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.
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: Section 2 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 . Therefore, the SEIR model has the following equations system:
Here, is the recruitment rate; is the transmission rate of susceptible into exposed class; is the rate of transmission of exposed into infected class; and are natural and disease death rates; is the transmission rate of exposed into recovered class; and 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 to represent the incubation period; while 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).
, , , and are the transmission rates of susceptible into vaccinated class; vaccinated into exposed class; vaccinated into infected class; and vaccinated into recovered class, respectively.
The basic reproduction number,of model (2), has a significant impact in epidemiology since it decides whether an epidemic occurs or the disease dies out [29]. If , then model (2) has only a disease-free equilibrium and it is globally asymptotically stable; while if , then, is unstable and there is a unique endemic equilibrium 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:with
and , , are non-negative continuous initial functions on . , represent the independent Brownian motions defined on a complete probability space with a filtration satisfying the usual conditions (it is right continuous and contains all null sets), where , 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 on , and the solution will remain in 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 defineBy It’s formula on ,wherewhere is a positive constant. It follows that is bounded. Hence, the rest of the proof is standard [33], so it is omitted.
Theorem 2. Definewhere , , , , and . If , then, system (4) has a unique stationary distribution and it admits the ergodic property.
Proof. Let is a regular time-homogenous Markov process in , defined by the stochastic delay differential equation:The diffusion matrix of the process is
Lemma 1 ([see 34]). The Markov process has a unique ergodic stationary distribution if there exists a bounded domain with regular boundary and(i)there is a positive number such that .(ii)there exists a non-negative -function such that is negative for any .
With a view to prove Theorem 2, 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 , where, , and , , where is a positive constant so thatwhere such that
In addition, is continuous and tends to as approaches the boundary of and . Hence, must have a minimum point in the interior of .
We define a -function as
By It’s formula, we obtainsuch that . Hence,
Define a closed bounded set.
By Lemma 1, we need to show that for . such that , where
Case 1. If , thenfrom condition (10) and , we obtain .
Case 2. If , we havewhich is obtained from (13) and .
Case 3. If , we getsuch that .
Case 4. Let , one may obtainwhere .
Case 5. If , we havewhere .
Case 6. If , we havewhere .
Case 7. If , one may obtainwhere .
Cases 8, 9, and 10 are the same as Case 7.
Thus, condition (ii) of Lemma 1 holds; hence, system (4) identifies a unique stationary distribution .
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 that . First, we go through the following Lemmas [21, 32]
Lemma 2. Let be the solution of (4) with initial conditions (5), then
Lemma 3. Assume that . Let be the solution of (4) with initial conditions (4), we have
Theorem 3. Assume . Let be the solution of (4) with initial conditions (5). If , thenwhich means and tend to zero exponentially almost surely. In other words the disease dies out with probability one. Moreover,
Proof. Taking integration of the first and fifth equations of (4), we obtainTherefore,so that .
Additionally, we havewhere . Applying It’s formula to the second equation of system (4) yieldsIntegrating equation (34) from 0 to results inThen, from (32) and (33), we haveand a.s. If , from (36),Therefore, . From model (4), we getTaking integration of (38) from 0 to , one obtainswhereOne can easily obtain that , and since , we havewhich implies that , 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:where , are mutually independent random variables, are integers such that the time-delays can be expressed in terms of the step-size as , , and . We choose a set of parameters , , , , , , , , , , and , with and .
Figure 2 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 with , . In Figure 3, we increase slightly the intensities of white noise to , , , , and 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 , , and such that . Figure 4 implies that the disease will ultimately tend to extinction under the relatively strong white noises , , , , and confirmed by Theorem 3. In Figure 5, we investigate the impact of the transition rate from susceptible into vaccinated class with different values of , which indicates that the number of susceptible, exposed, and infected individuals decrease as increases, while the recovered individuals increase as increases, other parameter values are the same as in Figure 2.
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 Table 1 and a mathematical model (2), the objective function (weighted least squares function) is as follows:
Here, represents the variables ; is the model parameter to be estimated. Thus, we then try to attain the optimum parameter that satisfies , where 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 are , , , , and ; therefore, , 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.
The steps of parameter estimations are summarized as follows:(1)Guess an initial parameter estimate ;(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 value 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 as (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 Figure 7.(iii)It is shown that the disease can be controlled efficiently if the level of vaccination is increased. Therefore, as is increased, the solution of model (4) fluctuates around the disease-free equilibrium.(iv)If the stochastic perturbations , , then, the threshold of the stochastic model (4) can be reduced to that of the deterministic counterpart. Therefore, 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.
Data Availability
The authors confirm that the data supporting the findings of this study are available within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was funded by the UAEU-ZU Joint Research, fund # 12S107- 2022.