Computers, Materials & Continua DOI:10.32604/cmc.2022.024406
Article

Stochastic Epidemic Model of Covid-19 via the Reservoir-People Transmission Network

Kazem Nouri1,*, Milad Fahimi1, Leila Torkzadeh1 and Dumitru Baleanu2,3

1Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
3Institute of Space Sciences, Magurele–Bucharest, Romania
*Corresponding Author: Kazem Nouri. Email: knouri@semnan.ac.ir; knouri.h@gmail.com
Received: 16 October 2021; Accepted: 17 December 2021

Abstract: The novel Coronavirus COVID-19 emerged in Wuhan, China in December 2019. COVID-19 has rapidly spread among human populations and other mammals. The outbreak of COVID-19 has become a global challenge. Mathematical models of epidemiological systems enable studying and predicting the potential spread of disease. Modeling and predicting the evolution of COVID-19 epidemics in near real-time is a scientific challenge, this requires a deep understanding of the dynamics of pandemics and the possibility that the diffusion process can be completely random. In this paper, we develop and analyze a model to simulate the Coronavirus transmission dynamics based on Reservoir-People transmission network. When faced with a potential outbreak, decision-makers need to be able to trust mathematical models for their decision-making processes. One of the most considerable characteristics of COVID-19 is its different behaviors in various countries and regions, or even in different individuals, which can be a sign of uncertain and accidental behavior in the disease outbreak. This trait reflects the existence of the capacity of transmitting perturbations across its domains. We construct a stochastic environment because of parameters random essence and introduce a stochastic version of the Reservoir-People model. Then we prove the uniqueness and existence of the solution on the stochastic model. Moreover, the equilibria of the system are considered. Also, we establish the extinction of the disease under some suitable conditions. Finally, some numerical simulation and comparison are carried out to validate the theoretical results and the possibility of comparability of the stochastic model with the deterministic model.

Keywords: Coronavirus; infectious diseases; stochastic modeling; brownian motion; reservoir-people model; transmission simulation; stochastic differential equation

1  Introduction

Mathematical biology is one of the most interesting research areas for applied mathematicians. In these research areas, modeling infectious diseases are considered as an important tool to describe the dynamics of epidemic transmission [1–11]. These models play an important role in quantifying the efficient control and preventive measures of infectious diseases. Different scholars used various types of mathematical models in their analysis.

The ongoing Coronavirus disease 2019 (COVID-19) outbreak, emerged in Wuhan, China at the end of 2019 and increased the attention worldwide [12]. COVID-19 has rapidly spread among human populations and other mammals. Additionally, COVID-19 can be capable of spreading from person to person and between cities. Modeling and predicting the evolution of COVID-19 epidemics in near real-time is a scientific challenge [13], this requires a deep understanding of the dynamics of pandemics and the possibility that the diffusion process can be completely random [14]. Nowadays, a large number of models are used to simulate the current COVID-19 pandemic. One of the common ways for analyzing the dynamic of COVID-19 is the use of ordinary differential equations (ODEs), but there are some limitations compared to stochastic models. Recently, many works related to the ODEs have been published (see, for example [15–19]). Furthermore, Tahir et al. [20] developed a mathematical model (for MERS) in form of a nonlinear system of differential equations. Chen et al. [21] developed a Bats-Hosts-Reservoir-People (BHRP) transmission network model for simulating the phase-based transmissibility of Coronavirus, which focus on calculatingR0. Additionally, Hashemizadeh et al. [22], presented a numerical solution for the mathematical model of the novel Coronavirus by the application of alternative Legendre polynomials to find the transmissibility of COVID-19 based on Reservoir-People network model. Also, Gao et al. [23] presented a numerical solution for the BHRP transmission network model for simulating the transmissibility of Coronavirus.

The deterministic models make assumptions about the expected value of parameters in the future, but they ignore the variation and fluctuation about the expected value of parameters. One of the main benefits of the stochastic models is that they allow the validity of assumptions to be tasted statistically and produce an estimate with additional degree of realism. However, there are times when stochastic output cannot be thoroughly reviewed and compared to expectations. Stochastic models have advantages but they are computationally quite complex to perform and need careful consideration of outputs. The parameters of deterministic models governing the equations are supposed to be known and therefore the solutions are often unique. This limitation poses a practical problem because nature is intrinsically heterogeneous and the system is only measured at a discrete (and often small) number of locations. Stochastic models can be regarded as a tool to combine deterministic models, statistics and uncertainty within a coherent theoretical framework. So, this idea leads researchers to consider the stochastic models [3, 24–35]. On the other hand, one of the most considerable characteristics of COVID-19 is its different behaviors in various countries and regions, or even in different individuals, which can be a sign of uncertain and accidental behavior in the disease outbreak. This trait reflects the existence of the capacity of transmitting perturbations across its domains. In past studies, [10–11, 29, 30, 35–39] have been considered a stochastic form of the COVID-19 outbreak.

In this paper, we establish a model for investigation of the COVID-19 transmission based on a stochastic version of the Reservoir-People network with additional degree of realism. Also, it is important to note that an implication of our work is that the numerical simulations demonstrate the efficiency of our model and the possibility of comparability of the stochastic model with the deterministic model. We believe that the stochastic model established by this study is useful for researchers and scientists in making informed decisions and taking appropriate steps to dominance the COVID-19 disease. We conduct numerical simulations to support the theoretical findings. Numerical simulations demonstrate the efficiency of our model. We hope that the stochastic model established by this study will be useful for researchers and scientists in making informed decisions and taking appropriate steps to dominance the COVID-19 disease. The mathematical analysis of this stochastic model can be investigated in future researches and the simulation results can be extended to other countries involved in the global outbreak. This paper could lead to other studies that have included random and uncertainty of the model and can be more consistent with the reality of Coronavirus transmission. This paper is organized as follows. In Section 2, preliminaries are presented. In Section 3, the stochastic version of the Reservoir-People network model of transmission is presented. Section 4 is devoted to present dynamical analysis of solution. Finally, in order to validate our analytical results, numerical simulations are presented in Sections 5 and 6.

2  Preliminaries

Let (Ω,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions. We also let R+d={x∈Rd:xi>0,1≤i≤d}, d−dimensional stochastic differential equation can be expressed as follows:

dX(t)=f(t,X(t))dt+g(t,X(t))dB(t),t≥t0,(1)

with initial value X(t0)=X0, where f(t,x) is a function in Rd defined on [t0,+∞)×Rd,g(t,x) is a d×m matrix and B(t) is an m−dimensional standard Brownian motion defined on the probability space (Ω,Ft,P), also f,g are locally Lipschitz functions in x.

We define the differential operator L associated with Eq. (1) by [39],

L=∂∂t+∑i=1d⁡fi(t,x)∂∂xi+12∑i,j=1d⁡[gT(t,x)g(t,x)]ij∂2∂xi∂xj.

If L acts on function V in C2×1(Rd×[t0,∞)), then we obtain

LV(t,x)=Vt(t,x)+Vx(t,x)f(t,x)+12trace[gT(t,x)Vxx(t,x)g(t,x)],

where

Vt=∂V∂t,Vx=(∂V∂x1∂x2…∂xd)T,Vxx=(∂2V∂xi∂xj)d×d.

The generalized Itô formula implies that

dV(t,x)=LV(t,x)dt+Vx(t,x)g(t,x)dB(t).

3  COVID-19 Model

In this work, we consider a model of 2019-novel Coronavirus (COVID-19) named as Reservoir-people model. This model is the simplified and normalized form of Bats-Hosts-Reservoir-People model [21]. Assuming Sp and Ep refer to the number of susceptible and exposed people, Ip denotes symptomatic infected people, Ap denotes asymptomatic infected people, Rp denotes removed people including recovered and dead people and W denotes the COVID in the reservoir (the seafood market). These variables have the unit of number which are considered at time t. Also, the model parameters and their definitions are given in Tab. 1. Furthermore, the pictorial diagram indicating of the model is shown in Fig. 1.

Figure 1: The model's flow diagram

In this paper, we assume np and mp as the rate of people moving into and moving out from a region per day and the unit of measuring is (individual/day). The incubation period (wp) is the time between infection or contact with the agent and the onset of symptoms or signs of infection and the unit of measuring is (day−1). In this model, the unit of measuringwp′, γp and γp′ is (day−1). The Coronavirus typically spreads via droplets from an infected person's coughs or sneezes. Coronavirus could survive on different types, ranging from 2 h to 14 days. In this model, the unit of measuring ϵ is (day−1). Furthermore, the unit of measuring transmission rates βp and βw is the number of infections per unit time. Also, we defined asymptomatic COVID-19 positive persons as those who did not present any symptoms. The proportion δp is calculated as the number of asymptomatic COVID-19 infections at initial testing over COVID-19 positive persons.

The process of normalization is implemented as follows [21]:

sp=SpNp,ep=EpNp,ip=IpNp,

ap=ApNp,rp=RpNp,w=ϵWμpNp,

μp′=cμp,bp=βpNp,bw=μpβwNpϵ.

Therefore the six-dimensional integer-order COVID-19 model can be expressed by the following ordinary differential equation:

s˙p=np−mpsp−spbp(ip+kap)−spwbw,

e˙p=spbp(ip+kap)+bwspw−wpep+δpwpep−δpwp′ep−mpep,

i˙p=wpep−δpwpep−γpip−ipmp,

a˙p=δpwp′ep−γp′ap−mpap,

r˙p=γpip+γp′ap−rpmp,

w˙=ϵ(ip+cap−w).

3.1 Implementation of Stochastic Description

We can provide an additional degree of realism by defining the white noise and Brownian motion and introducing a stochastic model. Therefore, we implement this idea by replacing random parameters

np→np+σ1B˙1(t),mp→mp+σ2B˙2(t),

bp→bp+σ3B˙3(t),bw→bw+σ4B˙4(t),

δp→δp+σ5B˙5(t),ϵ→ϵ+σ6B˙6(t),

where Bi(t) and σi,i=1,2,…,6 are the Brownian motions and the intensities of the white noises, respectively. These parameters are selected for the implementation of stochastic environment because of their random essence. So, the following stochastic differential equation for COVID-19 model is obtained

sp=(np−mpsp−spbp(ip+kap)−spwbw)dt+σ1dB1−spσ2dB2−sp(ip+kap)σ3dB3−spwσ4dB4,ep=(spbp(ip+kap)+bwspw−wpep+δpwpep−δpwp'ep−mpep)dt+spσ3(ip+kap)dB3+spwσ4dB4+wpepσ5dB5−wp'epσ5dB5−epσ2dB2,ip=(wpep−δpwpep−γpip−ipmp)dt−ipσ2dB2−σ5wpepdB5,(2)

ap=(δpwp′ep−γp′ap−mpap)dt−apσ2dB2,

rp=(γpip+γp′ap−rpmp)dt−rpσ2dB2,

w= ϵ(ip+cap−w)dt+σ6(ip+cap−w)dB6.

4  Dynamical Analysis of Solution

In this section, existence and uniqueness of the solution of the introduced model (2) and the equilibria of the system are presented.

4.1 Existence and Uniqueness of the Solution

Theorem 4.1. The coefficients of the stochastic differential Eq. (2) are locally Lipschitz.

Proof. We define the following kernel γ (sp,ip) = sp.ip. Now we should prove the Lipschitz condition for γ(sp,ip). First we set d= (sp,ip) therefore γ (d) = sp.ip and γ(d)∈C1(Ω,R). So we must prove

∀d1,d2∈Ω,∃K>0:||γ(d2)−γ(d1)||≤K||d2−d1||(3)

Now we choose suitable a subset of Ω and consider N(x,r) as an open ball centered at x and with radius r, such that

N(x,r)={y∣∥y−x∥≤r}

Then we continue our proof by applying: d1,d2∈N(x,r)⊂Ω⊂R2. Since γ (d) is unbounded therefore, due to (3), we replace K with Kx which depends on x.

So we define the path ψ:[0,1]→N(x,r) such that ψ(t)=(1−t)d1+td2. So. We consider the following increment from d1 to d2:

∥γ(d2)−γ(d1)∥=∥γ(ψ(1))−γ(ψ(0))∥=∫01dγ(ψ(t))dtdt

where dγ(ψ(t))dt=∇γ(ψ(t))ψ′(t)=∇γ(ψ(t))(d2−d1). Thus we have:

‖∫01dγ(ψ(t))dtdt‖=‖∫01∇γ(ψ(t))(d2−d1)dt‖≤∫01∥∇γ(ψ(t))∥‖d2−d1‖dt

=||d2−d1||∫01∥∇γ(ψ(t))∥dt≤Kx||d2−d1||

Where Kx>sup{∥∇Υ(d)∣d∈N(x,r)∥} .

As a result, there is no global Lipschitz constant K valid on all of \varOmega =R2 in  this case, thus  γ (d) is in fact locally Lipschitz.

Lemma 4.1. Suppose thatX⊂Rn, and let f, g: X⊂Rn satisfy Lipschitz conditions, then f+g satisfies Lipschitz condition.

Proof. The proof is presented in [40].

Therefore, we have local Lipschitz condition for the coefficient of dB1(t) and similarly, this condition will be proved for the coefficient of dB2(t),dB3(t),dB4(t),dB5(t) and dB6(t). So our Model contains stochastic differential equations with Lipschitz coefficient and we can obtain a strong solution and for each initial value, the solution will be unique.

Theorem 4.2. For any given initial value (sp(0),ep(0),ip(0),ap(0),rp(0),w(0))∈R6+, there exists the solution (sp(t), ep(t), ip(t), ap(t), rp(t), w(t)) for model (2) and the solution will remain in R6+ with probability one.

Proof. There exists a local solution on the interval [0,τe) where τe is the explosion time [39]. Now, we need to prove τe=+∞. Let m0≥0 be sufficiently large such that each component of initial value (sp(0), ep(0), ip(0), ap(0), rp(0),w(0)) remain within the interval [1m0,m0]. For each integerm≥m0, we define

τm=inf{t∈[0,τe):min{(sp(t),ep(t),ip(t),ap(t),rp(t),w(t))}≤1m}(4)

or

τm=inf{t∈[0,τe):max{(sp(t),ep(t),ip(t),ap(t),rp(t),w(t))}≥m}(5)

where τm is stopping time. We denote τ+∞=limm→∞τm and we will show that τ+∞=+∞  is valid almost surely. Then the proof goes by contradiction. If this statement is false, then there exist constants T≥0 and \eta ∈(0,1)such that P{τm≤T}≥ηforanym≥m0.

Let f be a C2 - function, by defining f:R+6→R such that

f(sp(t),ep(t),ip(t),ap(t),rp(t),w(t))=lnsp(t)+lnep(t)+lnip(t)+lnap(t)+lnrp(t)+lnw(t)

Generalized Itô’s formula yields that

df(t,sp,ep,ip,ap,w)=fspdsp+fepdep+fipdip+fapdap+frpdrp+fwdw++12(fspspdspdsp+fepepdepdep+fipipdipdip+fapapdapdap+frprpdrpdrp+fwwdwdw)+…

So we obtain

df=npns−5mp−bp(ip+k⋅ap)−w⋅bw+spbpep(ip+kap)+bw⋅sp⋅wep−wp+δpwp−δpwp′+wpepip−σpwpepip−γp+δpwp′epap−γp′+γpiprp+γp′aprp+ε(ip+C⋅apw−1)−12(σ12sp2+5σ22+σ32(ip+k⋅ap)2+w2σ42+w2σ42sp2ep2+wp2σ52+wp′σ52+σ52wp2ep2ip2+σ62(ip+C⋅ap−w)2w2+sp2σ32(ip+k⋅ap)2ep2)+σ1spdB1−5σ2dB2+(spep−1)(ip+Kap)σ3dB3+wσ4(spep−1)dB4+(wp(1−epip)−wp′)σ5dB5+σ6ε(ip+C⋅apw−1)dB6

Therefore we obtain

df=Lfdt+σ1spdB1−5σ2dB2+(spep−1)(ip+Kap)σ3dB3+wσ4(spep−1)dB4 +(wp(1−epip)−wp')σ5dB5+σ6ε(ip+C⋅apw−1)dB6 (6)

where K is a positive constant and we have

Lf=npns−5mp−bp(ip+k⋅ap)−w⋅bw+spbpep(ip+k⋅ap)+bw⋅sp⋅wep−wp+σpwp−σpwp+wpepip−δpwpepip−γp+δpwpepap−γ′p+γpiprp+γ′paprp+ε(ip+C.apw−1)−12( σ12sp2+5σ22+σ32(ip+k⋅ap)2+w2σ42+w2σ42sp2ep2+wp2σ52+wpσ52+σ52wp2ep2ip2+σ62(ip+C⋅ap−w)2w2 +sp2σ32(ip+k⋅ap)2ep2 )≤npns+spbpep(ip+K.ap)+bw⋅sp⋅wep+δpwp+wpepip+δpw′pepap+γpiprp+γ′paprp+ε(ip+C⋅apw−1)≤K(7)

Due to (7) and integrating both sides of (6) from to (τm∧T)and taking expectation deduce that Ef≤f(sp(0),ep(0),ip(0),ap(0),rp(0),w(0))+KT

Thus for every ω∈Ωm={τm≤T} we obtain

f(sp(0),ep(0),ip(0),ap(0),rp(0),w(0))+KT≥E[1Ωm(w).f(s(τm∧T), ep(τm∧T), ip(τm∧T)

 , ap(τm∧T),rp(τm∧T),w(τm∧T))]

≥p{τm≤T}⋅[(lnm)∧(−lnm)]

≥η[(lnm)∧(−lnm)]

where 1Ωm(ω)is indicator function of \varOmega . Letting m→+∞  leads to contradiction

+∞>f(sp(0),ep(0),ip(0),ap(0),rp(0),w(0))+KT≥+∞.

So the statement τ+∞=+∞ is being satisfied and the proof is complete.

4.2 Extinction of the Disease

In this section, we establish sufficient conditions for the extinction of the disease. In order to form these conditions, we consider variation in the number of symptomatic infected people (Ip). Chen et al. [21] have proved that the deterministic model admits the basic reproduction number R0 as follows:

R0=bpψ+κbpψ′+bwψ+bwcψ′ where ψ=bpnpmp(1−δp)wP(1−δp)wp+δpwp′+mp(γp+mp) and ψ′=κbpnpmpδpwp′(1−δp)wP+δpwp′+mp(γp′+mp)

In deterministic epidemiological models, basic reproduction number is planned to be a criterion of transmissibility of infectious agents. IfR0>1, then the outbreak is expected to continue if R0<1 then the outbreak is expected to become extinct. In this stochastic COVID-19 model, we consider the effect of high intensity white noises on the prevalence or extinction of the COVID-19. So we present the following theorem to demonstrate that random effect may lead the disease to extinct under suitable conditions, although the disease may be immanent for the deterministic model. Before presenting the main theorem of this section, we refer to the following lemma presented in [39].

Lemma 4.2. Let Z(t): t≥0 be a continuous local martingale [41] and < Z(t), Z(t) > be its quadratic variation. Let 0 <c<1 and k be a random integer. Then for almost all ω∈Ω, there exit a random integer k0(ω) such that for allk>k0:

P{sup0≤t≤k⁡[Z(t)−c2<Z(t),Z(t)>]≥2cln⁡k}≤1k2c

Theorem 4.3. Let (sp(0), ep(0), ip(0), ap(0), rp(0), w(0))∈ R6+ be the solution of model (2) with initial value (sp(0), ep(0), ip(0), ap(0), rp(0), w(0))∈ R6+, we obtain

limt→+∞⁡supln⁡I(t)t≤12σ52−(12σ22+Υp+mp) a.s.

Furthermore, if the assumptions 12σ52≤12σ22+Υp+mp and δp>1 hold, then I(t) will tend to zero exponentially with probability one.

Proof. Itô’s formula yields that

d(ln⁡ip)=(ωpepip−δpωpepip−Υp−mp)dt−σ2dB2(t)−σ5ωpepipdB5(t)

So we obtain

ln⁡(ip)=ln⁡(ip(0))+∫0t⁡(ωpepip−δpωpepip−Υp−mp−12σ22−12 σ52ωp2ep2ip2)dt−σ2B2(t)+Z(t). (8)

where Z(t)=−∫⁡σ5ωpepipdB5(t) is a continuous local martingale and its quadratic variation is

<Z(t),Z(t)> = σ 52∫⁡ωp2ep2ip2dt

According to Lemma 4.2, we obtain

P{sup0≤t≤k⁡[(Z(t)−c2⟨Z(t),Z(t)⟩)≥2cln⁡k]} ≤ 1k2c

where 0<c<1 and k is a random integer. Applying Borel-Cantelli Lemma [41] leads to that for almost all ω∈Ω, there exit a random integer k0(ω) such that for k>k0:

sup0≤t≤k⁡[(Z(t)−c2⟨Z(t),Z(t)⟩]≤2cln⁡k

Therefore

∫0t⁡σ5ωpepipdB5(t)≤12σ52∫0t⁡ωp2ep2ip2dt+2cln⁡k(9)

for 0≤t≤k. Substituting (9) into (8) yields

ln⁡I(t)≤ln⁡ip(0)+∫0t⁡(ωpepip−δpωpepip−Υp−mp)dt+σ2B2(t)+12cσ52∫⁡ωp2ep2ip2 dt+2cln⁡k−12δ22−12σ52ωp2ep2ip2=∫0t⁡(ωpepip−δpωpepip−Υp−mp−12(δ22+δ52ωp2ep2ip2)+12cδ 52ωp2ep2ip2)dt+δ2B2(t)+2cln⁡k =∫0t⁡(ωpepip−δpωpepip−Υp−mp−12σ22−12(1−c)σ52ωp2ep2ip2)dt+σ2B2(t)+2clnk.

If δp>1, we obtain

ωpepip−δpωpepip−12(1−c)σ52ωp2ep2ip2=(1−δp)ωpepip−12(1−c)σ22ωp2ep2ip2

≤−σ522(1−c)(−2ωpep(1−c)ipσ52+ωp2ep2ip2)

=−σ222(1−c)(ωp2ep2ip2−(2ωpep(1−c)ipσ52+1σ54(1−c)2−1σ54(1−c)2)

=−σ522(1−c)(ωpepip−1σ52(1−c))2 +12σ52(1−c)≤12σ52(1−c)

So we have

lnI(t)≤lnI(o)+∫0t(−12σ22−Υp−mp+12σ52(1−c))dt+σ2B2(t)+2clnk=lnI(0)+(−12σ22−Υp−mp+12σ52(1−c))t+σ2B2(t)+2clnk.(10)

By dividing both sides of Eq. (10) by t such that k−1≤t≤k, we achive

ln⁡I(t)t≤ln⁡I(0)t+(−12σ22−Υp−mp+12σ52(1−c))+σ2B2(t)t+2ctln⁡k.