key: cord-351343-zdh8ms1z
authors: Din, Anwarud; Khan, Amir; Baleanu, Dumitru
title: STATIONARY DISTRIBUTION AND EXTINCTION OF STOCHASTIC CORONAVIRUS (COVID-19) EPIDEMIC MODEL
date: 2020-06-24
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110036
sha: 
doc_id: 351343
cord_uid: zdh8ms1z

Similar to other epidemics, the novel coronavirus (COVID-19) spread very fast and infected almost two hundreds countries around the globe since December 2019. The unique characteristics of the COVID-19 include its ability of faster expansion through freely existed viruses or air molecules in the atmosphere. Assuming that the spread of virus follows a random process instead of deterministic. The continuous time Markov Chain (CTMC) through stochastic model approach has been utilized for predicting the impending states with the use of random variables. The proposed study is devoted to investigate a model consist of three exclusive compartments. The first class includes white nose based transmission rate (termed as susceptible individuals), the second one pertains to the infected population having the same perturbation occurrence and the last one isolated (quarantined) individuals. We discuss the model’s extinction as well as the stationary distribution in order to derive the the sufficient criterion for the persistence and disease’ extinction. Lastly, the numerical simulation is executed for supporting the theoretical findings.

The 2019-novel coronavirus has been known to the virologists community as Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) [1] . The COVID-19 refers to the virus associated syndrome. SARS-CoV-2 being previously unrecognized novel-strain of the coronavirus in humans [2, 3] . Coronaviruses in general circulate among various animals with some being highly susceptible for infecting humans. Among these animals, naturally bats are thought to be proven hosts of such novel coronaviruses, nevertheless, various species of other animals are also considered an active cause for such spreads [4] . At present, the Middle East Respiratory Syndrome Coronavirus (MERS-CoV)" which is much similar to COVID-19 was spread from camels to humans, while the civet cats have been considered as source of "Severe Acute Respiratory Syndrome Coronavirus-1 (SARS-CoV-1)" for transmission into human. Bunch of information are presented in the ECDC factsheet on coronaviruses [4, 5] .

Though the animals are understood to be a proven source, however, currently, human-tohuman transmission is also considered as one of the spread source. At present, the epidemiological information are sparse for the determination of an effortless spread of this virus among the people, nonetheless, currently, on average, it is estimated that, infection in one person can cause the spread among 2-3 more people [1, 5] . The virus appears to be transferred mostly through narrow respiratory droplets by coughing, sneezing, or peoples interaction in close proximity (usually less than one meter) with each other for a certain time frame. These droplets can further be inhaled, or can stay on the surfaces being came in contact by the infected person, that can cause infection in others by touching their nose, mouth or eyes. The virus possesses ability to survive on various surfaces commencing several hours (e.g. copper, cardboard) up to a few days (e.g. plastic and stainless steel). Nonetheless, the quantity of the viable virus certainly decays over a time span and might not be present in sufficient quantity for causing the infection. It is currently estimated that the appearance of symptoms and initial infections in case of COVID-19 almost lies between 1-14 days [1, 5, 6] . Moreover till today there is no proper treatment in term of vaccine etc. However, the scientists are working faster to develop vaccine for the novel COVID-19, which will take enough time. Therefore the only way to stop the spread of this disease is to quarantine or isolate the initially infected population as showed by the Chinese Govt and also the guide line of WHO.

It could be also noted that most of the real world phenomenon are not simply deterministic, because in deterministic models, the output of the model is fully determined by the parameter values and the initial conditions. Stochastic models possess some inherent randomness. The same set of parameter values and initial conditions will lead to an ensemble of different outputs or we can say in simple world a deterministic model is one that uses numbers as inputs, and produces numbers as outputs. A stochastic model includes a random component that uses a distribution as one of the inputs, and results in a distribution for the output. These distributions may reflect the uncertainty in what the input should be (e.g. a deterministic input plus noise), or may reflect a random process (i.e. a stochastic input) [7] [8] [9] .

For describing the changing behavior of several epidemic diseases in a realistic sense, the mathematical modeling is considered as an influential tool. Several epidemic models have been developed by various mathematicians and ecologists for comprehending and controlling various epidemic diseases in a region. In last twenty years, mathematical modeling is widely used for characterizing the communication of various infectious diseases (see e.g. [10, 11] . Recently various comprehensions have been made to deepen the understanding about the novel coronavirus (COVID-19) particularly grasping the valuable inferences through mathematical modeling [12] [13] [14] . The modelsdescribe the dynamics of infectious diseases, however, for modeling biological phenomenon, it is appropriate to use the stochastic differential equations due to its realistic approach. Compared to deterministic models, the stochastic models can generally result in more valuable output, by several times execution, a distribution of the expected results can be built, such as the average infections at any time t, whereas the deterministic models result in a single predicted value [15] [16] [17] [18] . Numerous approaches and methods exist for studying stochastic models (such as Binomial moment equation etc.) [19, 20] .

The most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. How can a model be defined explaining the sometimes-observed scenario of frequent mid-sized epidemic outbreaks? How can evolution of the infectious agent transmission rates be modeled and fitted to data in a robust way? In this paper we understand the transmission mechanism of the COVID-19 mathematically, we have formulated a model using the available literature on modeling epidemics, we propose a stochastic epidemic model for the transmission dynamics of COVID-19 virus with a varying population environment for a longterm behavior. We categorize the total population into three different classes. The first class is the susceptible individuals in which the transmission rate is distributed by white noise. The second class includes the infected individuals in which the same transmission occurs. The third class consists of the quarantine individuals with white noise.

In the recent study, we proposed a stochastic epidemic model for the transmission dynamics of the COVID-19 with a changing environment considering long term behavior. The overall population has been divided into three exclusive classes: the susceptible individuals with white noise transmission rate distribution, the infected individuals in which the same perturbation occur and quarantined individuals. Then, we will discuss the disease' extinction and stationary distribution and develop the sufficient condition for the COVID-19. Furthermore, sample simulations are find out with the help of stochastic Runge-Kutta method for supporting the theoretical results.

The present section is devoted to formulation of a model based on stochastic theory for studying the transmissions dynamic of the novel virus i.e., COVID-19 pandemic. We propose a susceptibleinfected-quarantined epidemic model as according to the characteristic of the disease. We also take the varying population environment to study the dynamics of the COVID-19 particularly its long-term behavior. Before to present the model, we put some assumption as given by the following assertions.

(A 1 ). The total population at any time t is symbolized by N(t) and it is stratified into three exclusive groups of individual: the susceptible class S(t), the COVID-19 infected people I(t) and the quarantined Q(t), i.e., S(t) + I(t) + Q(t) = N(t) which is changing with t.

(A 2 ). The state variables and parameters included in the model are assumed to be nonnegative.

(A 3 ) The initially infected individuals move to the quarantined class as performed by the Chinese in Wuhan city.

(A 4 ). Once the infection confirmed then the quarantined will go back to the infected compartment.

In the light of the above assumption (A 1 ) − (A 4 ), the proposed model leads to the following stochastic epidemic problem which consist of three stochastic differential equations

Here in the model, Λ represents the per capita constant fecundity rate. µ 0 , µ 1 and µ represents the natural mortality rate and disease-related mortality rate, respectively. γ 1 represents the constant rate at which people getting quarantined from COVID-19 infected class. B(t) is considered to be the usual Brownian motion with intensity η 1 ,η 2 and η 1 taken to be positive.

Let (Ω, {F t } t≥0 , P) be the complete probability space with filtration {F t } t≥0 which satisfy the normal conditions,

along with condition z(t 0 ) = z 0 ∈ R d , where B(t) denotes an m-dimensional usual Brownian motion. Define the operator L related to (4) by

By operating L on V (a function from the space

By generalized Itô's formula, we have

This section is about studying the existence and uniqueness of solution of the proposed stochastic COVID-19 model (1). (1) is unique for t ≥ 0 with initial condition (S(0), I(0), Q(0)) ∈ R 3 + . Further, the solution will always remains in R 3 + with unit probability, that is, (S(t), I(t), Q(t)) ∈ R 3 + ∀ t ≥ 0 almost surely (a.s). Proof: As for initial value of the state variables (S(0), I(0), Q(0)) ∈ R 3 + , the coefficients used in equations are continuous and locally lipschitz. Thus, there must exists a local unique solution (S(t), I(t), Q(t)) of the model over t ∈ [0, τ e ). For detail analysis of the explosion time τ e one must see the references [21, 22] . To prove the global nature of the solution, we must show that τ e = ∞ a.s. Assume that we have a sufficiently large nonnegative number k 0 such that all of the initial conditions on the state lie within [ 1 k 0 , k 0 ]. Let for each positive integer k ≥ k 0 , the finishing time be defined as

Throughout this manuscript, we must choose inf φ = ∞ where φ stand for the null set. Definition of τ k force us to say that it is increasing as k tends to ∞. Setting τ ∞ = lim k→∞ with τ e ≥ τ ∞ a.s. Upon showing τ ∞ = ∞ a.s., we will declare that τ e = ∞ and hence (S(t), I(t), Q(t)) will lies in R 3 + a.s. ∀t ≥ 0. Thus, it is suffice to prove that τ e = ∞ a.s. In otherwise case, there must exists two positive constants from (0, 1) and T such that

Hence there is an integer k 1 ≥ k 0 , such that

Next, we will define a C 2 -function H :

It is to be noted that the H is a nonnegative function, and it can be verified from the fact that 0 ≤ y − logy − 1, ∀0 < y. Assume that k 0 ≤ K and 0 < T are arbitrary. Upon applying Itô formula to Eq.(5) gives us

In Eq.(6), LH : R 3 + → R + is defined by the following equation

Thus,

Setting Ω k = {τ k ≤ T} for k ≥ k 1 and by Eq.(4), P(Ω k ) ≥ . Note that for each ω from Ω k there must exist one or more than one S(τ k , ω),

By using Eq.(4) and Eq. (8), we can write

Here 1 Ω(ω) represent the indicator function of Ω. Approaching k to ∞ will lead us to the contra- 

which implies that S(t) > 0. Solving the second equation of the model (1) gives us

which simply means that 0 ≤ I(t). It is handy to shown that 0 < Q(t). Hence (S(t), I(t), Q(t)) ∈ R 3 + , for all t ≥ 0, which proves the conclusion. Remark 1. Clearly, Theorem 1 and Theorem 2 guarantees that for the initial data (S(0),

By solving the differential inequality Eq. (12) yields

In upcoming study, we shall always assume that (S(0), I(0), Q(0)) ∈ Ω * unless otherwise stated.

As for as the stochastic systems are concerned, they have no endemic equlibria. Thus, the stability analysis cannot be used as a tool for studying the disease' persistence. As a result, one must turn his/her attention to the existence/uniqueness theory of the stationary distribution which in some sense, will work for persistence of the disease. For this purpose, we will cite a famous result from Hasminskii [23] . Let 

" Lemma 2. [16, 17] 

"

Suppose that X(t) is a regular Markov process (time-homogeneous) in R n + whose dynamics is given by

"The diffusion matrix is of the form

Lemma 3. ( [16, 17] 

Define a parameter

Theorem 3. The solution (S(t), I(t), Q(t)) of the model (1) is ergodic as well as there is a unique stationary distribution π(.) whenever R S 0 > 1.

In order to verify condition (2) of Lemma (3), we need to develop a non-negative C 2 −function V : R 3 + → R + . For this, we will first define

where c 1 , c 2 are the positive constant and need to be determined later on. By using the Ito's formula and the proposed model (1), we obtain

Therefore, we have

The above implies that

Let

Namely

Consequently

In addition, we can obtain

where the constant c 3 > 0 to be determined at later stages. It is handy to show that lim inf (S,I,Q)∈R 3 + \U k V 2 (S, I, Q) = +∞, as k → ∞,

where

. The next step is to prove that V 2 (S, I, Q) has one and only one minimum value V 2 (S 0 , I 0 , Q 0 ). The partial derivative of V 2 (S, I, Q) with respect to S, I, Q is as follow

It could be easily obtain that V 2 have unique stagnation point (S 0 , I 0 , Q 0 ) = 1+c 1 c 3 1+c 3 , c 3 c 2 1+c 3 , 1 1+c 3 . Moreover, the Hesse matrix of V 2 (S, I, R) at (S 0 , I 0 , Q 0 ) is

Obviously, the Hesse matrix is positive definite. Thus, V 2 (S, I, Q) has a minimum value V 2 (S 0 , I 0 , Q 0 ). According to Eq. (22) and from the continuity of V 2 (S, I, Q), we can say that V 2 (S, I, Q) has one and only one minimum value V 2 (S 0 , I 0 , Q 0 ) inside R 3 + . Next, we will define a non-negative C 2 −function V : R 3 + → R + as follows

Applying the Itô's formula and using the proposed model, we get

which leads to the following assertion

where δ i > 0 for (i = 1, · · · , 6) are infinitesimally small constants to be determined later. For the sake of simplicity, we will divide the whole R 3 + \D into the following regions

Next, we shall prove that LV(S, I, Q) < 0 on R 3 + \D which is the same as displaying it on the above-mentioned six regions. 

We can choose a small constant δ 1 > 0 in such a way that c 1 c 3 β + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 3 2 + µ + σ + Λ − Λ δ 1 ≤ 0 so we can get LV < 0 for each (S, I, Q) ∈ D 1 .

Case 2. If (S, I, Q) ∈ D 2 , then from Eq.(24), we can obtain

Let δ 2 1 = δ 2 , we can select enough large c 3 > 0 and as small as possibleδ 1 

By choosing small δ 3 > 0 such that c 1 c 3 β + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 

We can select enough small δ 4 > 0 such that 

We can choose sufficiently small δ 5 > 0 such that c 1 c 3 βδ 1 + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 

We can choose sufficiently small δ 6 > 0 such that

Thus, we reach to the conclusion that there exist a constant W > 0 such that

Assume that (S(0),

and τ x is that time at which a path starting from x reach to the set D, τ n = in f {t : |X(t)| = n} and τ (n) (t) = min{τ x , t, τ n }.

Upon integration of both sides of the inequality (25) from zero to τ (n) (t), taking expectation, and then by applying Dynkins formula, we obtain EV(S(τ (n) (t)), I(τ (n) (t)), Q(τ (n) (t)))V(x) = E τ(n)(t) 0 LV(S(u), I(u), Q(u))du,

Following the proof of Theorem (3) we have P{τ e = ∞} = 1. Alternatively, one can say that the system (1) is regular. Thus, if we let t → ∞ and n → ∞ then we have τ(n)(t) → τ x almost surely. Accordingly, with the help of Fatou's lemma we get

Obviously, sup x∈K Eτ x < ∞, where K being a compact subset of R 3 + . It directly proves the condition (ii) of Lemma 3.

Moreover, the diffusion matrix for system (1) is given by

It means, condition (1) of Lemma (3) also holds. Concluding the above discussion, we can say that Lemma (3) guarantees that system (1) is ergodic as well as it has one and only one stationary distribution. Hence the proof. (1) along with initial data (S(0), I(0), Q(0)) ∈ R 3 + , then lim sup t→∞ (S(t)

Proof: From the proposed model (1) we can write

The integration of both sides yields

We define

where

, Setting

Because of the quadratic variation, we can write

By using Lemma(1) (see for detail [16] [17] [18] ) and (31), we get

Similarly, we also get

which proves Eq. (26) and hence the Lemma 4.1.

For the purpose of disease' extinction, we have to state and prove the following theorem. (1) 

a.s., (I(t) approaches zero exponentially a.s., i.e., the COVID-19 infection will dies out from the community with unit probability). Moreover

Proof: To prove the theorem, we shall apply direct integration to the proposed stochastic COVID-19 model (1) . First of all, we will apply the Itô formula to the second equation of system (1)

By integrating relation (35) from zero to t and dividing it by t leads to

By using the theorem related to large number for local martingales, we obtain lim t→∞ B 2 (t) t = 0 a.s.

By taking the limit superior of both sides In fig. 9 the total cases of covid-19 has been depicted from 9 th April to 2 nd June 2020, which becomes one month and 24 days. In fig. 10 we fitted the real data with the infected class of our covid-19 model which clearly shows the appropriateness of behavior of the infected class. Fig.  11 shows long time behavior of the covid-19 cases vs time (months). We can see that the data is accurately fit to the model curve and further, one can observe that the cases with time on long term behavior grows exponentially. This case could be alarming that the incidence may increases further in the coming months if the government not applied the proper optimal strategies. Covid-19 incidence data 

In the current section, we shall perform the numerical simulation of the developed coronavirus stochastic epidemic model. The well know stochastic Runge-Kutta (RK) method for the purposes of numerical findings will be used. This analysis will verify our derived analytical results and will show the influence and effect of noise intensity. We assume the numerical value of the parameters with biological feasibility to verify the extinction result are as: i.e., the infection of novel corona virus vanishes exponentially with increasing the value of white noise intensity. However there will be always susceptible population in the case of extinction. In a similar fashion, we assume the following parameter value and the strong effect of white noise to show the permanence or stationary distribution i.e., Λ = 0.5, µ 0 = 0.2, β = 0.6, γ 1 = 0.3, µ 1 = 0.2, σ = 0.1, µ = 0.2, η 1 = 0.5, η 3 = 0.7, and η 2 = 0.6 while the initial population sizes will be taken as above. The simulation carried out for this are presented in Fig.4b . Again the three trajectories in Fig.4b , which represent the dynamics of susceptible (red dashed), infected (purple solid) and quarantined population (red solid), which show that the model maintain the persistence i.e., there will be always susceptible, infected and quarantine individuals. Hence it could be noted from the simulation analysis that the white noise intensity have a great influence on the dynamics of the disease: as when the value of the white noise intensity increases the infection will decreases, while on the other hand if the value of the white noise intensity decrease, the infection will increases. 

The novel COVID-19 is one of the severe disease in the world and till today there is no proper treatment. It could be also noted that majority of real world phenomenon are not simply deterministic, and contain randomness. With the help of stochastic theory, we developed a model for the novel COVID-19 keeping in view the characteristic of the disease to investigate the transmission dynamics with changing population environment. By adopting the idea of stochastic Lyapunov functions theory, the existence and positivity are shown. We established a suitable stochastic Lyapunov function to perform the above activity. The extinction as well as the stationary distribution have been further discussed to find the conditions that how to extinct the disease. It could be noted that the there is a great influence of noise intensity on the COVID-19 transmission. Clearly it has been observed, that the extinction of COVID-19 infected individuals increases with increasing the noise strength, while decreases disease persisting. All the above analytical findings are supported graphically with the help of numerical simulation and therefore concluded that the work reveals stochastic analysis is a better approach to study the dynamics of infectious disease particularly novel COVID-19 etc, because there are many factor which varies time to time and place to place. In future, the model can be further extended by adding an exposed class. One can also fractionalize the model by using Atangana-Baleanu, Caputo or Caputo-Fabrizo operator. Not only this can but researcher may apply optimal control technique to minimize the infected people by choosing suitable optimal control variables.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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This work was supported by the HEC.

Adding respective sides of equations (40), we getCalculation leads towhereFrom the last equation of system (40) we havewhich implieswe thus obtain lim