rate r>0). Recently, Ding et al. in [2] gave some results to deal with the case of R0=(pB+βC)/(α+β)<1; for example, they showed that the zero solution of (1.3) was asymptotically stable provided that the intensity of random perturbation is sufficiently small.In the real world, at the same time, mortality rate due to disease has often been affected by some factors, such as age, seasons, and food supply. Thus, in order to describe epidemic more reasonable, we suppose that the increase of the death rateα is subjected by random disturbance also, and the model (1.3) becomes
(1.4)dZ=[(p-1)BZ+(βC-α)(1-Z)Z]dt+ρ1C(1-Z)ZdB1(t)-ρ2(1-Z)ZdB2(t).Here,B1(t) and B2(t) are independent standard Brownian motions; ρ1 and ρ2 are the intensity of the environmental disturbance. In this paper, for the new model (1.4) we carried out complete parameters analysis for R0>1 (i.e., intrinsic growth rate r>0), R0=1 (i.e., r=0), and R0<1 (i.e., r<0), respectively. Our results are generalizations of the ones in [1, 2]. Some interesting details about the system are revealed. For instance, we proved that when R0<1, the zero solution of (1.4) is globally asymptotically stable, no matter how large the intensity of stochastic fluctuation is. Moreover, the explicit expectation and variance of steady solution of (1.4) are also given. Thus, the statistic properties of dynamics of (1.4) are completely clear.The main argument used in this paper is the classical Fokker-Planck equation method. Of course, to deal with stochastic differential equations without linear growth condition, there are some other effective methods; we refer the reader to the papers [3–6].


## 2. Nonnegative Solutions

Following, we show that the solution of the SDE model (1.4) is global existent and unique.Theorem 2.1.
Assume thatB,β,α, and C are positive real numbers. Then for any initial value Z0(0≤Z0<1), there is a unique solution Z(t) to (1.4) on t≥0 and 0≤Z(t)<1 almost surely; moreover, the solution Z(t) to (1.4) will remain in ℝ+ with probability 1.Proof.
It is obvious that the coefficients of (1.4) are locally Lipschitz continuous; hence, for any initial value Z0(0<Z0<1), (1.4) has a unique local solution Z(t), t∈[0,τe). Moreover, by Gardiner [7, page 132], we obtain that system with reflecting barrier; hence, Z(t)<1 almost surely for any initial value Z0(0<Z0<1). And, 0 is the solution of model (1.4); hence, the solution Z(t) to (1.4) for any initial value Z0(0≤Z0<1) will remain in ℝ+ with probability 1. The proof is complete.Remark 2.2.
To prove thatZ(t)<1, Gong gives other effective method [8, page 371].Remark 2.3.
To prove that global existence, there is other effective method also, (cf. [3]).


## 3. The Fokker-Planck Equation

We rewrite (1.4) as
(3.1)dZ=F(Z)dt+G1(Z)dB1(t)+G2(Z)dB2(t),
where
(3.2)F(Z)=(p-1)BZ+(βC-α)(1-Z)Z;G1(Z)=ρ1C(1-Z)Z;G2(Z)=-ρ2(1-Z)Z.
By Gard [9, page 144], the Fokker-Planck equation of this model is
(3.3)d2dZ2{[G12(Z)+G22(Z)]p(Z)}-2ddZ[F(Z)p(Z)]=0,
where p(Z) is the steady density function, due to system with reflecting barrier [7, page 129], thus p(Z) is proportional to
(3.4)(Z1-Z)2r/ρ21Z2(1-Z)2exp{2(p-1)Bρ2(1-Z)},
where
(3.5)ρ2=ρ12C+ρ22,
and the constant of proportionality is determined by ∫01‍p(Z)dZ=1.By differentiating (3.4), it is clear that the extremum of p(Z) is related to the zeros of quadratic function:
(3.6)-2ρ2Z2+(3ρ2-r+(p-1)B)Z+r-ρ2.


## 4. The Asymptotic Behaviour of Solution

To continue, we will regard theρ2=ρ12C+ρ22 as a parameter to separately discuss the asymptotic behaviour of the solutions of model (1.4) in threecases: intrinsic growth rate r>0, r<0, and r=0, respectively.


### 4.1. Asymptotic Behaviour in the Caser>0

Case 1 (p∈ [0,1)).
In this case, obviously,limZ→1-p(Z)=0.     (I)
Ifρ2<r, then limZ→0p(Z)=0. In (3.6), the function is negative when Z=1, positive when Z=0, and has unique root in (0,1). Hence, the distribution p(Z) tends to zero as Z→0,1 and is unimodal. The mode of p(Z) to K is ρ→0, and to zero is ρ2→r. (see, e.g., Figure 1(a).)(II)
Ifρ2=r, then limZ→0p(Z)≠0. In (3.6), the function is negative when Z=1, and is zero when Z=0, and zero is unique root in [0,1). Hence, the distribution p(Z) is strictly monotonic decreasing function in [0,1) with a positive maximum at Z=0 (see Figure 1(b).)(III)
Ifr<ρ2<2r in (3.6), the function is negative when Z=1 and Z=0, and the p(Z) is singular at Z=0 but integrable in (0,1).
(i)
If3ρ2-2ρ2(ρ2-r)<r-(p-1)B, then function (3.6) is strictly negative for 0<Z<1; hence the distribution p(Z) is a monotonic decreasing function (see Figure 1(c).)(ii)
If3ρ2-2ρ2(ρ2-r)=r-(p-1)B, then (3.6) has one zero for 0<Z<1, the system occur a pitchfork bifurcation (see Figure 2(a).)(iii)
If3ρ2-2ρ2(ρ2-r)>r-(p-1)B, then (3.6) has two zeros for 0<Z<1, and the function p(Z) is bimodal, with a singular peak at Z=0 and a positive maximum for some 0<Z<1 (see Figure 2(b).)(iv)
Ifρ2≥2r, then p(Z) may be represented by a delta function at Z=0. That is, if perturbation sufficient is larger, the disease becomes extinct (see Figure 2(c).)(a)(b)(c)(a)(b)(c)Case 2 (p = 1).
(i) Ifρ2<2r, then p(Z) is a delta function at Z=1.
(ii) Ifρ2≥2r, then p(Z) consists of delta function at Z=0 and Z=1. The former corresponds to limt→∞Z(t)=1 a.s., while for the later the limit for larger t is either zero or one, the probability of each outcome being determined by the magnitudes of the residues of the density functions (3.3) at its extremes.


### 4.2. Asymptotic Behaviour in the Caser<0

Case 1 (p∈ [0,1)).
In this case,limZ→1-p(Z)=0. Moreover, for any ρ, obviously, ρ2≥2r, then p(Z) may be represented by a delta function at Z=0. That is, for any perturbation, the disease becomes extinct; this is the same when there is no stochastic perturbation.Case 2 (p = 1).
In this case,p(Z) consists of delta function at Z=0 and Z=1.


### 4.3. Asymptotic Behaviour in the Caser=0

Case 1 (p∈ [0,1)).
Equation (3.4) becomes
(4.1)1Z2(1-Z)2exp{2(p-1)Bρ2(1-Z)},
and then p(Z) may be represented by a delta function at Z=0.Case 2 (p = 1).
In this case,p(Z) consists of delta function at Z=0 and Z=1.


## 4.1. Asymptotic Behaviour in the Caser>0

Case 1 (p∈ [0,1)).
In this case, obviously,limZ→1-p(Z)=0.     (I)
Ifρ2<r, then limZ→0p(Z)=0. In (3.6), the function is negative when Z=1, positive when Z=0, and has unique root in (0,1). Hence, the distribution p(Z) tends to zero as Z→0,1 and is unimodal. The mode of p(Z) to K is ρ→0, and to zero is ρ2→r. (see, e.g., Figure 1(a).)(II)
Ifρ2=r, then limZ→0p(Z)≠0. In (3.6), the function is negative when Z=1, and is zero when Z=0, and zero is unique root in [0,1). Hence, the distribution p(Z) is strictly monotonic decreasing function in [0,1) with a positive maximum at Z=0 (see Figure 1(b).)(III)
Ifr<ρ2<2r in (3.6), the function is negative when Z=1 and Z=0, and the p(Z) is singular at Z=0 but integrable in (0,1).
(i)
If3ρ2-2ρ2(ρ2-r)<r-(p-1)B, then function (3.6) is strictly negative for 0<Z<1; hence the distribution p(Z) is a monotonic decreasing function (see Figure 1(c).)(ii)
If3ρ2-2ρ2(ρ2-r)=r-(p-1)B, then (3.6) has one zero for 0<Z<1, the system occur a pitchfork bifurcation (see Figure 2(a).)(iii)
If3ρ2-2ρ2(ρ2-r)>r-(p-1)B, then (3.6) has two zeros for 0<Z<1, and the function p(Z) is bimodal, with a singular peak at Z=0 and a positive maximum for some 0<Z<1 (see Figure 2(b).)(iv)
Ifρ2≥2r, then p(Z) may be represented by a delta function at Z=0. That is, if perturbation sufficient is larger, the disease becomes extinct (see Figure 2(c).)(a)(b)(c)(a)(b)(c)Case 2 (p = 1).
(i) Ifρ2<2r, then p(Z) is a delta function at Z=1.
(ii) Ifρ2≥2r, then p(Z) consists of delta function at Z=0 and Z=1. The former corresponds to limt→∞Z(t)=1 a.s., while for the later the limit for larger t is either zero or one, the probability of each outcome being determined by the magnitudes of the residues of the density functions (3.3) at its extremes.


## 4.2. Asymptotic Behaviour in the Caser<0

Case 1 (p∈ [0,1)).
In this case,limZ→1-p(Z)=0. Moreover, for any ρ, obviously, ρ2≥2r, then p(Z) may be represented by a delta function at Z=0. That is, for any perturbation, the disease becomes extinct; this is the same when there is no stochastic perturbation.Case 2 (p = 1).
In this case,p(Z) consists of delta function at Z=0 and Z=1.


## 4.3. Asymptotic Behaviour in the Caser=0

Case 1 (p∈ [0,1)).
Equation (3.4) becomes
(4.1)1Z2(1-Z)2exp{2(p-1)Bρ2(1-Z)},
and then p(Z) may be represented by a delta function at Z=0.Case 2 (p = 1).
In this case,p(Z) consists of delta function at Z=0 and Z=1.


## 5. The Mean and Variance

The first two moments of the distribution may be found directly from the differential equation by noting thatG1(Z)=G2(Z)=G1′(Z)=G2′(Z)=0 if Z=0 or 1; from (3.3) we have
(5.1)ddZ[(G12(Z)+G22(Z))p(Z)]-2[F(Z)p(z)]=0.
Integrating again we obtain
(5.2)∫01F(Z)p(Z)dZ=0,
or multiplying (5.1) by Z-1 and integrating we obtain
(5.3)2∫01Z-1F(Z)p(Z)dZ-∫01Z-2(G12(Z)+G22(Z))p(Z)dZ=0.
Hence, solving (5.1) and (5.3), we obtain expression for the first moment and second moment of Z:
(5.4)EZ=(2r-ρ2)(βc-α)ρ2r+2(βc-α)2-2ρ2(βc-α),EZ2=rβc-αEZ.
Hence, we obtain expression for the mean μ and variance σ2(5.5)μ=EZ,σ2=EZ2-μ2
and embody
(5.6)μ=(2r-ρ2)(βc-α)ρ2r+2(βc-α)2-2ρ2(βc-α),σ2=r(2r-ρ2)ρ2r+2(βc-α)2-2ρ2(βc-α)-[(2r-ρ2)(βc-α)ρ2r+2(βc-α)2-2ρ2(βc-α)]2.



---

*Source: 904383-2009-11-10.xml*