COVID-19 Model with High- and Low-Risk Susceptible Population Incorporating the Effect of Vaccines

It is a known fact that there are a particular set of people who are at higher risk of getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model with high-risk and low-risk susceptible individuals and their respective intervention strategies. We find two equilibrium solutions and we investigate the basic reproduction number. We also carry out the stability analysis of the equilibria. Further, this model is extended by considering the vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses and numerical simulations are carried out. From the results, we are able to obtain disease-free and endemic equilibrium solutions by solving the system of equations in the model and show their global stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis shows that reducing the hospitals’ imperfect efficacy can have a positive impact on the control of COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately control or eliminate COVID-19.


Introduction
The COVID-19 pandemic is still among the most devastating infectious diseases in the world. It is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and is transmitted within individuals through respiratory droplets created when someone that is infected coughs or sneezes or via direct contact with saliva, nasal discharge, and sputum of an infected individual [1,2]. The COVID-19 virus has been responsible for over 6 million deaths around the world by the end of March 2022, as a result, it has become the most significant global health disaster since the 1918 influenza pandemic [3,4].
The fight against COVID-19 made significant strides in early 2022 when the reported daily number of cases plunged, and hospitalizations were down by almost 27.9%. These changes are attributed to the wide implementation of COVID-19 control strategies on a large scale, which includes keeping in touch with the authorities, getting vaccinated, properly wearing a recommended facemask, avoiding crowded areas, or maintaining distance between the self and others, yet an infection is still detected [5,6]. Therefore, it is reasonable to speculate that transmission cannot be ruled out completely, and given the potential importance of such transmission, urgent research on this subject is imperative [7,8].
trix, calculating disease-free equilibrium points, identifying endemic equilibrium points, analysing the stability of these equilibrium points on a local and global scale. In Section 4, we perform sensitivity analysis and numerical simulations of the model; in Section 5, we extend the model by considering movement from the high-risk to low-risk susceptible population which is caused due to vaccination of some unvaccinated individuals in the high-risk population. We provide discussion and conclusions regarding our results in Section 6.

Model Formulation
In this part, we modify an existing COVID-19 model proposed by Pal Bajiya et al. in [22]. The modification procedure is presented below. We define N(t) as the total human population at time t divided into six sub-populations: high-risk susceptible S 1 , low-risk susceptible S 2 , exposed E, infected I, hospitalized H, and recovered R individuals: (1) Our model also accounts for certain demographic impacts by assuming that all subpopulations have a natural death rate (µ > 0) and a net inflow of humans to the two susceptible population at a rate Λ. Next, we define the incident fraction as follows: where β is the effective contact rate, and is a modification factor that measures hospital inefficacy. The high-risk susceptible population S 1 (t) are those that are unvaccinated, human with underlying medical conditions demonstrated to have a higher risk of death due to COVID-19 [23], as well as the elderly [24]. Undocumented migrants associated with limited access to healthcare because of legal, administrative, social barriers, etc., are also part of this population. The high-risk susceptible population S 1 (t) is reduced by the natural death rate µ and the force of infection η. The parameter σ 1 incorporates the impact of the non-pharmaceutical intervention (wearing masks, physical or social distancing, and washing hands regularly) by individuals in S 1 on the number of contacts, Similarly, the low-risk susceptible population S 2 (t) includes those that are not in S 1 (t), and are recruited either by birth, or screened of COVID-19 by the authorities at the point of entry through any of the borders. S 2 (t) also reduced by natural death rate µ and force of infection η, following effective contacts with an infected individual. Here, σ 2 represents the same interventions with S 1 . High-risk individuals must take additional precautions in addition to the above-mentioned actions to reduce transmission, this implies 0 ≤ σ 2 < σ 1 , Exposed individuals E(t) stands for the number of people who were exposed to the virus. They are generated by the population of susceptible (high or low risk) humans that become exposed or contact with the infected persons, and decreases as a result of infection at a rate α and natural death at the rate µ, Infected individuals I(t) stands for the number of people who are infected but have not been detected. It is reduced by either hospitalization at the rate of ν 2 , recovery without being hospitalized at a rate ν 1 , the natural death µ, and COVID-19 induced death at a rate δ, Hospitalized individuals H(t) are those who have been diagnosed with COVID-19 and have been isolated by the authorities. They are reduced due to recovery at the rate τ, the natural death µ, and COVID-19 induced death at a rate δ, Recovered individuals R(t) represents the number of recovered, undiagnosed individuals, who are not being officially identified and those that recovered due to the impact of isolation. The only decline in this population is due to the natural death rate µ,  Table 2 lists the model parameters and their descriptions. Based on the assumptions stated above, Figure 1, below, describes the flow transmission of the infection from one compartment to another. The interactions is represented by the system of non-linear ordinary differential equations below: (3) Rate of reduction in infectiousness in S 1 σ 2 Rate of reduction in infectiousness in S 2 µ The natural death rate α The rate of progression from exposed population to infected population The rate of hospital inefficacy ν 1 The recovery rate of infected in ν 2 The hospitalization rate of infected individuals τ The recovery rate of the hospitalized individuals δ The COVID-19 induced mortality rate The basic dynamic characteristics of the model will now be discussed. Since the model tracks populations of humans. In the same way as [25], the following theorem shows that all the state variables are non-negative for all time t > 0. Theorem 1. Consider the model (3) with initial conditions S 1 (0) > 0, S 2 (0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, H(0) ≥ 0, and R(0) ≥ 0 then the solution is positive in R 6 + .

Proof. Using the first equation of the model (3) given by
which implies that Similarly, S 2 (t), E(t), I(t), H(t), R(t) can be demonstrated to be positive. Therefore, the solution of the model (3) is a positive quantity in R 6 + for all t ≥ 0.

Theorem 2. The model system's (3) solution is bounded in
Proof. By combining all equations in (3) we obtain Following from [26], it can be observed that The proposed model is well presented epidemiologically and mathematically from above in the region D. As a result, analysing the qualitative dynamics of model in D is adequate.

Model Analysis
The model system (3) admits two equilibria: disease free equilibrium (DFE) and endemic equilibrium (EE).

Disease Free Equilibrium
If there is no disease, the DFE of the model system (3) is calculated as follows: Setting E = 0, I = 0, H = 0, R = 0 and denoted as We computed the basic reproduction number in a similar manner as in [27]. Considering the infected block x = {E, I, H} we obtain the matrix F and V as follows: Therefore, the basic reproduction R 0 is as follows The basic reproduction number R 0 defined as the number of new infections in a population induced by a single infected person within a given period of time. If the R 0 < 1, the DFE is said to be locally stable, otherwise is said to be unstable [17]. Proof. From the Jacobian matrix evaluated at the DFE defined as . A reduced Jacobian matrix defined as J r DFE below has three (3) eigenvalues, which correspond to the remaining eigenvalues of J DFE . Define The characteristic polynomial of the matrix Jr DFE is determined as where According to Routh-Hurwitz criterion, the sufficient and necessary condition for stability is Because all of the model parameters are positive, the first inequality in (10) is automatically satisfied, the second inequality is satisfied when R 0 < 1. As a result c 1 c 2 − c 3 > 0 as long as c 3 > 0 holds. Hence, the disease free equilibrium DFE of the model (3) is locally asymptotically stable when R 0 < 1 and unstable whenever R 0 > 1.
For global stability of the DFE, we have the following theorem Theorem 4. The disease-free equilibrium (DFE) of the model (3) is globally asymptotically stable (GAS) if R 0 < 1.

Proof. Considering the following Voltera-type Lyapunov function
where f 1 , f 2 , and f 3 > 0 are the Lyapunov coefficient. The corresponding derivative of L 0 ( dL 0 dt ) is given byL At the DFE, we linearize the (14). We note that near the DFE, Using this relation, we havė We choose Clearly, dL 0 dt ≤ 0 whenever R 0 ≤ 0. As a result, and according to the Lasalle invariance principle [28], the DFE is said to be globally asymptotically stable.

Endemic Equilibrium
Finding the endemic equilibrium solutions S * 1 , S * 2 , E * , I * , H * , and R * defined as EE helps to further analyze the model. In order to obtain this endemic equilibrium, we solve (3) substituting k 1 , k 2 , k 3 , and k 4 as defined in Section 3.1 simultaneously in terms η (i.e., force of infection) and obtained , , where and N * = S * 1 + S * 2 + E * + I * + H * + R * From Equation (20) we obtained η * = 0 as one of the solutions (which corresponds to DFE) and the following equation which is quadratic: where a = k 5 k 6 (αµk 4 + αµν 2 + ατν 2 + αk 4 . Clearly, a is greater than zero. When R 0 > 1, we have the following: This shows that there is a unique endemic equilibrium point. otherwise no real roots of (21).

Global Stability Analysis of the Endemic Equilibrium
Following [29][30][31], Theorem 6 below is established Theorem 5. The unique endemic equilibrium of (3) is GAS in D if R 0 > 1, provided that and are satisfied.
Proof. In this case, we use the approach in [29][30][31] to establish the proof. If we consider a Lyapunov function: where ω i > 0(i = 1, 2, 3, 4, 5) are constants to be determined. It is easy to see that L 1 ≥ 0 for all S 1 , S 2 , E, I, H > 0, and L 1 = 0 ⇐⇒ (S 1 , S 2 , E, I, H) = (S * 1 , S * 2 , E * , I * , H * ). From (3), we have the solution below: We can differentiate L 1 along these solutions which is given bẏ By direct computation from (26) , we have and Likewise, 2 − S * existing literature to simulate and perform a sensitivity analysis of our model. Occasionally, we assumed the values which can be seen in Table 3. We begin the numerical simulation of our model by first letting β = 0.2693 so that R 0 = 0.9380 < 1 for various initial conditions. When R 0 < 1 Figure 2a-d shows that the system has a DFE that is asymptotically stable which supports the result stated in Theorem 3. In Figure 3, we examine the scenario in which R 0 = 2.3986 > 1 for various initial conditions. When R 0 > 1 Figure 3a-d shows that the system has a DFE and it is unstable whenever R 0 > 1.  Table 3 with β = 2693 so that R 0 = 0.9380 < 1; (a) the high-risk susceptible, (b) the low-risk susceptible, (c) exposed, and (d) infectious.  Table 3 with β = 0.6886 so that R 0 = 2.3986 > 1; (a) the high-risk susceptible, (b) the low-risk susceptible, (c) exposed, and (d) infectious.

Sensitivity Analysis
In mathematical modeling, numerous parameter values are uncertain, this could be due to incorrect parameter estimation and uncertainty regarding the accurate values of the parameters. Therefore, it is reasonable to carry out sampling and sensitivity analysis to identify the parameters that significantly affect the output of our model. We used a Sampling and Sensitivity Analysis Tool (SaSAT) in [36] to recognise these parameters. This is an efficient tool that enables us to analyze the sensitivity. We first have values and boundaries assigned to our parameters as in Table 3. We then applied uniform probability distributions to each of the parameters, in accordance with the suggestion of [36]. We evaluated Partial Rank Correlation Coefficients (PRCC) with 1000 samples for each parameter per run, which was resulted from the Latin hypercube sampling (LHS). Figure 4 shows how varying the parameters of model (3) changes the behavior of R 0 . The five (5) most sensitive parameters affecting R 0 are β, τ, σ 2 , ν 1 , and . Rising the value of the recovery rate of the hospitalized individuals, rate of reduction in infectiousness of the low-risk susceptible which is proportional to rising the value of σ 1 might cause the value of R 0 to drop. Meanwhile, reducing the contact rate, hospital inefficacy might also cause the value of R 0 to drop significantly. This study demonstrate that isolating and hospitalizing the infected persons reduces the spread of infection. Figure 5 shows the change in behavior of R 0 while varying the value of the most sensitive parameters.

Extended Model with Vaccination
Vaccine effectiveness has so far been the driving force in the dynamics of COVID-19 transmission. In this section, we carried out numerical simulations to test for various situations in relation to the parameters under close examination. We propose two main parameters ω, which indicates the level of vaccination for the high-risk population, and ν 3 , which represents the effectiveness of the vaccine among the vaccinated individuals in the low-risk population.
In summary, as a result of incorporating an imperfect COVID-19 vaccine we believe: i.
The number of high-risk individuals reduces. ii. Infections can be prevented with some degree of efficacy.
Next, we see the effect of vaccination and its effectiveness on the dynamics of COVID-19 by simulating the related vaccine parameters of the model (3) over time.
In Figure 6, we can see that as vaccination coverage increases, there is a striking decline in the number of infected individuals at each level of effectiveness of the vaccine. In Figure 7, we can see that the hospitalized population decreases relatively quickly over time at various levels of ν 3 . Therefore, improving the vaccination rates for people at high risk will result in a decrease in the transmission of COVID-19 in the future.
A look at Figure 8 illustrates, on the other hand, what the effects of vaccination may be on the recovered class, it can be concluded that the population of the recovered class increases as the value of ω increases from 0.01 to 0.3.

Discussion and Conclusions
As it stands now, there is still no specific drug for COVID-19, so many treatment guidelines were developed in the literature based on patient treatment records for physicians and healthcare personnel as a guide for decision-making in the treatment of patients with COVID-19. Understanding the influence of the NPIs on both low and high-risk individuals, the impact of the vaccine on the high-risk population can help us inform public health policy as it was discussed in [37]. It is equally vital to evaluate the impact of hospital efficacy/inefficacy.
In this paper, we present a model which explains the transmission and spread of COVID-19, incorporating a fraction ρ low-risk and the remaining (1 − ρ) high-risk of the human population. The model also captures NPIs by both low-and high-risk individuals, to reduce infection transmission and insisted that the high-risk individuals take more intervention than the low-risk, similar to what is mentioned in [38]. The model was later extended by incorporating the vaccination of some high-risk individuals and vaccine efficacy. This study has been able to draw some important conclusions both mathematically and biologically, which are summarized below.
Many models of the SEIR form in literature, see [21,22,39,40], did not take into consideration the segregation in the susceptible class into low-and high-risk individuals. However, considering this division is essential in providing more insight into the transmission dynamics of the disease, as well as guiding the relevant authorities in introducing and providing plans to curtail the spread of the disease.
(i) A GAS DFE occurs in the model without vaccination every time the associated basic reproduction number is less than 1 and there are multiple endemic equilibria, which is a GAS in a particular case. (ii) Based on numerical simulations it is indicated that if some high-risk individuals were vaccinated and moved to low-risk, the disease could be reduced to a minimum. However, this is dependent on the coverage of vaccinations. (iii) The Latin Hypercube Sampling to create 1000 samples and the resulting Partial Rank Correlation Coefficients (PRCCs) were used to perform a sensitivity study of the model parameter values. A tornado plot is used to visually display the results. The parameters τ, (hospitalized recovery rate), σ 1,2 , (reduction in infectiousness of risk individuals), according to sensitivity analysis, greatly lessen an epidemic if increased. However, if the rates of person-to-person interaction and hospital inefficacy are reduced, transmission also decreases.
In order to stop the epidemic, it is crucial to improve vaccination of the high-risk individuals. Law enforcement must also be strengthened in order to restrict undocumented immigration.

Future Work Directions
• Some new models with a different approach of incidence fraction can be proposed. • Several dynamical features of COVID-19 were captured by our model, though other population compartments might be added and furthermore implement optimal control strategies when having access to more detailed and authentic COVID-19 data.

Data Availability Statement:
There are no underlying data.

Acknowledgments:
The authors are grateful to the editors and reviewers for their anticipated thoughtful and insightful comment that will improved the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.