Dynamical Analysis of the Stochastic COVID-19 Model Using Piecewise Differential Equation Technique

1  Introduction

Coronavirus disease 2019 is a contagious infection transmitted by the serious acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Headaches, congestion, weariness, muscle aches, breathlessness, diminished appetite, taste and aroma are typical problems. Effects including bronchitis, multi-organ failure, chronic pulmonary disruption phenomenon, septicaemia, tachycardia, cardiogenic shock, thrombosis, cardiac arrest, convulsions, meningitis, dementia and Guillain Barré syndrome may occur in certain people [1]. The implantation phase might last anywhere from two to fourteen days.

The new coronavirus transmits through fluids created by inhaling, breathing or conversing. Liquids emitted by sick individuals are breathed into the trachea of others, developing quality infestations. If individuals contact infected materials and then their faces with unhygienic fingers after the particles tumble on them, they can transmit the disease. Whereas saliva and mucous are key infection transmitters, new evidence indicates that the infection is transferred via faecal contamination pathways. Aerosol-generating methods (AGMs) potentially assist in simpler pathogen propagation than standard pathogen propagation.

SARS-CoV-2 shares characteristics with the severe acute respiratory syndrome coronavirus (SARS-CoV or SARS-CoV1), an encapsulated, newly infected cause that primarily attacks the trachea after entering the host organism and binding to angiotensin-converting-enzyme 2 (ACE2), which is highly prevalent in atelectasis type II (AT2) keratinocytes of the respiratory system [2,3]. It is classified as a congenital illness since it is spread to people via bats as biological transmitters. According to chromosomal investigation, the infection belongs to the Betacoronavirus species and includes two bat-derived viruses [4].

COVID-19 was reported on December 31, 2019, in Wuhan, China’s Hubei region, and has since proliferated globally. On January 30, 2020, the WHO labelled the epidemic a Public Health Emergency of International Concern (PHEIC), and on March 11, 2020, it designated the disease a global epidemic [4,5]. As of August 20, 2020, 217 nations and entities, such as Pakistan, had recorded 30.6 million documented infections and 890,000 fatalities. The very first COVID-19 incidence in Pakistan was detected on February 26, 2020. On September 20, 2020, the government had 306,304 documented favourable patients, 6420 fatalities, and 292,869 recovered [6].

COVID-19 has had a negative impact on the world’s economic development, basic necessities, price volatility, cultural activities and entertainment domains, religious ceremonies, recreation, film, hospitality, academia, care facilities, and democracy. Several specifics about the spread, mitigation, and therapy of this novel ailment are being investigated by mathematicians and virologists [7–9]. Numerous studies using mathematical simulations to understand infection processes and epidemic interventions have been presented [10–14]. Tang et al. [15] developed a cohort system in which the community was classified into nine ordinary differential equations. Considering evidence from scientifically verified COVID-19 occurrences in central China during the month of January, they approximated the virus’s fundamental reproductive rate. It was discovered that prevention strategies can successfully diminish the reproductive capacity. Tang et al. [16] examined a simplified form of their earlier approach, anticipating time-dependent interaction and identification outcomes. This resulted in a lower reproductive rate than had been forecasted in their prior investigation. Li et al. [17] discussed coronavirus dissemination utilizing an SEIR approach for the incidents reported in Wuhan as well as the proportion of exported infections. It is demonstrated that implementing preventive actions, including immigration prohibitions, might be critical in understanding epidemic patterns and the chances of controlling their dissemination. In [18], it seems to be other noteworthy research where the propagation speed is regarded to be a time-dependent variable. The structure is created by adding three additional compartments to the SEIR system. Other significant breakthroughs in innovative coronavirus modelling methodologies can be discovered in [19–24].

Scientists in numerous classifications and scientific disciplines have gravitated toward using fractional systems of differential equations (DEs) in the majority of their novel evidence and investigations as a consequence of the emergence of fractional derivatives [25–27]. In addition, to see this realistically, we can resort to numerical techniques of the proliferation and evolution of numerous infections and communicable conditions, which have emerged as an intriguing issue for scholars in past centuries, employing fractional frameworks of initial value problems. While reviewing the articles, we discovered that various scholars have proposed kernels that can be employed to create fractional differential formulations. The major motivation behind this is that serious challenges exhibit signs of mechanisms that are similar to the behaviours of precise scientific expressions. Fractional calculus incorporating a power law kernel is led by the contributions of Riemann, Liouville, Cauchy, and Abel. Caputo later improved their approach, and this form has been employed in several scientific disciplines owing to its capacity to enable classical initial conditions (ICs) [28]. Prabhakar proposed an appropriate kernel containing three components as a combo of index-law and the generalized Mittag-Leffler (GML) kernel [29–31]. This form has likewise piqued the interest of numerous scholars and investigations into both concepts and implementations have been conducted [32,33].

Furthermore, the various kernels have distinctive features; for instance, index-kernel only aids in the replication of systems that indicate index-kernel tendencies. GML, the combination of the index kernel and the generalized three distinct, has its own applicability domain [34]. Because the phenomenon is multifaceted, Caputo and Fabrizio developed a novel kernel, a particular exponential kernel exhibiting Delta Dirac characteristics. A differential formulation that is becoming increasingly popular because of its capacity to repeat processes after fading memory [35–38]. Furthermore, the notion of fractional derivative having a nonsingular kernel was pioneered by this kernel, marking the inauguration of a revolutionary era in fractional calculus [39]. A scientist’s observation regarding the kernel’s non-fractionality resulted in the creation of a novel kernel, the GML function, including one component. Atangana et al. [40] proposed this formulation, which represents another advancement in the discipline of fractional calculus. The formulations have been employed successfully in a variety of fields of study. Glancing at reality and its intricacies, it is clear that these proposed kernels are insufficient to forecast all of our universe’s complicated characteristics. Following the remark, one will look for a different kernel or modified kernel, or a set of procedures that will be used to add novel differential formulations. Sabatier has proposed various kernel variants that will additionally lead to novel avenues of inquiry [41]. In addition to these remarkable breakthroughs, numerous additional notions were proposed, such as the conception of short memory and the definition of a fractional derivative in the Caputo interpretation for distinct characteristics of fractional orders. Notwithstanding the well-known formulation, which takes a fractional order to be time-dependent, the goal is to achieve a different form of variable order derivative. Wu et al. [42] proposed and implemented this scenario in chaotic theory. However, researchers have discovered that some real-world phenomena demonstrate mechanisms exhibiting varying behaviours as a factor of space and time. A transition from deterministic to stochastic, either from index-law to exponential decay, is an example. Because conventional differential formulations may be incapable of accounting for these tendencies, piecewise differential/integral formulations were devised to cope with issues manifesting crossover phenomena [43]. The primary goal of this article is to present a detailed evaluation, potential implementations, strengths and shortcomings of these two notions.

Mathematical modelling is a critical technique for understanding the transmission of a contagious illness, making predictions, and developing prevention and extinction tactics. Inspired by the aforesaid research, we propose to construct a stochastic framework to examine the impact of quarantine and isolation in reducing the transmission of the coronavirus epidemic in Pakistan via the piecewise differential operators. Pakistan is an emerging nation in the midst of an economic, environmental, and strategic crisis. The region’s health sector is indeed inefficient and faces several issues. Atangana et al. [44] recently proposed the notions of piecewise differential/integral formulations. This revolutionary notion may represent the direction of modelling, as we propose in this work a progressive frontier to analyze epidemiological challenges involving crossover behaviours. A COVID-19 model in Pakistan will be discussed in this presentation stochastically. We will presume that real world propagation reveals waves exhibiting diverse chaotic structures, classical, local/nonlocal, randomness and that a permutation depending on the aforementioned mechanisms can lead to various tendencies. Furthermore, the qualitative characteristics of the aforesaid model are displayed in terms of Brownian motion, such as the existence-uniqueness of the global positive solutions, ergodic stationary distribution, extinction and persistence of the epidemic.

The remainder of the article is organized as follows. Section 2 discusses the preliminaries, framework construction, which is constructed on a variety of assumptions. Then we analyze the strength number of the deterministic COVID-19 model. Section 3 investigates the existence-uniqueness of the global non-negative solution of a stochastic system and calculates the crucial parameter that can readily decide the extermination and permanence of the infection. We additionally show that an ergodic stationary distribution emerges in the stochastic COVID-19 model. Section 4 introduces several simulation studies to illustrate the conceptual framework and illustrate the impacts of environmental white noise. Section 5 describes the results and discussion related to numerical findings. To conclude this work, several findings and appendices are offered.

2  Model Configuration

Taking into account the underlying hypotheses (Fig. 1), we construct a quantitative framework to investigate the behaviors of six classifications of people: susceptible S(t), exposed E(t), quarantined Q(t), infected I(t), separated J(t), and recovered R(t).

Figure 1: Schematic diagram of COVID-19 epidemic model

The vulnerable group expands by Λ due to consistent new recruits and a massive influx of people from the forced to evacuate and healed classifications at valuations of φ2(1−ϑ), where ϑ∈(0,1) is the proportion of people transmitted to the separated category due to therapeutic signs and symptoms, respectively. The vulnerable people get the virus and acquire transmitters at the pace ψ, causing a reduction in the vulnerable individual, which is additionally reduced by spontaneous mortality at the rate χ. The vulnerable people who have still not exhibited disease manifestations but are transmitters of the pathogen have boosted the number in the endangered category. The unprotected group is diminished at a pace of one by clinical manifestations η1, two by isolation, and three by spontaneous mortality η2.

The vulnerable people who have not yet manifested pathological changes but are susceptible to infection have increased the proportion of people in the η2 classification. Random fatality reduces uncovered cohort seclusion by φ2ϑ, while initial symptoms reduce it by φ2(1−ϑ).

People in the contaminated group acquire COVID-19 indications, resulting in an estimated prevalence. It diminishes at a rate of φ1 due to quarantine of affected patients, recuperation at a rate ζ1, and lingering death. People in the separated group have experienced COVID-19 indications and have been separated to receive healthcare attention. These people are drawn at φ1 and φ2ϑ rates from the infectious and confined categories, respectively. The segregated community shrinks as a consequence of the rate of ζ1 recuperation and fatal disease. There is apparently no indication that people acquire lifetime sensitivity to COVID-19. As a result, it is expected that the restored people are vulnerable at a rate ξ. The populace grows at rates ζ1 and ζ2 due to the recovery of infectious and segregated people and shrinks due to spontaneous mortality.

The associated dynamic scheme of ordinary DEs COVID-19 propagation mechanism was proposed by [45] as follows:

{dSdt=Λ+φ2(1−ϑ)Q+ξR−(ψI+χ)S,dEdt=ψIS−(η1+η2+χ)E,dQdt=η2E−(χ+φ2)Q,dIdt=η1E−(φ1+ζ1+χ)I,dJdt=φ1I+φ2ϑQ−(χ+ζ2)J,dRdt=ζ1I+ζ2J−(ξ+χ)R,(1)

supplemented to the positive ICs S≥0, E≥0, Q≥0, I≥0, J≥0, R≥0.

Furthermore, current outcomes indicate that white noise can disrupt the transmission of contagious diseases, population movement, and the formulation of prevention mechanisms. As a result, an increasing number of researchers have researched the appropriate stochastic models (see [46–48]). In [49], Atangana et al. presented deterministic-stochastic modelling with crossover effects. Rashid et al. [50] proposed the novel dynamics of a stochastic fractal-fractional immune effector response to viral infection via latently infectious tissues. Regarding the epidemiologist’s ideas, we assume that randomized white noise is independent and directly proportional to six cohorts. The stochastic form pertaining to scheme (1) can therefore be represented by the stochastic differential equations shown below:

{dS(t)=(Λ+φ2(1−ϑ)Q+ξR−(ψI+χ)S)+℘1S(t)d𝒲1(t),dE(t)=(ψIS−(η1+η2+χ)E)+℘2E(t)d𝒲2(t),dQ(t)=(η2E−(χ+φ2)Q)+℘3Q(t)d𝒲3(t),dI(t)=(η1E−(φ1+ζ1+χ)I)+℘4I(t)d𝒲4(t),dJ(t)=(φ1I+φ2ϑQ−(χ+ζ2)J)+℘5J(t)d𝒲5(t),dR(t)=(ζ1I+ζ2J−(ξ+χ)R)+℘6R(t)d𝒲6(t),(2)

where 𝒲ι,ι=1,…,6 are two independent standard Brownian motions described on a complete filtered probability space (Θ,ℱ,{ℱξ}ξ≥0,P) involving a ℘-filtration {ℱξ}ξ≥0 [51]. ℘ι≥0,  ι=1,…,6 represents the intensities of the system. Here, we provide the accompanying description to help readers who are acquainted with fractional calculus (see [34,39,40]).

0CDtβℱ(t)=1Γ(1−β)∫0tℱ′(r)(t−r)βdr,β∈(0,1].(3)

0CFDtβℱ(t)=ℳ(β)1−β∫0tℱ′(r)exp⁡[−β1−β(t−r)]dr,β∈(0,1],(4)

where ℳ(β) is defined to be normalized function having ℳ(0)=ℳ(1)=1.

The formulation of the Atangana-Baleanu derivative is represented below:

0ABCDtβℱ(t)=ABC(β)1−β∫0tℱ′(r)Eβ[−β1−β(t−r)β]dr,β∈(0,1],(5)

where ABC(β)=1−β+βΓ(β) signifies the normalization function.

3  Qualitative Aspects of COVID-19 Model

In this section, we will discuss some qualitative aspects of the deterministic and stochastic characteristics of the COVID-19 models (1) and (2), respectively.

3.1 Deterministic Case

For COVID-19 model (1), there are two kind of steady states. The first one is disease-free equilibrium point (DFEP) ℰ0=(Λχ,0,0,0,0,0) which always exists. According to [45], R0 is the basic reproduction number

R0=Λψη1χ(η1+η2+χ)(φ1+ζ1+χ).

Notice that if R0>1, in addition to the DFEP, model (1) has a fixed non-negative endemic equilibrium point (EEP) ℰ∗=(S∗,E∗,Q∗,I∗,J∗,R∗), where

ℰ∗=((η1+η2+χ)(φ1+ζ1+χ)ψη1,φ1+ζ1+χη1,η2(φ1+ζ1+χ)η1(φ2+χ)I∗,χ(η1+η2+χ)(φ1+ζ1+χ)−ψη1Λψη1(φ2(1−ϑ)Ψ1+ξΨ2)−ψ(η1+η2+χ)(φ1+ζ1+χ),φ1η1(φ2+χ)+φ2ϑη2(φ1+ζ1+χ)η1(φ2+χ)(ζ2+χ)I∗,η1(φ2+χ)(ζ1(ζ2+χ)+φ1)+φ2ϑη2(φ1+ζ1+χ)η1(φ2+χ)(ζ2+χ)I∗),(6)

where Ψ1=η2(φ1+ζ1+χ)β1(χ+φ2) and Ψ2=1ξ+χ(ζ1+ζ2(φ1+φ2ϑΨ1)χ+ζ2).

Theorem 3.1. (i)If R0<1, the DFEP ℰ0 is locally asymptotically stable.

(ii)If R0>1, the DFEP ℰ0 is unstable but EEP ℰ∗ is locally asymptotically stable.

Proof. The proof can be followed by [45].

3.2 Strength Number

In previous decades, the idea of reproduction has been extensively used in epidemiological modelling since it has been recognized as a valuable mathematical tool for evaluating reproduction in a specific illness. According to the concept proposed by Atangana [52], one will identify two components F and V~, then

(FV~−λI)=0

will be analyzed to generate reproductive number [53]. The component F is particularly intriguing because it is derived from the nonlinear part of the infected classes.

∂∂I(IN)=[N−I]N2

and

∂2∂I2((N−I)N2)=−2[N−I]N3=−2(S+E+Q+J+R)(S+E+Q+I+J+R)3.

At disease free equilibrium ℰ0=(Λχ,0,0,0,0,0), we have

∂2∂I2((N−I)N2)=−2(S0)(S0)3.

Therefore, we have

F𝒜=[00−2(η1S0)(S0)3000000000]=[00−2η1χ2Λ2000000000].

Then,

det(F𝒜V−1−λI)=0

gives

𝒜=−2η1ψΛχ(η1+η2+χ)(φ1+ζ1+χ)<0.

Also, 𝒜 indicates that the expansion will not repeat and will consequently have a single magnitude and wipe out. 𝒜>0 indicates that there is sufficient intensity to initiate the regeneration phase, implying that the dispersion will have more than one cycle. Consequently, researchers will supply a strong insight of the aforesaid number.

3.3 Dynamic of the Stochastic COVID-19 Model

In this paper, suppose a complete probability space (Θ,ℱ,{ℱt}t≥0,P) fulfilling the given assumptions (That is., it is nondecreasing and right continuous whilst ℱ0 have all empty sets P), indicating R+=[0,∞),  R+d={x=(x1,…,xd)∣xi>0,  i∈[1,d]}. Also, there is an integral mapping ℱ1(t) defined on [0,∞). Introducing ℱ1u=sup{ℱ1(t)∣t≥0},  ℱ1l=inf{ℱ1(t)∣t≥0}.

Next, we will examine at the d-dimensional stochastic DE

dY(t)=ℱ1(Y(t),t)dt+𝒢(Y(t),t)dB(t),t≥t0

subject to intial condition Y(0)=X0∈Rd, B(t) denotes a d-dimensional standard Brownian motion presented on the complete probability space (Θ,ℱ,{ℱt}t≥0,P). Suppose C2,1(Rd×[t0,∞];R+) the collection of all positive ℋ(x,t) on Rd×[t0,∞] such that continuous twice differentiable in Y and once in t. The differential operator L is proposed by [54]:

L=∂∂t+∑ι=1d1fι(Y,t)∂∂Xι+12∑ι,κ=1d1[𝒢T(Y,t)𝒢(Y,t)]ικ∂2∂Xι∂Xκ.

Now L imposed on a mapping ℋ∈C2,1(Rd×[t0,∞];R+), we have

Lℋ(Y,t)=ℋt(Y,t)+ℋx(Y,t)ℱ1(Y,t)+12trac[𝒢T(Y,t)ℋxx𝒢(Y,t)],

where ℋt=∂ℋ∂t,ℋx=(∂ℋ∂x1,…,∂ℋ∂xd),  ℋxx=(∂2ℋ∂xι∂xκ)d1×d1. By the Itô’s technique, if Y(t)∈Rd1, then

dℋ(Y(t),t)=Lℋ(Y(t),t)dt+ℋx(Y(t),t)𝒢(Y(t),t)dB(t).

3.4 Existence-Uniqueness of the Global Non-Negative Solution

Theorem 3.2. Suppose there is a unique solution (S(t),E(t),Q(t),I(t),J(t),R(t)) of model (2) on t≥0 for every initial data (S(t),E(t),Q(t),I(t),J(t),R(t))∈R+6 and the unique solution of stochastic model (2) will stay in R+6 having probability 1.

Proof. Our argument is predicated on the research of Mao et al. [55]. Because the parameters of scheme (2) are Lipschitz continuous locally. As a result, there is a unique local solution (S,E,Q,I,J,R) on t∈(0,ρ0) for every ICs (S(0),E(0),Q(0),I(0),J(0),R(0))∈R+6, where ρ0, where ρ0 is the moment of the explosive. We simply require to demonstrate ρ0=∞ (a.s) to show the local solution is global. Allow ℓ0 to be large enough for each factor of (S(0),E(0),Q(0),I(0),J(0),R(0)) inside this interval [1/ℓ0,ℓ0] for every integer ℓ≥ℓ0, now introducing the stopping time

ρℓ=inf{t∈[0,ρ0]∣min{S(t),E(t),Q(t),I(t),J(t),R(t)}≤1ℓ  or  max{S(t),E(t),Q(t),I(t),J(t),R(t)}≥ℓ}.

Setting inf{∅}=∞. Note that when ℓ↦∞ then ρℓ is nondecreasing. Taking ρ∞=limℓ↦∞ρℓ, therefore ρ∞≤ρ0  (a.s). We intend to prove ρ∞=∞  (a.s) then ρ0=∞  (a.s), which implies that (S,E,Q,I,J,R)∈R+6,  ∀t≥0. Also, if ρ∞  (a.s), then there are two constants T≥0 and ε∈(0,1) such that P{ρ∞≤T}≥ϵ. Therefore, there is an integer ℓ1≥ℓ0 such that

P{ρℓ≤T}≥ϵ,  ∀ℓ≥ℓ1.(7)

Introducing a C2-functional V^:R+6↦R+, i.e.,

V^(S,E,Q,I,J,R)=(S+E+Q+I+J+R)−7−ln⁡(S+E+Q+I+J+R).

The positivity of the C2-function V^ can be observed from u1−1−ln⁡u1≥0,  ∀u1>0.

implementing Ito’s rule, we can obtain that

dV^(S,E,Q,I,J,R)=LV^(S,E,Q,I,J,R)dt+℘1(S−1)d𝒲1(t)+℘2(E−1)d𝒲2(t)+℘3(Q−1)d𝒲3(t)+℘4(I−1)d𝒲4(t)+℘5(J−1)d𝒲5(t)+℘6(R−1)d𝒲6(t),

where

dV^(S,E,Q,I,J,R)=(1−1S){Λ+φ2(1−ϑ)Q+ξR−(ψI+χ)S}+℘122+(1−1E){ψIS−(η1+η2+χ)E}+℘222+(1−1Q){η2E−(χ+φ2)Q}+℘322+(1−1I){η1E−(φ1+ζ1+χ)I}+℘422+(1−1J){φ1I+φ2ϑQ−(χ+ζ2)J}+℘522+(1−1R){ζ1I+ζ2J−(ξ+χ)R}+℘622≤Λ+4χ+η1+η2+φ1+φ2+ζ1+ζ2−ξ−ΛS−φ2(1−ϑ)QS−ξRS−ψISE−η2EQ−η1EI−φ1IJ−φ2ϑQJ+ζ1IR−ζ2JR−χ(S+E+Q+I+J+R)+12(℘12+℘22+℘32+℘42+℘52+℘62)≤Λ+4χ+η1+η2+φ1+φ2+ζ1+ζ2−ξ+12(℘12+℘22+℘32+℘42+℘52+℘62):=Υ1.

Therefore, we derive

dV^(S,E,Q,I,J,R)≤Υ1dt+℘1(S−1)d𝒲1(t)+℘2(E−1)d𝒲2(t)+℘3(Q−1)d𝒲3(t)+℘4(I−1)d𝒲4(t)+℘5(J−1)d𝒲5(t)+℘6(R−1)d𝒲6(t),

Furthermore, we have

∫0ρℓ∧TdV^(S(t),E(t),Q(t),I(t),J(t),R(t))≤∫0ρℓ∧TΥ1dt+∫0ρℓ∧T℘1(S−1)d𝒲1(t)dt+∫0ρℓ∧T℘2(E−1)d𝒲2(t)dt+∫0ρℓ∧T℘3(Q−1)d𝒲3(t)dt+∫0ρℓ∧T℘4(I−1)d𝒲4(t)dt+∫0ρℓ∧T℘5(J−1)d𝒲5(t)dt+∫0ρℓ∧T℘6(R−1)d𝒲6(t)dt.

Implementing expectation gives

EV^(S(ρℓ∧T),E(ρℓ∧T),Q(ρℓ∧T),I(ρℓ∧T),J(ρℓ∧T),R(ρℓ∧T))≤Υ1E(ρℓ∧T)+V^(S(0),E(0),Q(0),I(0),J(0),R(0))≤Υ1T+V^(S(0),E(0),Q(0),I(0),J(0),R(0)).

For ℓ≥ℓ1, suppose Θℓ={ρℓ≤T}, from (7), then P(Θℓ)≥ϵ. Obviously, for each ν∈Θℓ, there is one or more of S(ρℓ,ν),E(ρℓ,ν),Q(ρℓ,ν),I(ρℓ,ν),J(ρℓ,ν) and R(ρℓ,ν) equals either 1/ℓ or ℓ, therefore V^(S(ρℓ,ν),E(ρℓ,ν),Q(ρℓ,ν),I(ρℓ,ν),J(ρℓ,ν),R(ρℓ,ν)) is not less than either ℓ−1ln⁡ℓ or 1/ℓ−1+ln⁡ℓ, then

Υ1T+V^(S(0),E(0),Q(0),I(0),J(0),R(0))≥E[ℐΘℓ(ν)V^(S(ρℓ,ν),E(ρℓ,ν),Q(ρℓ,ν),I(ρℓ,ν),J(ρℓ,ν),R(ρℓ,ν))]≥ϵ(ℓ−1−ln⁡ℓ)∧(1/ℓ−1+ln⁡ℓ),

where ℐΘℓ(.) represents the indicating mapping of Θℓ. Thus, applying limit ℓ↦∞ yields contradiction

∞=Υ1T+V^(S(0),E(0),Q(0),I(0),J(0),R(0))<∞.

Thus, we find ρ∞=∞,  (a.s). This completes the proof.

3.5 Basic Reproduction Number for Stochastic Model

Utilizing fourth compartment of the model (2), we have

dI(t)=(η1E−(φ1+ζ1+χ)I)+℘4I(t)d𝒲4(t).

Considering Ito’s formulation for a twice differentiable mapping ℱ(I)=ln⁡(I), and expanding by Taylor series

dℱ(t,I(t))=∂ℱ∂tdt+∂ℱ∂IdI+12∂2ℱ∂I2(dI)2+∂2ℱ∂t∂IdIdt+12∂2ℱ∂t2(dt)2, ⟹ dℱ(t,I(t))=0.dt+1I{(η1E−(φ1+ζ1+χ)I)dt+℘4Id𝒲4(t)}−12I2{(η1E−(φ1+ζ1+χ)I)dt+℘4Id𝒲4(t)}2 ⟹ dℱ(t,I(t))={(η1E−(φ1+ζ1+χ))dt+℘4d𝒲4(t)}−12I2{𝒬12(dt)2+2𝒬1𝒬2dtd𝒲4(t)+𝒬22(d𝒲4(t))2},

where 𝒬1=η1E−(φ1+ζ1+χ)I and 𝒬2=℘4I,

 ⟹ dℱ(t,I(t))={(η1E−(φ1+ζ1+χ))dt+℘4d𝒲4(t)}−12I2{𝒬22(d𝒲4(t))2} ⟹ dℱ(t,I(t))={(η1E−(φ1+ζ1+χ))dt+℘4d𝒲4(t)}−12I2{℘42I2dt}⏟(By  variance  of  Wiener  technique) ⟹ dℱ(t,I(t))={(η1E−(φ1+ζ1+χ)−12℘42)}dt+℘4d𝒲4(t).

The next generation matrices are F=[η1E−12℘42] and V=[φ1+ζ1+χ]. At DFEP and by the principal of next generation matrix, the dominant eigenvalue is the basic reproductive number. Hence R0S=℘422(φ1+ζ1+χ).

3.6 Extinction and Persistence of the Disease

One of the primary challenges in epidemiological data is how to control infection behaviours so that the infection becomes endangered and persists throughout time. In this part, we attempt to determine the important threshold for pathogen extermination and permanence.

Lemma 3.1. For the initial settings (S(0),E(0),Q(0),I(0),J(0),R(0))∈R+6, the solution of the stochastic system (2) admits

limt↦∞ln⁡S(t)t≤0,limt↦∞ln⁡E(t)t≤0,limt↦∞ln⁡Q(t)t≤0limt↦∞ln⁡I(t)t≤0,limt↦∞ln⁡J(t)t≤0,limt↦∞ln⁡R(t)t≤0.  (a.s).(8)

then

limt↦∞S(t)+E(t)+Q(t)+I(t)+J(t)+R(t)t=0,  a.s.(9)

Also, if χ>12(℘12∨℘22∨℘32∨℘42∨℘52∨℘62), we find

limt↦∞1t∫0tS(r)d𝒲1(r)=0,  limt↦∞1t∫0tE(r)d𝒲1(r)=0,  limt↦∞1t∫0tQ(r)d𝒲1(r)=0,limt↦∞1t∫0tI(r)d𝒲1(r)=0,  limt↦∞1t∫0tJ(r)d𝒲1(r)=0,  limt↦∞1t∫0tR(r)d𝒲1(r)=0.(10)

Theorem 3.3. Suppose that χ>12(℘12∨℘22∨℘32∨℘42∨℘52∨℘62) and assume that there is a positive solution of the system (S(t),E(t),Q(t),I(t),J(t),R(t)) having initial settings (S(0),E(0),Q(0),I(0),J(0),R(0)), we find

(i) If R0S<1, then

limt↦∞supln⁡I(t)t≤(φ1+ζ1+χ+12℘42)(R0S−1)<0,(a.s).(11)

which means the diseases will be eliminated from a community.

(ii) If R0S>1, then

limt↦∞inf1t∫0tI(r)dr≥(φ1+ζ1+χ+12℘42)(R0S−1)𝒦2>0(a.s).,(12)

which means the diseases will persist in the community.

Proof. Implementing the Ito’s formula to ln⁡I(t), we find

dln⁡I(t)={(η1EI−(φ1+ζ1+χ)−12℘42)}dt+℘4d𝒲4(t)(13)

Integrating the aforesaid equation from 0 to t on both sides, then