# Dynamics of a Stochastic Epidemic Model with Vaccination and Multiple Time-Delays for COVID-19 in the UAE

Authors: H. J. Alsakaji; F. A. Rihan; A. Hashish
Journal: Complexity
 (2022)

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## Abstract

In this paper, we study the dynamics of COVID-19 in the UAE with an extended SEIR epidemic model with vaccination, time-delays, and random noise. The stationary ergodic distribution of positive solutions is examined, in which the solution fluctuates around the equilibrium of the deterministic case, causing the disease to persist stochastically. It is possible to attain infection-free status (extinction) in some situations, in which diseases die out exponentially and with a probability of one. The numerical simulations and fit to real observations prove the effectiveness of the theoretical results. Combining stochastic perturbations with time-delays enhances the dynamics of the model, and white noise intensity is an important part of the treatment of infectious diseases.


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## Body

## 1. Introduction

COVID-19 is a disease caused by SARS-CoV-2 that can trigger a respiratory tract infection. It spreads likewise other coronaviruses do, basically through person-to-person contact. Infections range from mild to deadly [1, 2]. To combat the spreading of all infectious diseases, vaccination is one of the most important procedures [3, 4]. Vaccines generally expose the immune system to harmless parts of the pathogen so that the immune system learns to recognize it and may be able to tamp down the infection before any symptoms appear [5, 6]. COVID-19 vaccines, such as Pfizer, AstraZeneca, and Sinopharm, are now widely available for people aged five years and older, and all the currently authorized COVID-19 vaccines are effective and reduce the risk of severe illness [7]. It is normal for a virus to mutate as it infects people, and SARS-CoV-2 has mutated so [8–10]. There are various variants which are now spreading, such as Alpha, Beta, Gamma, Delta, and Omicron. An initial study showed Omicron variant reduced the antibody protection by some vaccines, but a booster shot is likely to protect people from severe disease, and research works are still in proceedings in this field [11].Up to date, more than 4.41 billion people worldwide have received a dose of the COVID-19 vaccine, equal to about 57.4 percent of the world population [12]. A vaccinated person refers to someone who has received at least one dose of a vaccine, and a fully vaccinated person has completed receiving the vaccine, whether that is one dose or two, and two weeks have passed. A COVID-19 booster shot is an additional dose of a vaccine given after the protection provided by the original shot(s) has begun to decline [13]. The booster is recommended to help people keep up their level of immunity for longer. In the UAE, more than 99 percent of the population at least have one dose of the vaccine, 91 percent of the population are fully vaccinated, and 32.3 percent of the population are booster given [14]; therefore, the number of confirmed cases of COVID-19 in the UAE has decreased significantly.Modeling infectious diseases provides a controlled environment in which complex relationships between environmental and biological factors can be examined. In public health science, mathematical models of infectious diseases can be used to analyze various scenarios, and the results can inform policy, programs, and practices [15, 16].Researchers are working to develop mathematical models that can be used to predict vaccination strategies for controlling epidemic diseases [3, 4, 17, 18].Human virus diseases are highly affected by stochastic perturbations. Because human contact can change from one person to another, epidemic growth and spread in human disease are normally random, and the population is subject to factors that are either not fully understood or difficult to model precisely. A model that ignores these phenomena will negatively affect the analysis of the studied biological systems. Stochastic differential equation models (SDEs) are more suitable for modeling epidemic dynamics under certain conditions [19–21]. Increasingly, deterministic models need to be extended to stochastic models that can account for more complex variations in dynamics [22]. Furthermore, delay differential equations (DDEs) are extensively used to describe the dynamics of infectious diseases. Due to the fact that time-delay is relevant to hidden mechanisms such as the incubation period and the recovery of infected individuals [23–25].In this paper, we study the dynamics of the COVID-19 epidemic in the UAE, using a modified stochastic delayed SEIRV (Susceptible-Exposed-Infected-Recovered-Vaccinated) model. The model incorporates white noise and time-delays. This model assumes that individuals can become infected during vaccination, but then become healthy afterwards. A stochastic Lyapunov function and Ito’s formula are used to determine the existing results of stationary distribution and extinction of the disease. Combining stochastic perturbations and time-delays can provide a more realistic view of disease dynamics. The rest of this paper is organized as follows: Section2 presents the model formulation. In Sections 3 and 4, this model derives the stationary distribution and extinction results. In Section 5, numerical simulations are presented to verify the theory. In Section 6, conclusions are provided.


## 2. The Model

For the dynamics of COVID-19 in the UAE, we propose an extended SEIR epidemic model with vaccination, time-delays, and random noise [26, 27]. The basic model categorized people into four classes: susceptible (S): individuals not yet infected; exposed (E): individuals experiencing incubation duration; infectious (I): confirmed cases; and removed (R): recovered individuals. We assume that the recovered individuals will remain in the class Rt. Therefore, the SEIR model has the following equations system:(1)dSdt=Λ−β1SE−αS,dEdt=β1SE−β2EI−κ+αE,dIdt=β2EI−d+r+αI,dRdt=rI+κE−αR.Here,Λ is the recruitment rate; β1 is the transmission rate of susceptible into exposed class; β2 is the rate of transmission of exposed into infected class; α and d are natural and disease death rates; κ is the transmission rate of exposed into recovered class; and r is the transmission rate of infected into recovered class. Many researchers develop the above model to include vaccination strategies to control epidemic diseases realistically [17, 18]. There is evidence that individuals can become infected during vaccination and go on to be healthy afterward [28]. Incorporating time lags in epidemic models makes the systems much more realistic and enriches the dynamics of the model. Therefore, we include time-delays τ1andτ2 to represent the incubation period; while τ3 stands for the time required for the infected individuals to become recovered. Hence, the deterministic SEIR model with vaccination and time-delays takes the form (see Figure 1).(2)dSdt=Λ−β1SE−β3+αS,dEdt=β4VE+β1SE−β2EIt−τ1−κ+αE,dIdt=β5VI0+β2EIt−τ1−d+r+αI,dRdt=β6V+rIt−τ3+κE−αR,dVdt=β3S−β6+αV−β4VE−β5VIt−τ2.Figure 1
Flow chart of the model (2).β3, β4, β5, and β6 are the transmission rates of susceptible into vaccinated class; vaccinated into exposed class; vaccinated into infected class; and β6 vaccinated into recovered class, respectively.The basic reproduction number,(3)ℛ0=β1Λβ3+ακ+α+Λβ3β4β3+ακ+αβ6+α,of model (2), has a significant impact in epidemiology since it decides whether an epidemic occurs or the disease dies out [29]. If ℛ0<1, then model (2) has only a disease-free equilibrium E0=S+,0,0,R+,V+=Λ/β3+α,0,0,Λβ3β6/αβ6+αβ3+α,Λβ3/β6+αβ3+α and it is globally asymptotically stable; while if ℛ0>1, then, E0 is unstable and there is a unique endemic equilibrium E∗=S∗,E∗,I∗,R∗,V∗ which is globally asymptotically stable [28].Because some factors cannot be measured precisely, stochastic models always provide an estimate of these uncertainties based on approximate estimates [1, 30–32]. Therefore, we introduce randomness into model (2) by adding white noise to the state of the SEIR model with vaccination and time-delays. The modified model takes the form:(4)dS=Λ−β1SE−β3+αSdt+ν1SdW1,dE=β4VE+β1SE−β2EIt−τ1−κ+αEdt+ν2EdW2,dI=β5VIt−τ2+β2EIt−τ1−d+r+αIdt+ν3IdW3,dR=β6V+rIt−τ3+κE−αRdt+ν4RdW4,dV=β3S−β6+αV−β4VE−β5VIt−τ2dt+ν5VdW5,with(5)Sθ=ϕ1θ,Eθ=ϕ2θ,Iθ=ϕ3θ,Rθ=ϕ4θ,Vθ=ϕ5θ,θ∈−τ,0,τ=maxτ1,τ2,τ3,ϕi0>0 and ϕiθ, i=1,…,5, are non-negative continuous initial functions on −τ,0. Wit, i=1,…,5, represent the independent Brownian motions defined on a complete probability space Ω,U,Ut≥0,P with a filtration Utt≥0 satisfying the usual conditions (it is right continuous and U0 contains all P− null sets), where νi, i=1,…,5 are the intensities of white noise.


## 3. Stationary Distribution and Ergodicity

Among the most important and significant characteristics of the stochastic epidemic model (4) is its ergodic property. Under some conditions of white noise, the stochastic model fluctuates in the neighborhood of the infected equilibrium of the corresponding deterministic model for all time regardless of the starting conditions. First, we need to show that there is a global non-negative solution of model (4), which is as follows:Theorem 1.
For any given initial value (5), system (4) has a unique solution St,Et,It,Rt,Vt on t≥−τ, and the solution will remain in ℝ+5 with probability one.Proof.
Since the system coefficients (4) satisfy linear growth and Lipschitzian conditions and based Khasminskii Lyapunov functional approach, we can show that system (4) has a global positive solution. The main challenge is to establish a Lyapunov function, so we define(6)GS,E,I,R,V=G.=S−1−lnS+E−1−lnE+I−1−lnI+R−1−lnR+V−1−lnV+β2∫tt+τ1Is−τ1ds+β5∫tt+τ2Is−τ2ds+r∫tt+τ3Is−τ3ds.
ByIto^’s formula on G,(7)dG.=ℒGdt+ν1S−1dW1t+ν2E−1dW2t+ν3I−1dW3t+ν4R−1dW4t+ν5V−1dW5t,where(8)ℒG=Λ−αS−ΛS+β1E+β3+α−κ+αE−β4V−β1S+β2I+κ+α−d+r+αI−β5VIt−τ2I−β2EIt−τ2I+d+r+α+β6V+rI+κE−αR−β6VR−rIt−τ3R−κER+α−β6+αV−β3SV+β6+α+β4E+β5I+ν12+ν22+ν32+ν42+ν522≤Λ+β3+κ+d+r+β6+5α−α+β1S+β1+β4−αE+β2+β5−d−αI−αR−β4+αV+ν12+ν22+ν32+ν42+ν522≤A,where A is a positive constant. It follows that ℒG is bounded. Hence, the rest of the proof is standard [33], so it is omitted.Theorem 2.
Define(9)ℛ^0=β1Λν^1ν^2+Λβ3β4ν^1ν^2ν^5,where ν^1=β3+α+ν12/2, ν^2=κ+α+ν22/2, ν^3=d+r+α+ν32/2, ν^4=α+ν42/2, and ν^5=β6+α+ν52/2. If ℛ^0>1, then, system (4) has a unique stationary distribution π. and it admits the ergodic property.Proof.
LetYt is a regular time-homogenous Markov process in ℝn, defined by the stochastic delay differential equation:(10)dYt=fYt,Yt−τ,tdt+∑r=1ngrYt,tdWrt.
The diffusion matrix of the processYt is(11)Πy=ςijy,ςijy=∑r=1ngriygrjy.Lemma 1 ([see 34]).
The Markov processYt has a unique ergodic stationary distribution π. if there exists a bounded domain ℬ⊂ℝn with regular boundary Δ and(i)
there is a positive numberX such that ∑i,j=1nςijyξiξj≥Xξ2,5y∈ℬ,ξ∈ℝn.(ii)
there exists a non-negativeC2-function V such that ℒV is negative for any ℝn∖ℬ.With a view to prove Theorem2, we need to guarantee the validity of conditions (i) and (ii) of Lemma 1. Clearly, condition (i) satisfies; we need to check condition (ii). Define ℱ5=ℱ3+ℱ4, where(12)ℱ1=−lnE−ω1lnS+β2∫tt+τ1Is−τ1ds,ℱ2=ℱ1−ω2lnS−ω3lnV+ω3β5∫tt+τ3Is−τ3ds,ℱ3=Nℱ2+−2lnS−lnR−lnI,ℱ4=1η+1S+E+I+R+Vη+1.ω1=β1Λ/ν^12, ω2=β3β4Λ/ν^12ν^5, and ω3=β3β4Λ/ν^1ν^52, 0<η<4α/ν12∨ν22∨ν32∨ν42∨ν52, where N is a positive constant so that(13)−Nν^2ℛ^0−1+D≤−2,where D=maxD1,D2 such that(14)D0=supS,E,I,R,V∈ℝ+5ΛS+E+I+R+Vη−α^2S+E+I+R+Vη+1,D1=supS,E,I,R,V∈ℝ+5Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^−α^4Eη+1+D0,D2=supS,E,I,R,V∈ℝ+5Nω1β1+ω2β1+ω3β4E+2β1E+2ν^1+ν^3+ν^−α^4Iη+1+D0,D3=supS,E,I,R,V∈ℝ+5Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^−α^4Eη+1+Iη+1+D0.In addition,ℱ5 is continuous and tends to +∞ as S,E,I,R,V approaches the boundary of ℝ+5 and S,E,I,R,V⟶∞. Hence, ℱ5 must have a minimum point in the interior of ℝ+5.We define aC2-function ℱ.:ℝ+5⟶ℝ+ as(15)ℱS,E,I,R,V=ℱ5S,E,I,R,V−ℱ5S0,E0,I0,R0,V0.ByIto^’s formula, we obtain(16)ℒℱ1=−β4V−β1S+β2I+ν^2+ω1−ΛS+β1E+ν^1≤−β1Λν^1+ν^2−β4V+β2I+ω1β1E,ℒℱ2≤−β1Λν^1+ν^2−β4V+β2I+ω1β1E+ω2−ΛS+β1E+ν^1+ω3−β3SV+ν^5+β4E+β5I≤−β1Λν^1+ν^2+ω1β1+ω2β1+ω3β4E+β2+ω3β5I+ω2ν^1+ω3ν^5−3Λβ3β4ω2ω33≤−ν^2β1Λν^1ν^2+Λβ3β4ν^1ν^2ν^3−1+ω1β1+ω2β1+ω3β4E+β2+ω3β5I≔−ν^2ℛ^0−1+ω1β1+ω2β1+ω3β4E+β2+ω3β5Iℒℱ3≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I−2ΛS+2β1E+2ν^1−β5VIt−τ2I−β2EIt−τ1I+ν^3−β6VR−rIt−τ3R−κER+ν^4,ℒℱ4=S+E+I+R+VηΛ−αS−αE−d+αI−αR−αV+η2S+E+I+R+Vη−112ν12S2+12ν22E2+12ν32I2+12ν42R2+12ν52V2≤ΛS+E+I+R+Vη−αS+E+I+R+Vη+1+η4ν12∨ν22∨ν32∨ν42∨ν52S+E+I+R+Vη+1=ΛS+E+I+R+Vη−α−η4ν12∨ν22∨ν32∨ν42∨ν52S+E+I+R+Vη+1≔ΛS+E+I+R+Vη−α^S+E+I+R+Vη+1,such that α^=α−η/4ν12∨ν22∨ν32∨ν42∨ν52. Hence,(17)ℒℱ=Lℱ5≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I−2ΛS+2β1E+2ν^1+ν^3−β6VR−rIR−κER+ν^4+ΛS+E+I+R+Vη−α^S+E+I+R+Vη+1.Define a closed bounded set.(18)ℬ=S,E,I,R,V∈ℝ+5:ϵ≤S≤1ϵ,ϵ≤E≤1ϵ,ϵ≤I≤1ϵ,ϵ2≤R≤1ϵ2,ϵ≤V≤1ϵ.By Lemma1, we need to show that ℒℱ≤−1 for S,E,I,R,V∈ℝ+5∖ℬϵ. such that ℝ+5∖ℬϵ=∪i=19ℬi, where(19)ℬ1=S,E,I,R,V∈ℝ+5:0<E<ϵ,ℬ2=S,E,I,R,V∈ℝ+5:0<I<ϵ,ℬ3=S,E,I,R,V∈ℝ+5:0<S<ϵ,ℬ4=S,E,I,R,V∈ℝ+5:0<V<ϵ,R>ϵ2,ℬ5=S,E,I,R,V∈ℝ+5:0<R<ϵ2,V≤1ϵ,ℬ6=S,E,I,R,V∈ℝ+5:R>1ϵ2,ℬ7=S,E,I,R,V∈ℝ+5:I>1ϵ,ℬ8=S,E,I,R,V∈ℝ+5:S>1ϵ,ℬ9=S,E,I,R,V∈ℝ+5:V>1ϵ,ℬ10=S,E,I,R,V∈ℝ+5:E>1ϵ.Case 1.
IfS,E,I,R,V∈ℬ1, then(20)ℒℱ≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^4−α^4Eη+1+D0≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4E+D1≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4ϵ+D1≤−1,from condition (10) and −Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4ϵ+D1≤−1, we obtain ℒℱ≤−1.Case 2.
IfS,E,I,R,V∈ℬ2, we have(21)ℒℱ≤−Nν^2ℛ^0−1+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^4−α^4Iη+1+D0≤−Nν^2ℛ^0−1+Nβ2+ω3β5I+D2≤−Nν^2ℛ^0−1+Nβ2+ω3β5ϵ+D2≤−1,which is obtained from (13) and −Nν^2ℛ^0−1+Nβ2+ω3β5ϵ+D2≤−1.Case 3.
IfS,E,I,R,V∈ℬ3, we get(22)ℒℱ≤Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I−2ΛS+2β1E+2ν^1+ν^3+ν^4+D0−α^4Iη+1+Eη+1≤−2ΛS+D3≤−2Λϵ+D3≤−1,such that 0<ϵ≤2Λ/D3+1.Case 4.
LetS,E,I,R,V∈ℬ4, one may obtain(23)ℒℱ≤−β6VR+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^4+D0−α^4Iη+1+Eη+1≤D3−β6ϵ≤−1,where 0<ϵ≤β6/D3+1.Case 5.
IfS,E,I,R,V∈ℬ5, we have(24)ℒℱ≤−β6VR+Nω1β1+ω2β1+ω3β4E+Nβ2+ω3β5I+2β1E+2ν^1+ν^3+ν^4+D0−α^4Iη+1+Eη+1≤D3−β6ϵ3≤−1,where 0<ϵ≤β6/D3+13.Case 6.
IfS,E,I,R,V∈ℬ6, we have(25)ℒℱ≤−α^4Rη+1+D3≤D3−α^4ϵ−2η+1≤−1,where 0<ϵ≤α^/4D3+11/21+η.Case 7.
IfS,E,I,R,V∈ℬ7, one may obtain(26)ℒℱ≤−α^4Iη+1+D3≤D3−α^4ϵ−η+1≤−1.where 0<ϵ≤α^/4D3+11/1+η.
Cases 8, 9, and 10 are the same as Case7.
Thus, condition (ii) of Lemma1 holds; hence, system (4) identifies a unique stationary distribution π..


## 4. Extinction of the Disease

In this section, we discuss conditions that predict the extinction of the disease. From the formula of the reproduction number, we can conclude thatℛ0^<ℛ0. First, we go through the following Lemmas [21, 32]Lemma 2.
LetSt,Et,It,Rt,Vt be the solution of (4) with initial conditions (5), then(27)limt⟶∞Stt=0,limt⟶∞Ett=0,limt⟶∞Itt=0,limt⟶∞Rtt=0,limt⟶∞Vtt=0.Lemma 3.
Assume thatα>1/2ν12∨ν22∨ν32∨ν42∨ν52. Let St,Et,It,Rt,Vt be the solution of (4) with initial conditions (4), we have(28)limt⟶∞∫0tSrdW1rt=0,limt⟶∞∫0tErdW2rt=0,limt⟶∞∫0tIrdW3rt=0,limt⟶∞∫0tRrdW4rt=0,limt⟶∞∫0tVrdW5rt=0.Theorem 3.
Assumeα>1/2ν12∨ν22∨ν32∨ν42∨ν52. Let St,Et,It,Rt,Vt be the solution of (4) with initial conditions (5). If ℛ0<1, then(29)limt⟶∞logEtt+β2It≤κ+αℛ0−1<0a.s.,which means Et and It tend to zero exponentially almost surely. In other words the disease dies out with probability one. Moreover,(30)limt⟶∞St+Rt+Vt=Λα=S++R++V+.Proof.
Taking integration of the first and fifth equations of (4), we obtain(31)St−S0t=Λ−β1StEt−β3+αSt+ν1∫0tSrdW1rt,Vt−V0t=β3St−β6+αVt−β4VtEt−β5VtIt+ν5∫0tVrdW5rt.
Therefore,(32)St=1β3+αΛ−β1StEt+ν1∫0tSrdW1rt−St−S0t≤Λβ3+α+1β3+αν1∫0tSrdW1rt−St−S0t≔Λβ3+α+φ1t,so that limt⟶∞φ1t=0.
Additionally, we have(33)Vt=1β6+αβ3St−β4VtEt−β5VtIt+ν5∫0tVrdW5rt−Vt−V0t≤β3Stβ6+α+1β6+αν5∫0tVrdW5rt−Vt−V0t≤β3Λβ6+αβ3+α+β3β6+αφ1t+1β6+αν5∫0tVrdW5rt−Vt−V0t≔β3Λβ6+αβ3+α+β3β6+αφ1t+φ2t,where limt⟶∞φ2t=0. Applying Ito^’s formula to the second equation of system (4) yields(34)dlogEt=β4Vt+β1St−β2It−κ+α+ν222dt+ν2dW2t.
Integrating equation (34) from 0 to t results in(35)logEt−logE0t=β4Vt+β1St−β2It−κ+α+ν222+ν2dW2tt.
Then, from (32) and (33), we have(36)logEtt+β2It≤β4β3Λβ6+αβ3+α+β3β6+αφ1t+φ2t+β1Λβ3+α+φ1t−κ+α+ν2dW2tt+logE0t=β4β3Λβ6+αβ3+α+β1Λβ3+α−κ+α+β3β4β6+α+β1φ1t+β4φ2t+ν2dW2tt+logE0t≔β4β3Λβ6+αβ3+α+β1Λβ3+α−κ+α+φ3t,and limt⟶∞φ3t=0 a.s. If ℛ0<1, from (36),(37)limt⟶∞logEtt+β2It≤β4β3Λβ6+αβ3+α+β1Λβ3+α−κ+α=κ+αℛ0−1<0.
Therefore,limt⟶∞Et=limt⟶∞It=0. From model (4), we get(38)dSt+Et+It+Rt+Vt=Λ−αSt+Et+It+Rt+Vt−dItdt+ν1StdW1t+ν2EtdW2t+ν3ItdW3t+ν4RtdW4t+ν5VtdW5t.
Taking integration of (38) from 0 to t, one obtains(39)St+Et+It+Rt+Vt=Λα+φ4t,where(40)φ4t=1α1tS0+E0+I0+R0+V0−1tSt+Et+It+Rt+Vt−dIt+ν1∫0tSrdW1rt+ν2∫0tErdW2rt+ν3∫0tIrdW3rt+ν4∫0tRrdW4rt+ν5∫0tVrdW5rt.
One can easily obtain thatlimt⟶∞φ4t=0, and since limt⟶∞Et=limt⟶∞It=0, we have(41)limsupt⟶∞St+Et+It+Rt+Vt=Λα,which implies that limt⟶∞St+Rt+Vt=Λ/α=S++R++V+, as required; hence, the proof is completed.


## 5. Numerical Simulations

In this section, we numerically simulate the solution of the stochastic system (4) using Milstein’s higher-order method [35]. The discretization transformation takes the form:(42)Sj+1=Sj+Λ−β1SjEj−β3+αSjh+ν1Sjξ1,jh+ν122Sjξ1,j2−1h,Ej+1=Ej+β4VjEj+β1SjEj−β2EjIj−n1−κ+αEjh+ν2Ejξ2,jh+ν222Ejξ2,j2−1h,Ij+1=Ij+β5VjIj−n2+β2EjIj−n1−d+r+αIjh+ν3Ijξ3,jh+ν322Ijξ3,j2−1h,Rj+1=Rj+β6Vj+rIj−n3+κEj−αRjh+ν4Rjξ4,jh+ν422Rjξ4,j2−1h,Vj+1=Vj+β3Sj−β6+αVj−β4VjEj−β5VjIj−n2h+ν5Vjξ5,jh+ν522Vjξ5,j2−1h,where ξi,j, i=1,…,5 are mutually independent N0,1 random variables, n1,n2,andn3 are integers such that the time-delays can be expressed in terms of the step-size as τ1=n1h, τ2=n2h, and τ3=n3h. We choose a set of parameters Λ=0.5, β1=0.9905, β2=1.9, β3=0.6, β4=0.928, β5=1.93, β6=0.9092, α=0.5, κ=0.003, d=0.01, and r=0.001, with τ1=τ2=3 and τ3=4.Figure2 indicates that the system has a unique stationary distribution and the disease is persistent by Theorem 2, such that the intensities of white noise are relatively small where ℛ^0>1 with νi=0.01, i=1,…,5. In Figure 3, we increase slightly the intensities of white noise to ν1=0.09, ν2=0.05, ν3=0.08, ν4=0.09, and ν5=0.1 with the the same set of parameters as in Figure 2, that is the positive equilibrium is globally asymptotically stable such that the stochastic solution fluctuates around the deterministic steady state value and the disease still persistent. However, we increase the intensities of white noise ν1=0.1, ν2=0.5, and ν3=ν4=ν5=0.2 such that ℛ^0<1. Figure 4 implies that the disease will ultimately tend to extinction under the relatively strong white noises ν1=0.1, ν2=0.5, ν3=0.3, ν4=0.4, and ν5=0.2 confirmed by Theorem 3. In Figure 5, we investigate the impact of the transition rate from susceptible into vaccinated class with different values of β3, which indicates that the number of susceptible, exposed, and infected individuals decrease as β3 increases, while the recovered individuals increase as β3 increases, other parameter values are the same as in Figure 2.Figure 2
Numerical simulations of model (4), which shows that model (4) has a unique ergodic stationary distribution where the disease is persistent and ℛ^0>1 with νi=0.01, i=1,…,5.Figure 3
The solutions of the stochastic system (4) and the undisturbed system (2) left banners such that ℛ^0>1. While right banners show the density function diagram of St, Rt, and Vt.Figure 4
Numerical simulations of model (4) shows that the disease dies out when the white noise is relatively large such that ℛ^0<1.Figure 5
The impact of the transition rate from susceptible into vaccinated class with different values ofβ3, which indicates that the number of susceptible, exposed, and infected individuals decrease as β3 increases, while the recovered individuals increase as β3 increases.Remark 1.
Under certain criteria with a large magnitude of white noises, the disease can be eradicated, whereas the small intensity of white noises can preserve a stationary distribution.


### 5.1. Fitting the DDEs Model to Real Data

To investigate the reality of the deterministic model (2), we fit real data for the number of the confirmed cases of COVID-19 in the UAE during June 22, 2021, to August 11, 2021 [36] with model (2) using least-square approach [37, 38].Given a set of real data in Table1 and a mathematical model (2), the objective function (weighted least squares function) is as follows:(43)ΦHp=∑i=15∑j=1Mxitj,p−Xji2hij.Table 1
Number of recorded COVID-19 cases in the UAE, from June 21, 2021, to August 11, 2021 [36].
Time (days)June 21June 22June 23June 24June 25June 26June 27Infected cases1,8501,9642,1671,9882,1612,2232,282Time (days)June 28June 29June 30July 1July 2July 3July 4Infected cases2,1221,7471,6751,6631,6321,5991,573Time (days)July 5July 6July 7July 8July 9July 10July 11Infected cases1,5521,5132,042,1841,5391,5291,52Time (days)July 12July 13July 14July 15July 16July 17July 18Infected cases1,5181,5421,5221,5291,5081,5411,506Time (days)July 19July 20July 21July 22July 23July 24July 25Infected cases1,5471,5211,5071,5291,5491,5391,55Time (days)July 26July 27July 28July 29July 30July 31August 1Infected cases1,521,5371,5191,5371,5481,521,537Time (days)August 2August 3August 4August 5August 6August 7August 8Infected cases1,5191,5371,5481,5191,5081,521,545Time (days)August 9August 10August 11Infected cases1,411,3211,334Here,xi,i=1,…,5 represents the variables S,E,I,R,V; p is the model parameter to be estimated. Thus, we then try to attain the optimum parameter p^ that satisfies Φp^≤minpΦp≡maxpℒp, where ℒp is the likelihood function [37, 38]. However, the estimation of the parameters that appear in the undisturbed model (2) is considered as an optimization problem. Herein, the data are scaled in ten thousands.Parameters estimates areβ2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047; therefore, ℛ˜0=1.54>1, see Figure 6; while Figure 7 illustrates the response of the stochastic model (4) with the estimated parameters; therefore, the stochastic fluctuations enhance the consistency of the model with the real data.Figure 6
The fitted curve of DDEs model (2) and the confirmed COVID-19 cases in the UAE from June 22, 2021 to August 11, 2021. The estimated parameters are β2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047.Figure 7
The response of the stochastic model (4) with the estimated parameters β2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047, such that the intensities of white noises are ν1=0.03, ν2=0.02, ν3=0.02, ν4=0.04, and ν5=0.04.The steps of parameter estimations are summarized as follows:(1)
Guess an initial parameter estimatep0;(2)
We then solve the system using a deterministic model (2) using the current parameters;(3)
A minimization routine, such as OPTIMTOOL in Matlab, is then used to adjust the parameter values;(4)
When the valueΦp cannot be further reduced, the best fit parameter values have been determined;(5)
Determine if the chosen set of parameters is acceptable or not.


## 5.1. Fitting the DDEs Model to Real Data

To investigate the reality of the deterministic model (2), we fit real data for the number of the confirmed cases of COVID-19 in the UAE during June 22, 2021, to August 11, 2021 [36] with model (2) using least-square approach [37, 38].Given a set of real data in Table1 and a mathematical model (2), the objective function (weighted least squares function) is as follows:(43)ΦHp=∑i=15∑j=1Mxitj,p−Xji2hij.Table 1
Number of recorded COVID-19 cases in the UAE, from June 21, 2021, to August 11, 2021 [36].
Time (days)June 21June 22June 23June 24June 25June 26June 27Infected cases1,8501,9642,1671,9882,1612,2232,282Time (days)June 28June 29June 30July 1July 2July 3July 4Infected cases2,1221,7471,6751,6631,6321,5991,573Time (days)July 5July 6July 7July 8July 9July 10July 11Infected cases1,5521,5132,042,1841,5391,5291,52Time (days)July 12July 13July 14July 15July 16July 17July 18Infected cases1,5181,5421,5221,5291,5081,5411,506Time (days)July 19July 20July 21July 22July 23July 24July 25Infected cases1,5471,5211,5071,5291,5491,5391,55Time (days)July 26July 27July 28July 29July 30July 31August 1Infected cases1,521,5371,5191,5371,5481,521,537Time (days)August 2August 3August 4August 5August 6August 7August 8Infected cases1,5191,5371,5481,5191,5081,521,545Time (days)August 9August 10August 11Infected cases1,411,3211,334Here,xi,i=1,…,5 represents the variables S,E,I,R,V; p is the model parameter to be estimated. Thus, we then try to attain the optimum parameter p^ that satisfies Φp^≤minpΦp≡maxpℒp, where ℒp is the likelihood function [37, 38]. However, the estimation of the parameters that appear in the undisturbed model (2) is considered as an optimization problem. Herein, the data are scaled in ten thousands.Parameters estimates areβ2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047; therefore, ℛ˜0=1.54>1, see Figure 6; while Figure 7 illustrates the response of the stochastic model (4) with the estimated parameters; therefore, the stochastic fluctuations enhance the consistency of the model with the real data.Figure 6
The fitted curve of DDEs model (2) and the confirmed COVID-19 cases in the UAE from June 22, 2021 to August 11, 2021. The estimated parameters are β2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047.Figure 7
The response of the stochastic model (4) with the estimated parameters β2^=1.99854, β5^=1.9813, β6^=0.219092, κ^=0.01099, and r^=0.047, such that the intensities of white noises are ν1=0.03, ν2=0.02, ν3=0.02, ν4=0.04, and ν5=0.04.The steps of parameter estimations are summarized as follows:(1)
Guess an initial parameter estimatep0;(2)
We then solve the system using a deterministic model (2) using the current parameters;(3)
A minimization routine, such as OPTIMTOOL in Matlab, is then used to adjust the parameter values;(4)
When the valueΦp cannot be further reduced, the best fit parameter values have been determined;(5)
Determine if the chosen set of parameters is acceptable or not.


## 6. Concluding Remarks

In this paper, we extended the classical SEIR epidemic model to include vaccination and time-delays that incorporate randomness into the equations by including white noise perturbations on some parameters. The model has been examined by fitting to real observations in UAE, during June 22, 2021, to August 11, 2021. The study found that disease extinction is more likely if the noise intensity is high, and this can be used to develop some effective control strategies. Biological systems models should include random influences as they deal with real-life subsystems, which cannot be adequately isolated from factors outside the system. The addition of white noise and time-delays adds complexity to the model and enriches its dynamics.Our conclusions are as follows.(i)
When the intensity of white noise is relatively low, the disease will persist as long asℛ^0>1 (see Figures 2 and 3) and will die out with greater white noise; see Figure 4.(ii)
The stochastic fluctuations improve the consistency of the model with the real data; see Figure7.(iii)
It is shown that the disease can be controlled efficiently if the level of vaccination is increased. Therefore, asβ3 is increased, the solution of model (4) fluctuates around the disease-free equilibrium.(iv)
If the stochastic perturbationsνi=0, i=1,…,5, then, the threshold of the stochastic model (4) can be reduced to that of the deterministic counterpart. Therefore, ℛ^0>1 is a generalized result indicating the persistence of the disease.(v)
Using mathematical models to develop, manufacture, and deliver vaccines is more efficient and results in safer and more efficient vaccines.Future research will focus on stochastic epidemic models with Markovian switching and time-delays.



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*Source: 4247800-2022-04-26.xml*