Research article

Modeling the impact of sanitation and awareness on the spread of infectious diseases

  • Received: 08 June 2018 Accepted: 22 October 2018 Published: 14 January 2019
  • Sanitation and awareness programs play a fundamental role and are much effective public health interventions to control the spread of infectious diseases. In this paper, a nonlinear mathematical model for the control of infectious diseases, such as typhoid fever is proposed and analyzed by considering budget required for sanitation and awareness programs as a dynamic variable. It is assumed that the budget allocation regarding the protection against the disease to warn people and for sanitation increases logistically and its per-capita growth rate increases with the increase in number of infected individuals. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals as well as indirectly through bacteria shed in the environment. It is further assumed that a fraction of budget is used to warn people via propagating awareness whereas the remaining part is used for sanitation to reduce the density of bacteria. The condition when budget should spend on sanitation/awareness to reduce the number of infected individuals is obtained. Model analysis reveals that the sanitation and awareness programs have capability to reduce the epidemic threshold and thus control the spread of infection. However, delay in providing funds destabilizes the system and may cause stability switches through Hopf-bifurcation. Numerical simulations are also carried out to support analytical findings.

    Citation: Rajanish Kumar Rai, Arvind Kumar Misra, Yasuhiro Takeuchi. Modeling the impact of sanitation and awareness on the spread of infectious diseases[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 667-700. doi: 10.3934/mbe.2019032

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  • Sanitation and awareness programs play a fundamental role and are much effective public health interventions to control the spread of infectious diseases. In this paper, a nonlinear mathematical model for the control of infectious diseases, such as typhoid fever is proposed and analyzed by considering budget required for sanitation and awareness programs as a dynamic variable. It is assumed that the budget allocation regarding the protection against the disease to warn people and for sanitation increases logistically and its per-capita growth rate increases with the increase in number of infected individuals. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals as well as indirectly through bacteria shed in the environment. It is further assumed that a fraction of budget is used to warn people via propagating awareness whereas the remaining part is used for sanitation to reduce the density of bacteria. The condition when budget should spend on sanitation/awareness to reduce the number of infected individuals is obtained. Model analysis reveals that the sanitation and awareness programs have capability to reduce the epidemic threshold and thus control the spread of infection. However, delay in providing funds destabilizes the system and may cause stability switches through Hopf-bifurcation. Numerical simulations are also carried out to support analytical findings.


    Rumor spreading as a social contagion process is very similar to the epidemic diffusion, so most rumor spreading models have evolved from epidemic models, such as SI, SIR and SIS. A classic rumor model was DK model proposed by Daley and Kendall in 1964 [2], in which the population was divided into three classes: people who did not know the rumor, people who spread the rumor and people who know but will never spread the rumor. Maki and Thomson modified the DK model into the MK model [8]. Since then, a number of scholars have proposed various rumor spreading models by improving the traditional epidemic models [1,3,14,15,16]. In 2019, Tian and Ding [11] formulated an ordinary differential equation (ODE) compartmental model for rumor, where the population was divided into five disjoint classes, namely the ignorants, the latents, the rumor-spreaders, the debunkers and the stiflers. At time t, the numbers in each of these classes are denoted by I(t),L(t),R(t),D(t) and S(t), respectively. The rumor spreading model considered by Tian and Ding [11] can be described by a system of ODEs

    {dI(t)dt=ΛμIRkIDρI,dL(t)dt=μIR+kID(α+β+γ+ρ)L,dR(t)dt=αL(δ+ξ+ρ)R,dD(t)dt=βL+ξR(θ+ρ)D,dS(t)dt=δR+θD+γLρS, (1.1)

    where Λ is the constant immigration rate of the population, μ is the rumor-contacting rate, k is the debunker-contacting rate, ρ is the rate at which all existing users exit from the five classes (i.e. emigration rate), α is the rumor-spreading rate, β is the debunking rate, γ is the silent rate, δ is the rumor-stifling, ξ is the reversal rate and θ represents the debunking-stifling rate. All parameters are assumed to be independent of time t and positive.

    The model used in the above study to describe rumor propagation behavior is deterministic model, whereas the random models used to study rumor propagation are few[3]. But in the real world, rumor models are often affected by environmental noise. Especially in emergency events, when rumors are widely spread, the propagation process is affected by many uncertain factors, which increase the volatility of the propagation process. Therefore, it would be necessary and interesting to reveal how the environmental noise affects the rumor spreading model. Following the idea of Jia et al.[3], in this paper, we assume that the stochastic perturbations are of white noise type which are proportional to the system variable, respectively. Then we obtain a stochastic analogue of the deterministic model (1.1) as follows

    {dI=[ΛμIRkIDρI]dt+σ1IdB1(t),dL=[μIR+kID(α+β+γ+ρ)L]dt+σ2LdB2(t),dR=[αL(δ+ξ+ρ)R]dt+σ3RdB3(t),dD=[βL+ξR(θ+ρ)D]dt+σ4DdB4(t),dS=[δR+θD+γLρS]dt+σ5SdB5(t), (1.2)

    where Bi(t)(i=1,2,3,4,5) are independent Brownian motions and σi>0(i=1,2,3,4,5) are their intensities. All the other parameters in system (1.2) have the same meaning as in system (1.1). Throughout this paper, unless otherwise specified, let (Ω,F,{Ft}t0,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions, that is, it is rightly continuous and increasing while F0 contains all Pnull sets, and let Bi(t)(i=1,2,3,4,5) be scalar Brownian motions defined on the probability space. For the sake of simplicity, we introduce the following notations: R5+={(x1,x2,x3,x4,x5)R5:xi>0,i=1,2,3,4,5},ab=max{a,b},ab=min{a,b},σ=σ1σ2σ3σ4σ5.

    With the help of the Lyapunov function methods and the inequality techniques, the existence and uniqueness of an ergodic stationary distribution of the positive solutions to system (1.2) are presented. The main difficulties lies in the construction of Lyapunov function and the construction of a bounded closed domain. The main contribution of this paper are highlighted as follows: (ⅰ) a stochastic rumor spreading model is proposed and investigated; (ⅱ) some sufficient conditions for the existence of an ergodic stationary distribution; and (ⅲ) the stationary distribution implies that the rumor can be persistent in the mean. The subsequent part of this paper is as follows: In Section 2, we prove the existence and uniqueness of a global positive solution to system (1.2) with any positive initial value. In Section 3, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to model (1.2). In Section 4, some numerical simulations are provided to illustrate our theoretical results. Finally, some concluding remarks are presented to end this paper.

    To analyze the dynamical behavior of a rumor spreading model, the first concerning thing is whether the solution is global and positive. In this section, motivated by the methods in [9] and we show that there is a unique global positive solution of system (1.2). The key is to construct a Lyapunov function.

    Theorem 2.1. For any given initial value (I(0),L(0),R(0),D(0),S(0))R5+, system (1.2) admits a unique solution (I(t),L(t),R(t),D(t),S(t))R5+ on t0 and the solution will remain in R5+ with probability one, namely (I(t),L(t),R(t),D(t),S(t))R5+ for all t0 almost surely (a.s.).

    Proof. Since the coefficients of system (1.2) satisfy the local Lipschitz condition, we know that, for any initial value (I(0),L(0),R(0),D(0),S(0))R5+, there is a unique local solution (I(t),L(t),R(t),D(t),S(t))R5+ on t[0,τe], where τe is the explosion time [10]. Now we prove the solution is global, i.e. to prove τe= a.s. To this end, let m0>0 be sufficiently large such that each component of (I(0),L(0),R(0),D(0),S(0)) all lies in the interval [1m0,m0]. For each integer mm0, define the following stopping time

    τm=inf{t[0,τe):min{I(t),L(t),R(t),D(t),S(t)}1mormax{I(t),L(t),R(t),D(t),S(t)}m}.

    Throughout this paper, we set inf= (as usual denotes the empty set). Obviously, τm is an increasing function as m. Set τ=limmτm. Then ττe a.s. If τ= a.s. is true, then τe= a.s. and (I(t),L(t),R(t),D(t),S(t))R5+ a.s. for all t0. In other words, in order to show this assertion, we only need to prove τ= a.s. If the assertion is false, then there is a pair of constants T>0 and ˉϵ(0,1) such that P{τmT}ˉϵ for each integer mm0. Define a C2-function V:R5+R+{0} by

    V(I,L,R,D,S)=(IaalnIa)+(L1lnL)+(R1lnR)+(D1lnD)+(S1lnS),

    where a is a positive constant to be determined later. The nonnegativity of this function can be seen from u1lnu0,u>0. According to the general Itô formula (see, for example, Theorem 4.2.1 of [10]), we have

    dV(I,L,R,D,S)=LV(I,L,R,D,S)dt+σ1(Ia)dB1(t)+σ2(L1)dB2(t)+σ3(R1)dB3(t)+σ4(D1)dB4(t)+σ5(S1)dB5(t),

    where LV:R5+R is defined by

    LV(I,L,R,D,S)=(1aI)(ΛμIRkIDρI)+(11L)(μIR+kID(α+β+γ+ρ)L)+(11R)(αL(δ+ξ+ρ)R)+(11D)(βL+ξR(θ+ρ)D)+(11S)(δR+θD+γLρS)+aσ21+σ22+σ23+σ24+σ252=ΛρIρLρRρDρSaΛI+aμR+akD+aρμIRLkIDLαLR+(α+β+γ+ρ)+(δ+ξ+ρ)βLDξRD+(θ+ρ)δRSθDSγLS+ρ+aσ21+σ22+σ23+σ24+σ252Λ+(aμρ)R+(akρ)D+(a+4)ρ+α+β+γ+δ+ξ+θ+aσ21+σ22+σ23+σ24+σ252.

    Choose a=min{ρμ,ρk}, then we obtain

    LV(I,L,R,D,S)Λ+(a+4)ρ+α+β+γ+δ+ξ+θ+aσ21+σ22+σ23+σ24+σ252:=K.

    and K is a positive constant. Thus,

    dV(I,L,R,D,S)Kdt+σ1(Ia)dB1(t)+σ2(L1)dB2(t)+σ3(R1)dB3(t)+σ4(D1)dB4(t)+σ5(S1)dB5(t). (2.1)

    For any mm0, integrating (2.1) on both sides from 0 to τmT and then taking expectation yield

    EV(I(τmT),L(τmT),R(τmT),D(τmT),S(τmT))V(I(0),L(0),R(0),D(0),S(0))+KT.

    Let Ωm={ωΩ:τm=τm(ω)T} for mm0. Then we have P(Ωm)ˉϵ. Note that, for every ωΩm, there exists I(τm,ω) or L(τm,ω) or R(τm,ω) or D(τm,ω) or S(τm,ω) equaling either m or 1m. Thus V(I(τm,ω),L(τm,ω)),R(τm,ω),D(τm,ω),S(τm,ω) is no less than either

    maalnma or 1maaln1ma=1ma+aln(ma) or m1lnm or 1m1ln1m=1m1+lnm.

    So we have

    V(I(τm,ω),L(τm,ω),R(τm,ω),D(τm,ω),S(τm,ω))(maalnma)(1ma+aln(ma))(m1lnm)(1m1+lnm).

    Consequently,

    V(I(0),L(0),R(0),D(0),S(0))+KTE[1Ωm(ω)V(I(τm,ω),L(τm,ω),R(τm,ω),D(τm,ω),S(τm,ω))]ˉϵ(maalnma)(1ma+aln(ma))(m1lnm)(1m1+lnm),

    where 1Ωm denotes the indicator function of Ωm. Letting m leads to the contradiction

    >V(I(0),L(0),R(0),D(0),S(0))+KT=.

    This completes the proof.

    When considering rumor propagation model, we are also interested to know when the rumor will persist and prevail in a population. In the deterministic models, it can be solved by proving the rumor-epidemic equilibrium of the corresponding model is a global attractor or globally asymptotically stable. But there is no rumor-epidemic equilibrium in system (1.2). In this section, based on the theory of Khasminskii([7]), we prove that there is a stationary distribution which reveals that the rumor will persist in the mean. Here we present some theory about the stationary distribution which is introduced in ([7]).

    Definition 3.1. ([7]) The transition probability function P(s,x,t,A) is said to be time-homogeneous (and the corresponding Markov process is called time-homogeneous) if the function P(s,x,t+s,A) is independent of s, where 0st, xRl and AB and B denotes the σ algebra of Borel sets in Rl.

    Definition 3.2. ([7]) Let X(t) be a regular time-homogeneous Markov process in Rl described by the stochastic differential equation:

    dX(t)=f(X(t))dt+kr=1gr(X(t))dBr(t).

    The diffusion matrix of the process X(t) is defined as follows:

    A(x)=(aij(x)),aij(x)=kr=1gir(x)gjr(x).

    Lemma 3.1. ([7]) The Markov process X(t) has a unique ergodic stationary distribution π() if there exists a bounded domain DRn with regular boundary Γ and

    A1: there is a positive number M such that

    di,j=1aij(x)ξiξjM|ξ|2,xD,ξRn.

    A2: there exists a nonnegative C2-function V such that LV is negative for any RnD, where L denotes the differential operator defined by

    L=ni=1bi(x)xi+12ni,j=1aij(x)2xixj.

    Then

    Px{limT1TT0f(X(t))dt=Rnf(x)π(dx)}=1

    for all xRn, where f() is a function integrable with respect to the measure π.

    Theorem 3.1. Assume

    Rs0:=Λμα4(ρ+σ212)(α+β+γ+ρ+σ222)(δ+ξ+ρ+σ232)>1.

    Then, for any initial value (I(0),L(0),R(0),D(0),S(0))R5+, system (1.2) has a unique stationary distribution π() and the ergodicity holds.

    Proof. Theorem 2.1 tells us that for any initial value (I(0),L(0),R(0),D(0),S(0))R5+, there is a unique global solution (I(t),L(t),R(t),D(t),S(t))R5+. In order to prove this Theorem, it suffices to validate A1 and A2 in Lemma 3.1. First, we verify A1. The diffusion matrix of system (1.2) is given by

    A=(σ21I200000σ22L200000σ23R200000σ24D200000σ25S2).

    It is easy to see that the matrix A is positive definite for any compact subset of R5+, so condition A1 in Lemma 3.1 is satisfied. Now we prove condition A2. Define a C2-function

    ˜Q(I,L,R,D,S)=M(n1lnIn2lnLn3lnRn4lnSn5D)+1ϱ+1(I+L+R+D+S)(ϱ+1)lnIlnLlnRlnS:=MQ1+Q2lnIlnLlnRlnS,

    where n1,n2,n3,n4,n5,ϱ and M are positive constants, which will be determined later. It is easy to check that

    lim infk,(I,L,R,D,S)R5+Uk˜Q(I,L,R,D,S)=+,

    where Uk=Π5i=1(1k,k). In addition, ˜Q(I,L,R,D,S) is a continuous function. Hence, ˜Q(I,L,R,D,S) must have a minimum point (I0,L0,R0,D0,S0)R5+. Therefore, we define a nonnegative C2-function Q:R5+R+

    Q(I,L,R,D,S)=˜Q(I,L,R,D,S)˜Q(I0,L0,R0,D0,S0).

    Applying the general Itô formula [10] to Q1, one obtains the differential operator L of Q1 as follows:

    LQ1=n1I[ΛμIRkIDρI]+n1σ212n2L[μIR+kID(α+β+γ+ρ)L]+n2σ222n3R[αL(δ+ξ+ρ)R]+n3σ232n4S[δR+θD+γLρS]+n4σ252n5[βL+ξR(θ+ρ)D](n1ΛI+n2μIRL+n3αLR)+(n1μn5ξ)R+(n1k+n5(θ+ρ))D+n1ρ+n2(α+β+γ+ρ)+n3(δ+ξ+ρ)+n4ρ+n1σ212+n2σ222+n3σ232+n4σ25233n1n2n3Λμα+(n1μn5ξ)R+(n1k+n5(θ+ρ))D+n1ρ+n2(α+β+γ+ρ)+n3(δ+ξ+ρ)+n4ρ+n1σ212+n2σ222+n3σ232+n4σ252=33n1n2n3Λμα+(n1μn5ξ)R+(n1k+n5(θ+ρ))D+n1(ρ+σ212)+n2(α+β+γ+ρ+σ222)+n3(δ+ξ+ρ+σ232)+n4(ρ+σ252).

    Let

    n1=1,n2=ρ+σ2122(α+β+γ+ρ+σ222),n3=ρ+σ2122(δ+ξ+ρ+σ232),n4=ρ+σ212ρ+σ252,n5=(1+1M)μξ.

    Then it follows that

    LQ13[Λμα(ρ+σ212)24(α+β+γ+ρ+σ222)(δ+ξ+ρ+σ232)]13μMR+[k+(1+1M)μ(θ+ρ)ξ]D+3(ρ+σ212)=3(ρ+σ212)(Rs1301)μMR+[k+(1+1M)μ(θ+ρ)ξ]D. (3.1)

    Similarly, one has

    LQ2=(I+L+R+D+S)ϱ[Λρ(I+L+R+D+S)]+ϱ2(I+L+R+D+S)ϱ1(σ21I2+σ22L2+σ23R2+σ24D2+σ25S2)Λ(I+L+R+D+S)ϱρ(I+L+R+D+S)ϱ+1+ϱ2(σ21σ22σ23σ24σ25)(I+L+R+D+S)ϱ+1=Λ(I+L+R+D+S)ϱϕ(I+L+R+D+S)ϱ+1C012ϕ(I+L+R+D+S)ϱ+1C012ϕ(Iϱ+1+Lϱ+1+Rϱ+1+Dϱ+1+Sϱ+1), (3.2)

    where we choose ϱ sufficiently small such that ϕ=ρϱσ22>0 and

    C0=sup(I,L,R,D,S)R5+{Λ(I+L+R+D+S)ϱϕ2(I+L+R+D+S)ϱ+1}<.

    Moreover, one has

    {L(lnI)=ΛI+μR+kD+ρ+σ212,L(lnL)=μIRLkIDL+(α+β+γ+ρ)+σ222,L(lnR)=αLR+(δ+ξ+ρ)+σ232,L(lnS)=δRSθDSγLS+ρ+σ252. (3.3)

    Making use of (2.1)–(3.3), we then derive that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D+C0ΛI+ρ12ϕ(Iϱ+1+Lϱ+1+Rϱ+1+Dϱ+1+Sϱ+1)+σ212μIRLkIDL+(α+β+γ+ρ)+σ222αLR+(δ+ξ+ρ)+σ232δRSθDSγLS+ρ+σ2523M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)DΛIkIDLαLRθDS+C0+4ρ+α+β+γ+δ+ξ+σ21+σ22+σ23+σ252ϕ2(Iϱ+1+Lϱ+1+Rϱ+1+Dϱ+1+Sϱ+1) (3.4)

    For the convenience, we define

    H1=supDR+{3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D14ϕDϱ+1}<

    and

    H2=C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25).

    Now we are in the position to construct a bounded closed domain Uϵ such that the condition A2 in Lemma 3.1 holds. To this end, we define a compact set as follows

    Uϵ={(I,L,R,D,S)R5+:ϵI1ϵ,ϵ3L1ϵ3,ϵ4R1ϵ4,ϵD1ϵ,ϵ2S1ϵ2},

    where ϵ>0 is a sufficiently small constant such that

    3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)ϵ+H2<1 (3.5)

    and

    (Λϵθϵkϵαϵϕ2ϵ4(ϱ+1)ϕ4ϵϱ+1)+H1+H2<1, (3.6)

    For convenience, we can divide R5+Uϵ into ten domains, where

    U1={(I,L,R,D,S)R5+|0<I<ϵ},U2={(I,L,R,D,S)R5+|0<D<ϵ},
    U3={(I,L,R,D,S)R5+|ϵD,0<S<ϵ2},U4={(I,L,R,D,S)R5+|ϵI,ϵD,0<L<ϵ3},
    U5={(I,L,R,D,S)R5+|ϵ3L,0<R<ϵ4},U6={(I,L,R,D,S)R5+|I>1ϵ},
    U7={(I,L,R,D,S)R5+|L>1ϵ3},U8={(I,L,R,D,S)R5+|R>1ϵ4},
    U9={(I,L,R,D,S)R5+|D>1ϵ},U10={(I,L,R,D,S)R5+|S>1ϵ2}.

    Obviously, R5+Uϵ=10i=1Ui. Next, we will prove that LQ(I,L,R,D,S)1 for any (I,L,R,D,S)R5+Uϵ, which is equivalent to proving it on the above ten domains, respectively.

    Case 1. For any (I,L,R,D,S)U1, then (3.4) implies that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+1ΛI+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)ΛI+H1+H2Λϵ+H1+H2. (3.7)

    Case 2. For any (I,L,R,D,S)U2, using (3.4) one obtains

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)ϵ+H2. (3.8)

    Case 3. For any (I,L,R,D,S)U3, in view of (3.4), we get

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+1θDS+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)θDS+H1+H2θϵ+H1+H2. (3.9)

    Case 4. For any (I,L,R,D,S)U4, it follows from (3.4) that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+1kIDL+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)kϵ+H1+H2. (3.10)

    Case 5. For any (I,L,R,D,S)U5, according to (3.4), we derive

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+1αLR+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)αϵ+H1+H2. (3.11)

    Case 6. For any (I,L,R,D,S)U6, by (3.4), we obtain

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+112ϕIϱ+1+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)12ϕ1ϵϱ+1+H1+H2. (3.12)

    Case 7. For any (I,L,R,D,S)U7, note from (3.4) that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+112ϕLϱ+1+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)12ϕ1ϵ3(ϱ+1)+H1+H2. (3.13)

    Case 8. For any (I,L,R,D,S)U8, making use of (3.4) one obtains that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+112ϕRϱ+1+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)12ϕ1ϵ4(ϱ+1)+H1+H2. (3.14)

    Case 9. For any (I,L,R,D,S)U9, we know from (3.4) that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D14ϕDϱ+114ϕDϱ+1+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)14ϕ1ϵϱ+1+H1+H2. (3.15)

    Case 10. For any (I,L,R,D,S)U10, using (3.4), we can show that

    LQ3M(ρ+σ212)(Rs1301)+(M+1)(k+μ(θ+ρ)ξ)D12ϕDϱ+112ϕSϱ+1+C0+4ρ+α+β+γ+δ+ξ+12(σ21+σ22+σ23+σ25)12ϕ1ϵ2(ϱ+1)+H1+H2. (3.16)

    It follows from (3.5)–(3.16) that

    LQ<1,(I,L,R,D,S)R5+Uϵ,

    which proves that condition A2 holds. Thus the conditions in Lemma 3.1 are verified, the proof is completed.

    Remark 3.1. Comparing with the threshold parameter R0 in [11], the parameter Rs0 in stochastic system (1.2) is less than R0, which reveals that the extinction of the rumor is much easier than in the corresponding deterministic model (1.1). Moreover, taking attention to the expression of Rs0, we can control the rumor propagation by environmental white noise.

    In this section, we give some examples to illustrate the obtained theoretical results. We illustrate our findings by the Milstein's Higher Order Method developed in [5]. According to this method, we can get the following discretization equation of system (1.2):

    {Ij+1=Ij+[ΛμIjRjkIjDjρIj]Δt+σ1IjΔtξ1,j+σ212IjΔt(ξ21,j1),Lj+1=Lj+[μIjRj+kIjDj(α+β+γ+ρ)Lj]Δt+σ2LjΔtξ2,j+σ222LjΔt(ξ22,j1),Rj+1=Rj+[αLj(δ+ξ+ρ)Rj]Δt+σ3RjΔtξ3,j+σ232RjΔt(ξ23,j1),Dj+1=Dj+[βLj+ξRj(θ+ρ)Dj]Δt+σ4DjΔtξ4,j+σ242DjΔt(ξ24,j1),Sj+1=Sj+[δRj+θDj+γLjρSj]Δt+σ5SjΔtξ5,j+σ252SjΔt(ξ25,j1),

    where time increment Δt>0, and ξ1,k,ξ2,k,ξ3,k,ξ4,k,ξ5,k are independent Gaussian random variables which follows N(0,1). However, we would like to point out that the values of parameters of system (1.2) and the initial values in the following numerical simulations are chosen for illustration purposes and are not taken from any real life data for any rumors. To this end, we set(I(0),L(0),R(0),D(0),S(0))=(0.3,0.6,0.2,0.9,0.6).

    Example 1. Choose the parameters: Λ=5,μ=0.045,k=0.2,α=0.05,β=0.05,γ=0.05,ρ=0.01,δ=0.01,ξ=0.01,θ=0.2.

    By simple calculation, we have Rs0=1.213>1, which means that the conditions of Theorem 3.1 hold. Therefore, system (1.2) has a unique ergodic stationary distribution. Figures 1 and 2 illustrate this fact.

    Figure 1.  σ1=0.9,σ2=σ3=0.09,σ4=σ5=0.05.
    Figure 2.  Relative frequency density.

    Example 2. Choose the parameters: Λ=0.6,μ=0.04,k=0.8,α=0.5,β=0.05,γ=0.5,ρ=0.01,δ=0.1,ξ=0.2,θ=0.2.

    In what follows, we start comparing the stochastic system and deterministic system. In Figure 3, the rumor-spreaders in stochastic system are shown in red lines, compared with the rumor-spreaders in deterministic system which are shown in blue lines. It reveals that the environmental disturbance may help to curb the outbreak of rumors. Further, if we increase the environmental noise, the simulation results in Figure 4 suggests that the extinction of rumor-spreaders in stochastic system is much more easier than that in the corresponding deterministic system.

    Figure 3.  Red line: Rumor-spreaders in stochastic system. Blue line: Rumor-spreaders in deterministic system. The intensities of the white noise are σ1=2.3,σ2=1,σ3=0.25,σ4=0.3,σ5=0.5, and other parameter values are presented in Example 2.
    Figure 4.  Red line: Rumor-spreaders in stochastic system. Blue line: Rumor-spreaders in deterministic system. The intensities of the white noise are σ1=2.7,σ2=1.5,σ3=0.3,σ4=0.4,σ5=0.6, and other parameter values are presented in Example 2.

    In the current paper, we have studied a stochastic rumor spreading model. We have established sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to system (1.2) by using the stochastic Lyapunov function method. The ergodic property can help us better understand cycling phenomena of a rumor spreading model, and so describe the persistence of a rumor spreading model in practice. More precisely, we have obtained the following result:

    ● Assume

    Rs0:=Λμα4(ρ+σ212)(α+β+γ+ρ+σ222)(δ+ξ+ρ+σ232)>1.

    Then, for any initial value (I(0),L(0),R(0),D(0),S(0))R5+, system (1.2) has a unique stationary distribution π() and the ergodicity holds. The stationary distribution indicates that the rumor can become persistent in vivo. The theoretic work extended the results of the corresponding deterministic system. The results show that the rumors will maintain its persistence if the environmental noise is sufficiently small, while large stochastic noise can suppress the spread of rumors.

    Some interesting topics deserve further consideration. As we all know, time-delay occurs frequently in many practical engineering systems, which is usually the source of oscillation, instability and poor performance of the systems. Now time-delay has been considered into many stochastic models (see, for example, [4,6,12,13]). We leave time-delay case for our future work.

    This work was supported by the National Natural Science Foundation of China (11701460), the Natural Science Foundation of Sichuan Provincial Education Department (Grant No. 18ZB0581), the Meritocracy Research Funds of China West Normal University (17YC373) and the research startup foundation of China West Normal University (Grant No. 18Q060).

    The authors declare that there are no conflicts of interest.



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