key: cord-349841-eigcqb1b
authors: Boukanjime, Brahim; Caraballo, Tomas; Fatini, Mohamed El; Khalifi, Mohamed El
title: Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switching
date: 2020-10-16
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110361
sha: 
doc_id: 349841
cord_uid: eigcqb1b

In this paper, we analyze a stochastic coronavirus (COVID-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. Firstly, we investigate the existence and uniqueness of a global positive solution. Then, we establish the stochastic threshold for the extinction and the persistence of the disease. The data from Indian states, are used to confirm the results established along this paper.

Today, the world is facing the ongoing COVID-19 pandemic, caused by the SARS-CoV2 coronavirus. The novel coronavirus has been a serious threat to public health [4] . In late December 2019, the disease COVID-19 was first discovered in Wuhan (Hubei province) and caused the first pandemic of this century. The virus appears to be transferred mostly through narrow respiratory 5 droplets by coughing, sneezing, or peoples interaction in close proximity (usually less than one meter) with each other for a certain time frame. However, it might be possible that other unobserved environmental exposures may have facilitated the rate the disease spreads through human-to-human transmission. In [13] , it is reported that COVID-19 infected individuals generally develop symptoms, including mild respiratory symptoms and fever, on an average of 5-6 days after infection (mean 5-6 days, range 1-14 days). At the present, there is no effective treatment for COVID-19 in the world. Therefore the only way to stop the spread of this disease is to quarantine or isolate the initially infected population as showed by guide line of World Health Organization. By the end of June 2020, the COVID-19 virus has infected more than 10 927 025 people and died at least 521 512 in all over the world [27] . However, since the randomness of population mobility and uncertainty of 15 control measures, methods of predicting COVID-19 and then preventing and controlling the disease for public health departments still remain unclear.

Recently, the novel coronavirus COVID-19 has attracted much attention from many researchers and various comprehensions have been made to deepen understanding and grasping the valuable inferences through mathematical modeling [14, 15, 16] . Therefore, it is of great significance to 20 establish and study the model of infectious diseases. Mathematical modelling is an important decision tool that can be useful to analyze the spread and understand the level of manageability and the effect of prevention and control mechanisms applied to the pandemic. A numerous number of models are being used to project the current COVID-19 pandemic. Wang et al. [17] developed an SEIR model to estimate the epidemic trends in Wuhan, assuming the prevention and control 25 measures were either sufficient or insufficient to control the epidemic. Hellewell et al.

[18] developed a transmission model and found that highly effective contact tracing and case isolation are enough to control a new outbreak of COVID-19 within three months in most scenarios. In another recent work, Chakraborty and Ghosh [28] have considered a hybrid ARIMA-WBF model to forecast various COVID-19 affected countries throughout the globe. Several other models established 30 a stochastic transition model to evaluate the transmission of COVID-19 and also emphasized the necessity of interventions such as social-distancing, isolation and quarantine [19, 21] .

In [24] , Mandal et al. consider a mathematical model of COVID-19 where human populations are subdivided into five time-dependant classes, namely, Susceptible S(t), Exposed E(t), Quarantined Q(t), Hospitalized infected I(t) and Recovered or Removed R(t). They have assumed that the virus COVID-19 is spreading when a vulnerable person comes into contact with an exposed person. The model is a system of five first order ordinary differential equations shown as below:

where all parameters are positive numbers. A is the constant recruitment rate to the susceptible population; β stands for the disease transmission rate; ρ 1 (0 < ρ 1 < 1) is portion of susceptible human would maintain proper precaution measure and ρ 2 (0 < ρ 2 < 1) represents portion of the exposed class would take proper precaution measure for disease transmission (i.e, use of face mask, social distancing and implementing hygiene). Therefore (1 − ρ 1 )S denotes portion of susceptible individuals due to the contact of (1 − ρ 2 )E portion of exposed individuals; α and b 2 are the portions of the exposed class going to the infected class and quarantine class, respectively. However, b 1 and c represent the portions of the quarantine class moving to susceptible class and infected class, respectively. η and ν stand for the recovery rate of hospitalized infected population I and exposed class E; µ denotes the natural death rate and δ is the COVID-19 induced death rate.

According to the theory in [24] , the system (1) always has a disease-free equilibrium

R 0 = Aβ(1 − ρ 1 )(1 − ρ 2 ) µ(b 2 + α + ν + µ) > 1, there exists a unique endemic 35 equilibrium E * = (S * d , E * d , Q * d , I * d , R * d ), where S * d = b2+α+ν+µ β(1−ρ1)(1−ρ2) , E * d = (b 1 + c + µ) Aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ) β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)} , Q * d = b 2 Aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ) β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)} , 40 I * d = {α(b1+c+µ)+b2c}{Aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ)} β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)}(η+µ+δ) , R * d = ηI * d +νE * d µ .

[24] established the following theoretical results about the stability of the equilibriums:

(i) If R 0 < 1, then the disease-free equilibrium E 0 of system (1) is locally asymptotically stable.

(ii) If R 0 = 1, the system (1) passes through a transcritical bifurcation around its disease-free equilibrium.

(iii) If R 0 > 1, then the endemic equilibrium E * of system (1) is locally asymptotically stable.

In fact, the COVID-19 epidemic model is unavoidably subjected to the environmental noise, which made the parameters involved in the system often fluctuate randomly around some average values as the surrounding environment fluctuation. See [1, 2, 3, 6, 9, 10, 20, 29] and references therein for epidemic models with environmental noise. Therefore, it is necessary to include random fluctuations in the process of COVID-19 modelling. In this paper, we propose a stochastic COVID-19 model adopting a generalized incidence function [25, 26] as follows:

where B(t) is a real-valued Brownian motion and σ 2 > 0 represents the intensity of the white noisė

. The function f (·) is generally assumed to be a non-negative twice continuously differentiable

x < f (0) for any x > 0). Note that the COVID-19 epidemic models may be perturbed by telegraph noise which can causes the system to switch from one environmental regime to another [22] . Mostly the switching between environmental regimes is often memoryless and the waiting time for the next switching follows the exponential distribution [23] . Hence the regime switching can be modelled by a continuous time Markov chain (r(t)) t 0 with values in a finite state space S = {1, 2, ..., N }. Then model (2) disturbed by white noise and telegraph noise develops to

where Markov chain r(.) is F t -adapted but independent of Brownian motion B(.) and defined on

of Markovian chain is defined by

In this paper, we assume that γ ij > 0 for i, j = 1, ..., N with j = i. This assumption assures that the Markov chain r(t) is irreducible, which implies that it has a unique stationary distribution π = (π 1 , π 2 , ..., π N ), which can be determined by solving the following equation

subject to N k=1 π k = 1, π k > 0, for any k ∈ S.

For convenience, we denote for any fixed vector

To begin the analysis of the model, we define the subsets

Throughout this paper, we carry out the case of small noises:

The structure of the rest of the paper is as follows: In section 3, we show the existence and uniqueness of a global positive solution to the system (3). In the sections 4 and 5, we study the 55 existence of a stochastic threshold for the extinction and the persistence in mean of the disease. In last section, we present some numerical simulations to demonstrate our main theoretical results.

To study the dynamical behaviour of an epidemic model, we firstly need to consider whether the solution is global and positive. In this section, we will prove there is a unique global positive Proof. Obviously, the coefficient of model (3) are locally Lipschitz continuous, so there is a unique lo-

where τ e is the explosion time [31] . If τ e = ∞ a.s., then this local solution is global. To this end, 

where throughout this paper we set inf ∅ = ∞ (∅ denotes the empty set). Clearly, τ n is increasing 

For any n ≥ n 1 and t ∈ [0, τ n ), we have

It then follows that

The non-negativity of this function can be obtained from u − 1 − log u 0 for any u > 0.

Let n n 1 and T > 0 be arbitrary. For any 0 t min{τ n , T }, applying the generalized Itô's formula [31] to V yields

where LV : R 5 + → R is defined by

)Ǎ µ f (0) + 5μ +b 1 +b 2 +α +ν +č +η +δ

where the inequality f (E) E ≤ f (0) is used and K is a positive constant which is independent of 80 S, E, Q, I, R and k. The remaining part of the proof is similar to [30] and hence is omitted here.

This completes the proof.

It is most crucial to deal with the conditions for the extinction of diseases when their dynamics is under investigation. This section is devoted to establish sufficient conditions so that the COVID-19 85 goes out of the population. First, we need to define the following number

At this stage, we have the following theorem: Proof. Applying the generalized Itô's formula to log E, we have

Moreover, for any k ∈ S, the function defined by

Using the 90 boundedness of the solution and the fact that f (E) ≤ f (0)E, we obtain

Integrating (5) from 0 to t and then dividing by t on both sides lead to

It follows from the ergodic property of r(t) that lim sup

Taking the superior limit on the both sides of (6), (7) and making use of the large number theorem for local martingales, we get

which implies that Substituting this into the third equation of system (3), we obtain for all ω ∈ Λ, t > T,

Thus by the comparison theorem we get Recalling that P(Λ) = 1, hence we obtain

Similarly, when lim t→∞ E(t) = 0 a.s., and lim t→∞ Q(t) = 0 a.s., then by using the same approach as above, we can conclude that Remark 1. Obviously, the quantity R 0 s is smaller than R 0 . Hence, the extinction of the disease in system (3) could be ensured even if the condition R 0 < 1 is not verified.

To investigate epidemic dynamical system, we are also interested in when the disease persists 110 in host population. In this section we need to assume that the function f (·) · is C−Lipschitz and we have the following result on the prevailing behaviour of the COVID-19 disease. 

for some positive constants M i , i = 1, · · · , 4.

Proof. Applying the Itô's formula on the function E → log E, we get

where

Using the fact that S ≤Ǎ µ and f (E) E ≤ f (0), one can easily show that

From the first equation of (3), we can easily claim that

Now, let u and v in (0,Ǎ/μ). Without loss of generality, assume that u < v. The monotonocity and

Consequently,

Substituting this inequality into (11) and making use of (9), we obtain

where

Then, applying the generalized Itô's formula on the function V (E, k) = log E +ω(k) and using (12),

Since the generator Γ is irreducible, then for P 0 = (P (1), · · · , P (N )) with P (k) = F Ǎ µ f (0), k , there is a solutionω = (ω(1), · · · ,ω(N )) to the system Γω = −P 0 + N k=1 π k P (k)1, where 1 is the unit vector of R N . That is,

Substituting the above equality into the inequality (13), integrating from 0 to t and dividing by t,

we obtain

The large number theorem for local martingales and the boundedness of S implies that lim 

From the third equation of (3), we can establish that

Taking the inferior limit and making use of both the boundedness of Q and the inequality (14), we get lim inf

where M 2 =b 2 M 1 / b 1 +č +μ . Following the same way, we can easily claim that lim inf

where M 3 = (αM 1 +ĉM 2 ) / η +μ +δ and M 4 = (νM 1 +ηM 3 ) /μ. This makes finish of the proof. are assumed to be more likely than the first one to show the applicability of the analytical results established along this paper. The parameter values chosen are given in [24] and reported in the Table 1 .

To demonstrate the effect of telegraph noise on the dynamics of COVID-19 disease, in addition to data of Table 1 , we set the following settings: Then, by direct computation, we obtain R 0 s = 0.6303 < 1. In other words, the conditions of the Theorem 4.1 hold and the Fig. 2 shows the empirical means and the standard deviations of the solution to (3) in a 5 × 10 3 samples, as well as the trajectories of the deterministic system (1) without switching for different values of the three considered regimes. So the stochastic process (3) for COVID-19 disease switches over the states 1, 2 and 3 before going to extinction.

On the other hand, when the following parameter values are considered, 

This paper investigates a stochastic epidemic model describing COVID-19 dynamics affected 125 by mixture of environmental perturbations modeled by white and telegraph noises. By means of Lyapunov approach, the existence and positivity of a global solution is well proved. In terms of a stochastic threshold R 0 s , the extinction and the persistence in mean of the COVID-19 epidemic are investigated. Particularly, under small noises, the condition R 0 s < 1 is sufficient to reduce the daily number of confirmed infectives and make the coronavirus disease 2019 extinct. Reciprocally, the 130 persistence of this novel epidemic is inevitable once R 0 s stays away from unity. Based on the data from different states of India, we performed numerical simulations in order to support and illustrate the main results of this paper.

Although many important contributions are made in literature to draw the dynamical properties of the COVID-19, some of them still unidentified and much more efforts are recommended to 135 make it more comprehensible and help humanity to overcome the current pandemic. As a further suggestion, other improvements such us time varying parameters can be considered to make the studied COVID-19 model more realistic. A task which we leave for next works. 

☒ 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.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Brahim Boukanjime: Conceptualization, Writing -original draft, Writing -review & editing

Tomás Caraballo: Validation, Investigation, Conceptualization, Writing -original draft, Writing -review & editing

Methodology, Validation, Formal analysis, Writingoriginal draft, Writing -review & editing

Writing -original draft, Writing -review & editing

The authors would like to thank the editor and reviewers for their careful reading of the 
