Based on the advantages we previously mentioned of discrete-time mathematical modeling, as well as its compatibility with other mathematical theories, and its suitability for numerical simulation, in addition, noting the absence of the researches addressing this mathematical mechanism for monitoring the evolution and propagation of mutated strains of the COVID-19 virus. This study employed mathematical modeling in discrete time to track developments in the spread of COVID-19 variants. Due to the emergence of several COVID-19 variants that were mentioned previously, as mentioned on The World Health Organization website [24], in this study, the population is divided into n+2 compartments, S presents the susceptible, I_j presents the j^th strain, where j = 1, 2, 3, ..., n, and R represents the recovered people. In addition, we proposed two strategies: a vaccination strategy for the susceptible and a strategy of providing the appropriate treatment and medication for individuals infected with each strain separately to control and limit the spread of these strains at the minimum costs possible, deliberately avoiding the quarantine strategy because of its negative economic effects on individuals and different areas of economy, society, and life, and most studies used this strategy such that El Bhih et al. and Khajji et al. [9, 14, 18]. The goal was to evaluate the efficiency of the two strategies used to fight the spread of mutant strains by developing a discrete-time mathematical model that approximates a realistic picture of the spread of these strains by integrating the two aforementioned strategies into this model, which leads to an optimal control problem. In the followings sections, we used some mechanisms of optimal control theory, especially Pontryagin's maximum principle is used to obtain the desired results.

The remainder of this study is organized as follows. Section 2 introduces a discrete-time mathematical model, describes the model, and presents an optimal control problem. Section 3 describes the existence of optimal controls and characterizes them using Pontryagin's maximum principle in discrete time. Section 4 presents numerical simulations to further illustrate the model. Finally, Section 5 concludes the study.

This study aimed to evaluate the evolution of the spread of mutant strains using strategies that do not have negative effects on other areas to limit the excessive spread of these strains, such as the vaccination strategy and the strategy of providing appropriate treatment and medication while avoiding strategies that might have negative repercussions in other areas. In this study, we intentionally avoided the quarantine strategy due to its negative impacts on the economic sector for both individuals and society and the service sector and other areas. The important role of this study is to assess the efficiency of both strategies, vaccination u, and the provision of appropriate treatment and medication v_j for strain I_j in limiting the spread of mutant strains, without relying on quarantine strategy measures. We incorporate both strategies into this model (Equation 1). Then, we obtain the following:

The quantities S(0) = S₀, I_j(0) = I_{j, 0}, (j = 1, 2....n), and R(0) = R₀ are the initial conditions.

The objective function is defined, and the purpose of the optimal control problem is to find the control inputs that minimize this objective function while satisfying System Dynamic Equation (2) and its initial conditions.

where A_j > 0, B > 0, C > 0, and D_j > 0 for j ∈ {, 1, ...T} are cost coefficients. We chose these constants to weigh their relative importance of I₁(i), I₂(i), ..., I_n(i), R(i), u(i), v₁(i), v₂(i), ..., and v_n(i) at time i, and T is the final time. Our objective is to reduce the number of infected persons while also reducing systemic expenses by striving to maximize the number of people recovered from each I_j strain. In other words, we are looking for the optimal control (u*,v1*,v2*...,vn*) such that:

where U and V_j, j = 1, 2..., n are admissible control sets defined as follows:

U={u:u(i)∈[0, umax], i∈[0,  T−1]}.

Vj={vj:vj(i)∈[0, vjmax], i∈[0,  T−1]},j=1,2,.....n.

The sufficient condition for the existence of an optimal controls for problem (Equation 2) comes from the following theorem.

Theorem 2.1. There exist the optimal controls (u*,v1*,v2*...,vn*) such that:

subject to the discrete-time control system (Equation 2) with initial conditions.

Proof: Because the number of time steps is finite and the coefficients of all the states of the equations are bounded, S = (S₀, S₁, ..., S_T), for j = 1, 2, ..., n, we have Ij=(Ij0,Ij1,...,IjT) and R = (R₀, R₁, ..., R_T) are uniformly bounded for all u ∈ U and v_j ∈ V_j, j = 1, 2, ..., n. Thus, J(u, v₁, v₂..., v_n) is bounded for all u ∈ U and v_j ∈ V_j, j = 1, 2, ..., n. Because J(u, v₁, v₂..., v_n) is bounded,

Moreover, there exists a sequence (uk,v1k,v2k...,vnk) with u ∈ U and v_j∈V_j, j = 1, 2, ..., n such that:

and the corresponding sequences of the states Sk,I1k,I2k,...,Ink, and R^k. Because there are a finite number of uniformly bounded sequences, there exist (u*,v1*,v2*...,vn*), u ∈ U, v_j ∈ V_j with j = 1, 2, ..., n and S*,I1*,I2*,...,In*, and R^* ∈ ℝ^T+1 such that, on a subsequence, (uk,v1k,v2k...,vnk)→(u*,v1*,v2*...,vn*) ,Sk→S*,I1k→I1*,I2k→I2*,...,Ink→In*, and R^k→R^*. In the last, due to the finite dimensional structure of system (Equation 2) and the objective function, J(u, v₁, v₂..., v_n) and (u*,v1*,v2*...,vn*) are optimal controls with corresponding states S*,I1*,I2*,...,In*, and R^*. Therefore :

We utilized the discrete form of Pontryagin's maximum principle, as detailed in references [25–28]. The main concept involves introducing a co-state function that links the set of difference equations representing the system to the objective function. This linkage results in the creation of the Hamiltonian function. Essentially, this principle transforms the task of determining the optimal control to enhance the objective function (Equation 3) by considering the state difference equation with an initial condition. The objective is to identify the control that optimizes the Hamiltonian at each point concerning the control, so the Hamiltonian H at time step k is defined as follows:

where f_{j, k+1} represents the right side of the equation in a dynamic system (Equation 2) of the j^th state variable at time step k+1.

By replacing f_{j, k+1} with its expression found in system (Equation 2) in the Hamiltonian (Equation 4) we get:

The states S*,I1*,I2*,...,In*, and R^* describe the evolution of the system's state variables of (Equation 2) over time under the influence of the optimal controls (u*,v1*,v2*...,vn*) . These state variables represent different compartments of a population.

Theorem 2.2. Given optimal controls (u*,v1*,v2*...,vn*) and the associated solutions S*,I1*,I2*,...,In*, and R^* of the corresponding states system (Equation 2),

there exists co-states λ_{1, k}, λ_{2, k}, ..., λ_{n+2, k} satisfying:

i) The co-state equations:

ii) The transversality conditions at time T:

iii) The optimal controls:

For k = 0, 1, ..., T−1, the expressions of optimal controls u*(k),v1*(k),v2*(k),...,vn*(k) were obtained as follows:

Proof: i) For the co-state equations:

The Hamiltonian of the optimal control problem at time step k is given as follows:

According to Pontryagin's maximum principle version in discrete time, given in Ding et al., Zhang and Shi, Guibout and Bloch, Hwang and Fan [25–28], by taking the derivatives of the Hamiltonian with respect to the state variables as follows:

With calculation, the co-state variables (Equation 5) will be obtained.

ii) For the transversality conditions:

Transversality conditions are necessary for ensuring that the terminal states of the system aligns with the objective of the optimization problem. We put Ψ(T)=∑j=1nAjIj(T)-BR(T) by using also Pontryagin's maximum principle, in discrete time, such that:

iii) For the optimal controls:

For k = 1, 2, ..., T−1, the optimal controls u(k), v_j(k), j = 1, 2, ..., n, can be solved from the following optimality condition:

With calculation we obtained

therefore

And

With calculation, we obtained

Therefore,

This scenario depends on the vaccination strategy u(t) for susceptible people, where u(t) represents the number of individuals vaccinated at moment t, and the following figures show the evolution of individuals in all compartments under the influence of this scenario.

Figure 3 shows two curves representing the progression of strain I₁. The blue curve illustrates the evolution of the strain in a scenario with no medical intervention, peaking at approximately 50,000 cases on day 85, followed by a gradual decline in infection rates. In contrast, the yellow curve shows the progression of the same strain under the vaccination strategy. This curve initially mirrors the trajectory of the blue curve for the first 40 days. Subsequently, a noticeable slowdown in the spread of the strain was observed, reflecting the protective impact of the vaccination. Approximately 70 days after the implementation of vaccination, the infection rate of the strain was halved compared with the non-vaccination scenario, indicating the efficacy of vaccination in mitigating the spread of this particular strain.

Figure 4 shows two curves representing the progression of the Strain I₂. The blue curve illustrates the evolution of the strain in the absence of any intervention to limit its spread, peaking at ~40,000 cases on day 90, followed by a gradual decline in infection rates. In contrast, the green curve depicts the progression of the same strain under vaccination strategy u. This curve initially mirrors the trajectory of the blue curve for the first 30 days. Subsequently, a noticeable slowdown in the spread of the strain is observed, reflecting the protective impact of vaccination. Approximately 70 days after the implementation of the vaccination strategy, the infection rate of the strain is halved compared with the non-vaccination scenario, indicating the efficacy of vaccination in mitigating the spread of this particular strain.

Figure 5 contains two curves, where the Gold Curve shows the number of people who recovered from Strains I₁ and I₂ over time without using any treatment or infection control strategies. This curve indicates that the number of recoveries increased significantly over time; the explanation for this is that the lethality of these strains is not very dangerous. The brown curve represents the number of recoveries from these strains when a vaccination strategy is employed. This curve shows that the number of recoveries is significantly lower than that of the first curve. The explanation for this is that the vaccination strategy is effective and has prevented many people from contracting the disease, which might have been caused by the absence of vaccination and, after a number of them have recovered, would have been counted among the recoveries. Therefore, the number of recoveries was higher in the scenario without vaccination than in that with vaccination.

In addition to the vaccination strategy, this scenario depends on a strategy specific to the strain I₁ depends on providing the appropriate treatment and medications v₁(t) for individuals infected with strain I₁, and a strategy specific to the strain I₂ depends on providing the treatment and appropriate medications v₂(t) for individuals infected with strain I₂; the following figures show the evolution of number of infected people with each strain separately under the influence of scenario 2.

Figure 6 shows the two curves for Strain I1. The blue curve shows the natural development of Strain I1 in the absence of any intervention to limit its spread, as mentioned in the previous analysis. Conversely, the green curve displays the development of the strain under the influence of two strategies: vaccination and providing appropriate treatment and medication to those infected with this strain. From Figure 3, we observe that the period of overlap between the two curves becomes shorter than that in Figure 2. Additionally, there was a noticeable decrease in the number of infections, with a peak at ~16,000 cases on the 70th day, followed by a gradual decline. This pattern highlights the effectiveness of an integrated intervention strategy in reducing the spread of this strain.

Figure 7 shows two curves for Strain I₂. The curve colored in blue shows the natural development of this strain in the absence of any interventions to limit its spread, as mentioned in the previous analysis. In contrast, the olive curve displays the development of the strain under the influence of two strategies: vaccination u and appropriate treatment and medication v₂ for those infected with this strain. Figure 5 shows a shorter period of overlap between the two curves compared with Figure 4. The number of infections peaked at ~11,000 cases on the 60th day, followed by a significant reduction in the number of infected individuals, indicating the effectiveness of the integrated intervention strategy in reducing the spread of this strain.

In this scenario, all strategies were simultaneously implemented. The following figures illustrate the evolution of the number of individuals in all compartments under this scenario.

Figure 8 contains two curves: The Gold Curve shows the number of people who have recovered from Strains I₁ and I₂ over time without using any treatment or infection control strategies, as previously mentioned above. The purple curve represents the number of recoveries from these two strains, involving both vaccination (u) and the provision of appropriate medication and treatment (v₁ and v₂ ) for each strain. From this figure, we observe that the purple curve is above the gold curve, in contrast to what we observed in Figure 5. This is due to the use of the controls v₁ and v₂, which increased the number of recoveries.

Figure 9 presents four distinct curves illustrating the progression of all population compartments, encompassing susceptible individuals (S), individuals infected with each strain (I₁ and I₂), and the recovered individuals (R). These trajectories are depicted under the influence of a comprehensive vaccination strategy (u) and the administration of appropriate medications (v₁ and v₂) tailored to each I₁ and I₂ category. In comparison to the baseline scenario depicted in Figures 1, 8 unveils a stark reduction in the number of infections for both strains, along with a notable and continuously increasing count of recovered individuals. This unequivocally underscores the efficacy of the implemented intervention strategies within this context.