Optimal control analysis of Monkeypox disease with the impact of environmental transmission

1.   Introduction

Classified under the orthopoxvirus genus of the Poxviridae family, the monkeypox virus (MPXV) is responsible for the onset of the viral disease monkeypox (MPX) ^[1]. The first human case of MPX was reported in 1958 and was distinguished them from smallpox at the start of 1970s. Every continent, with the exception of Antarctica, has recorded more than 15,000 infected cases of MPX since May 2022 ^[2]. The smallpox-like MPXV causes MPX, a contagious zoonotic disease ^[3,4]. Transmission through animal-to-human, human-to-human, and environmental factors (inter- or intra-human) are the three considerable and possible routes that MPXV might spread. Animals (particularly rodents) and humans commonly come into direct contact and spread infections. Although, the infection cases have been observed in various non-endemic countries in the world, it is typically found in Africa. The typical symptoms of MPX include fever, chills, and a rash that develops after a few days. Animals of many different species can contract the MPXV. The incubation period of monkeypox in humans is commonly from 6 to 13 days, but can range from 5 to 21 days ^[1]. In order to identify the specific reservoir(s) and to understand how virus flow is regulated in nature, more research is required to understand the MPXV. Eating raw meat and other animal products that have been in contact with infected animals may pose a risk ^[5]. In non-endemic countries where cases have been discovered, additional public health measures are adopted, including thorough contact tracing and case discovery, laboratory analysis, diagnostic procedure, and isolation with supportive therapy. When possible, genomic sequencing has been employed to pinpoint the viral clade(s) causing this outbreak of MPX. Despite the fact that there are currently no specific treatments for MPX infection, outbreaks can be managed. While smallpox has been eradicated worldwide, the vaccine ST-246 and antiviral medications like tecovirimat, Cidofovir, and Brincidofovir designed for use in smallpox patients may help in the fight against MPX. However, these medications are not yet available commercially ^[6]. The Centers for Disease Control and Prevention (CDC) recommendation was developed using the most up-to-date data on the advantages and risks of smallpox immunization and treatments for MPX and other orthopoxvirus infections. The absence of a proper understanding of the MPXV disease's transmission dynamics and associated factors creates an environment that is conducive to the disease's incidence in both endemic and non-endemic regions.

Epidemic models have been used by numerous researchers to investigate the prevalence of infection outbreaks in different regions, and they have shown to be useful and reliable tools ^[7,8]. Many models have been created and investigated using a variety of approaches in order to better illustrate the dynamics of the disease's spread and control ^[9,10]. Researchers Somma ^[11], Lasisi ^[12], Usman, and Adamu ^[13], and Emeka ^[14] have formulated mathematical models of MPX with two host populations. To eradicate the spread of MPX, Somma ^[11] studied the impact of public awareness and quarantine strategy, whereas Lasisi ^[12] implemented an exposed population group for both human and animal populations, in addition to a vaccine class for migrants. To better understand the spread of MPXV infection Usman and Adamu ^[13] looked into the effectiveness of combined vaccine and therapy measures. On the other hand, Emeka et al. ^[14] developed a deterministic model where they examined how an incomplete vaccination might affect the dynamics of infection. For the purpose of analyzing the dynamics of the MPX in the human population, the authors of ^[15] formulated a deterministic mathematical model. These results imply that separating infected individuals from the broader community lessens the spread of disease. In ^[16], a novel approach based on fractional calculus and fractal theory was used to investigate human-to-animal transmission. Madubueze et al. ^[17] was the first to demonstrate how the dynamics of MPX are affected by a polluted environment (i.e., surfaces and materials that have been exposed to MPX by environmental viral shedding).

The optimal control approach is an effective tool used to find the best and optimum control strategy for a system to achieve a desired objective ^[18,19]. In optimal control theory, a system is typically converted to a set of differential equations describing the evolution of the system over time, and the control strategy is represented as a function that maps the current state of the system to a control input. The goal of optimal control theory is to find the control strategy that either minimizes or maximizes an objective function, subject to constraints on the system and the control input ^[20]. There are various approaches to solving optimal control problems, including analytical methods and numerical optimization techniques. Some common numerical methods used to solve the optimized problem include Pontryagin's maximum principle, dynamic programming, and model predictive control ^[21]. Application of optimal control theory to set the optimal strategies for an infectious disease eradication including MPX can be found in ^[22,23].

Despite the aforementioned discussion, MPX still persists in the population. We analyze the dynamics of MPX with the environment transmission routes in the presence of optimized preventive strategies. Moreover, the influence of some crucial factors on disease prevalence is identified and health officials are advised. This study is structured as follows: Construction of the epidemic model for the MPX dynamics and the estimation procedure are presented in section 2. The basic analysis of the proposed model is covered in section 3. Section 4, accomplishes the normalized sensitivity analysis of the parameters of the proposed model versus $\mathcal{R}_{0}$. In section 5, the construction of the optimal control system along with the solution and existence of the problem is presented. Simulation with comprehensive discussion are provided in section 6. The concluding remarks with future directions are mentioned in section 7.

2.   Mathematical formulation of the MPX dynamics

This section presents the construction of the mathematical model for the transmission dynamics of MPX using a system of nonlinear differential equations. Two populations i.e. humans and animals (those capable to transmit the infection) are considered to formulate the model. The MPX model for the human population consists of four compartments, $S_{h},$ $I_{h},$ $Q_{h},$ and $R_{h}$; the animal population is classified in two sub-classes, the susceptible $S_{r}$ and the infectious class $I_{r}$. The viral contamination due to the environment is denoted by $E_{n}.$ The cumulative population in both human and animal cases are shown as follows:

2.1.   Mathematical model for human sub-population

The susceptible human population recruits through the rate $\lambda_{h}$ and through the progression of the isolated population to susceptible at rate $\tau_{1}$. This class is decreased by the force of infection $\lambda_{h} = \frac{{\beta _1 I_r }}{{N_r }} + \frac{{\beta _2 I_h }}{{N_h }} + \frac{{\zeta E_n }}{{k + E_n }}$, where $\beta_{1}, \beta_{2},$ and $\zeta$ are the human to human, animals to humans, and environmental transmission rates, respectively. Moreover, the human natural mortality rate is denoted as $\mu_{h}.$

The class of infected humans is increased due to the progression of susceptible individuals at the rate $\lambda_{h}$ after getting the infection. The class of infected individuals is also reduced by the rates, $\mu_{h}$ (the natural death rate), $\phi$ (the progression of $I_{h}$ to $Q_{h}$), $\delta$ (the MPX-induced mortality rate), and $\gamma$ the infected humans recovery rate.

Progression of infected individuals $I_{h}$ to isolated individuals at the rate $\phi$ form the class $Q_{h}$, where $\tau_{1}$ represents the transfer of the isolated/quarantined individuals to susceptible people and $\tau_{2}$ is the recovery rate.

The infected and quarantine individuals join the recovered class after recovery from infection. This class is also decreased by the same natural death rate. Thus, the sub-model for the only human population is organized in the following system:

2.2.   Mathematical model for animals sub-population

The class of susceptible animals is recruited with a recruitment rate $\Lambda_{r}$. This class is decreased by $\lambda_{r}$ due to the interaction of susceptible animals with infected ones. This is further reduced at the rate $\mu_{r}$ due to natural death. The infected animals are increased by joining the susceptible animals at the infection force $\lambda_{r}$ and decreased by the natural mortality rates $\mu_{r}$. Thus, the sub-model describing the dynamics of only animals is described as follows:

2.3.   Environment concentration of MPXV

When the infected individuals (humans or animals) shed infection to the environment, it affects the infection force term $\lambda_{h}$. Therefore, the environmental viral concentration has an important role in disease transmission. Regarding the choice of the incidence function, there are a number of choices present in the literature ^[24,25]. In this study, we consider the saturated incidence rate for the concentration of the MPX disease within the environment. The following is one way that MPXV might spread indirectly and how long it can live in the environment.

The concentration of the MPXV in the environment increases when infected individuals and rodents release the pathogen at rates of $\rho_{1}$ and $\rho_{2}$, respectively, and decreases due to decay with rate $\nu_{n}$. Thus, the equation below describes the dynamics of the environmental viral concentration.

Combining Eqs (2.1) to (2.3), we have the following dynamical system:

where

subject to nonnegative initial conditions (ICs)

The transition among different population groups can be easily understood from Figure 1.

2.4.   Data fitting and parameter estimation

This section briefly addresses the parameter estimation procedure for the MPX model (2.4) using a nonlinear standard least square approach based on residual minimization. Furthermore, some of the demographic parameters (i.e., $\Lambda_{h},$ $\mu_{h}$) are estimated from the literature ^[26]. The values of parameters $\Lambda_{r},$ $\mu_{r}$ and $K$ are taken from ^[17]. For the rest of the model parameters, the actual reported cases in the recent outbreak in the USA during 2022 are taken into the account to provide a reasonable fitting by the model simulation. The best fit provided by the proposed epidemic model to the real data is depicted in Figure 2. The respective parameters with biological descriptions and numerical values are tabulated in Table 1.

3.   Basic analysis of the model

3.1.   Positivity and invariant region

Theorem 3.1. The solution $\mathcal{G}(t) = \left(S_{h}(t), I_{h}(t), Q_{h}(t), R_{h}(t), S_{r}(t), I_{r}(t), E_{n}(t)\right)$ with non-negative initial condition of the model (2.4) will be positive for $t\ge0,$ also $lim_{t\to {\infty}}Sup N_{h}\le \frac{\Lambda_{h}}{\mu_{h}}$, $lim_{t\to {\infty}}Sup N_{r} = \frac{\Lambda_{r}}{\mu_{r}}$ and $E_n \left(t \right) \le \frac{1}{{\nu _n }}\left({\frac{{\rho _1 \Lambda _r }}{{\mu _r }} + \frac{{\rho _2 \Lambda _h }}{{\mu _h }}} \right).$

Proof. Let the initial conation of the MPX problem is $\mathcal{G}(0),$ which is non-negative. In order to prove our required result, we consider the first the equation of system (2.4) and proceed as follows:

which can further be written as

By integrating, we have

Similarly, for other equations of the MPX model (2.4), we obtain the desired interpretation. To prove the subsequent part of theorem, consider the human and animal populations separately involved in the system (2.4), we have

By simple manipulation, we have

In a similar way, we can prove that $\mathop {\lim }\limits_{t \to \infty } Sup N_r \left(t \right) = \frac{{\Lambda _r }}{\mu_{r} }$. Since $I_{h}\le N_{h}$ and $I_{r}\le N_{r}$, we have from last equation of model (2.4) that

which implies that

Hence, the biological feasible region is given by

□

Proposition 3.2. For non-negative ICs, the region defined above in $\Theta$ is bounded as well as positively invariant in $\mathbb{R}_ + ^5 \times \mathbb{R}_ + ^2.$

Proof. We know form Theorem 3.1,

After some simplification, we have

Particularly, we have $N_{h}\le \frac{\Lambda_{h}}{\mu_{h}}$ and $N_{r}\le\frac{\Lambda_{r}}{\mu_{r}}$. Additionally, from the last equation of system (2.4) we have,

Hence, the set shown by $\Theta$ is positively invariant. Furthermore, all solution trajectories will attract in $\mathbb{R}^{5}_{+}\times \mathbb{R}^{2}_{+}.$ □

3.2.   Existence of equilibrium points and $\mathcal{R}_{0}$

This section presents the evaluation and existence of the model equilibria. The basic reproductive number $\mathcal{R}_{0}$, one of the essential and threshold parameters in epidemiology, is also evaluated in this part. Model (2.4) possess two equilibrium points: the monkeypox free equilibrium ($\texttt{MPXFE}$), and the monkeypox endemic equilibrium ($\texttt{MPXEE}$). The $\texttt{MPXFE}$ point is given by

The $\mathcal{R}_{0}$ is computed via the well-known next-generation technique ^[27]. The resulting expression of the $\mathcal{R}_{0}$ is as follows

where

3.2.1.   MPX endemic equilibrium point

By simultaneously solving the human, rodent, and environmental classes of the system (2.4) at steady state denoted by$\mathcal{M}_{ee} = \left({S_{h_{ee} }, I_{h_{ee} }, Q_{h_{ee} }, R_{h_{ee} }, S_{r_{ee} }, I_{r_{ee} }, E_{n_{ee} } } \right)$, where

Consider force of the infection term as follows

We obtain the following polynomial by using in $(3.6)$ in (3.7)

where

Furthermore, the expression in $b_i's$ are as follows

Since, $\left(l_{1}l_{2}-\phi \tau_{1}\right)>0$ here it should be noted that $b_{0}(<, >)0,$ iff $(\mathcal{R}_{h}^{0}, \mathcal{R}_{r}^{0})<, >1.$ Moreover, we summarize the the results as follows.

Theorem 3.3. ^[28] Every polynomial equation with an odd degree has at least one real root with a sign opposite to the sign of its last term.

When $\mathcal{R}_{r}^{0}>1$ then $b_{0}>0$, and we have two cases:

Case Ⅰ. When $\mathcal{R}_{0}>1,$ then using Theorem 4, Eq (3.8) possesses at least one positive root.

Again two cases arise:

Case Ⅱ. When $\mathcal{R}_{0}>1,$ and if $b_{1} < 0, b_{2} < 0$ then using Descartes rule of signs, (3.8) has exactly one root with positive sign.

Case Ⅲ. When $\mathcal{R}_{0}>1,$ and if $b_{1}>0, b_{2}>0$ then using Descartes rule of signs, (3.8) has at least one root with positive sign.

Case Ⅳ. When $\mathcal{R}_{0}\le 1,$ then using Descartes rule of signs Eq (3.8) has no positive root.

3.2.2.   Local stability at $\texttt{ MPXFE}$ point

Theorem 3.4. The MPX model (2.4) is locally asymptotically stable at $\texttt{MPXFE}$ iff $\mathcal{R}_{0} < 1.$

Proof. For the required proof, the necessary condition is that the Jacobian matrix evaluated at $\mathcal{M}_{0}$ has eigenvalues with negative signs. The subsequent Jacobian matrix $J(\mathcal{M}^{0})$ is obtained as

The eigenvalues of $J(\mathcal{M}^{0})$ are $- \mu_{r}, - l_2, - \mu_{h}, - \mu_{h},$ and $-\left(-\beta _3 + \mu_{r}\right)$ and the solution of second degree polynomial can be obtained from $c_0 \lambda ^2 + c_1 \lambda + c_2 = 0.$ Here, the coefficients $c_{i}(i = 0, 1, 2)$ given in terms of $\mathcal{R}_{0}$ as

Since $c_{1}, c_{2}>0$ if $\mathcal{R}_{h}^{0} < 1$, then by Routh-Hurwitz criteria for polynomial of degree two, the system (2.4) is stable locally asymptotically if $\mathcal{R}_{h}^{0} < 1$. □

3.2.3.   Global stability at $\texttt{MPXFE}$ point

Theorem 3.5. If $\mathcal{R}_{0} < 1$ then the $\texttt{MPXFE}$ of the proposed model $(2.4)$ is globally asymptotically stable (GAS).

Proof. According to the Castillo Chavez theorem ^[29], the basic epidemic model can be written as

with $G(X_{if}, 0) = 0$, $X_{if} = \left({S_h, R_h, S_r } \right)^t\,$and$\, Z_{ip} = \left({I_h, Q_h, I_r, E_{n} } \right)^t.$ According to the first condition of the theorem, non-infected compartments of MPX epidemic model ($2.4$) can be shown as

The respective jacobian matrix of $F(X_{if}, 0)$ is evaluated as

The problem in the system (3.11) will be GAS if $J\left({F\left({X_{if}, 0} \right)} \right)$ has negative eigenvalues. Since the eigenvalues of the given matrix (3.12) are clearly negative. Therefore, the system (3.11) is GAS.

By the second condition of the theorem, the infected compartments of the system (2.4) can be written as

where,

Clearly, $\bar G(X_{if}, Z_{ip})\ge0,$ since $\frac{{S_h^{0} }}{{N_r^0 }} \ge \frac{{S_h }}{{N_r }}, \, \frac{{S_h^{0} }}{{N_h^{0} }} \ge \frac{{S_h }}{{N_h }}, \,$and$\, \frac{{S_h^{0} }}{k} - \frac{{S_h }}{{k + E_n }} = \frac{{S_h^{0} E_n + \left({S_h^{0} - S_h } \right)k}}{{k\left({k + E_n } \right)}} \ge 0.$ Therefore, the Eq (3.13) becomes

where the eigenvalues are $- l_2, \mu _r \left({\mathcal{R}_r^0 - 1} \right),$ and the solution of the polynomial equation

where, $c_{i}, i = 0, 1, 2$ are given below

Clearly, (3.14) has negative eigenvalues and therefore, the matrix $A$ satisfies the second condition of the comparison theorem. Thus, the $\texttt{MPXFE}$ point is GAS if $\mathcal{R}_{h}^{0} < 1$ otherwise unstable. □

4.   Sensitivity analysis

Sensitivity analysis is a strong tool that helps in identifying the significantly influential factor on the reproductive number.?Moreover, this analysis is useful to set a suitable optimal control intervention for disease minimization. Due to possible errors in data collection and the assumed parameters, the sensitivity analysis determines the robustness of the model prediction to the variation of parameters values. In this regard, various approaches have been applied in the literature. In this study, we utilized the parametric approach of the normalized forward sensitivity indices introduced in ^[30]. A positive (or negative) index value indicates that the corresponding parameter directly (inversely) affects $\mathcal{R}_{0}$. The formulae for the normalized forward sensitivity index for the model key parameters is defined as follows:

Definition 4.1. The aforementioned technique to measure the relative change in $\mathcal{R}_{0}$ with respect to the relative change in the model parameter say $x$ is given by ^[30]:

Further, using the formula stated in (4.1), we evaluate the respective indices of various model parameters. These indices are given in Table 2. The parameters having considerable impact on $\mathcal{R}_{r}^{0}$ and $\mathcal{R}_{h}^{0}$ are $\beta_{3}, \mu_{r}, \beta_{2}, \zeta, \Lambda_{h}, \rho_{1}, \mu_{h}, \nu_{n}, K, \gamma, \delta,$ and $\phi$. The respective expressions evaluated by the formula mentioned above are given below. Because of their sensitivity indices, these factors identify potential preventive strategies. The following expressions are obtained for different parameters using the formula defined in (4.1)

where, $Y_{\beta _3 }, Y_{\mu _r }, Y_{\beta _2 }, Y_\zeta, Y_{\rho _1 }, Y_K, Y_{\nu _n }, Y_{\mu _h }, Y_\gamma, Y_\delta$, and $Y_\phi$ denote the normalizer of sensitivity index with respect to $\beta_{3}, \mu_{r}, \beta_{2}, \zeta, \rho_{1}, K, \nu_{n}, \mu_{h}, \gamma, \delta,$ and $\phi$, respectively. The corresponding numerical normalized sensitivity indices using the estimated parameters are given in Table 2, while the bar plot is depicted in Figure 3. It is observed that some of the parameters have positive signs and some have negative signs with indices. The parameters with positive indices directly relate to the basic reproduction number, which means that the reduction in these parameters reducesthe disease incidence. On the other hand, the parameters with negative indices have an inverse relation with reproduction numbers, revealing that enchasing these parameters will reduce the disease rate. Based on the sensitivity indices mentioned in the table below, the feasible and effective optimal control interventions are the use of personal protection (capable of reducing $\beta_1, \beta_2, \beta_3$, $\zeta$), proper contact tracing and isolation policy (enhancing $\phi$), reducing the shedding rate of infection in the environment (reducing $\rho_1$, $\rho_2$), and fumigating commercial areas (increasing the virus decay in the environment i.e., $\nu_n$).

5.   Optimal control analysis

This part presents the formulation of a control problem to suggest the optimized control intervention to reduce the disease incidence within a community. Moreover, this section includes the existence criteria and solution of the proposed control system. Based on normalized sensitivity indices, we construct an optimum control system by incorporating six time-dependent controls in the MPX model (2.4). The time-dependent control measures are represented by $u_{1}, u_{2}, u_{3}, u_{4}, u_{5}$, and $u_{6}$ with the details and biological meaning as follows:

● $u_{1}$ : use of personal protections minimizing infection transmission to susceptible humans from infected humans and animals

● $u_{2}$ : use personal protections minimizing infection transmission to susceptible humans from environmental routes (i.e., cleaning of contaminated surfaces/environment with home-based detergents, etc.)

● $u_{3}$ : isolation via contact tracing

● $u_{4}$ : use of protections measures minimizing infection transmission between susceptible and infected animals

● $u_{5}$ : $5^{th}$ control measure used for reducing the shedding rate of infection in the environment

● $u_{6}$ : $6^{th}$ fumigating commercial areas to increase the MPXV decay rate in the environment

With the use of the aforementioned controls, this part seeks to build an optimal control problem to explain how the variable control problem contributes to the eradication of the disease. The developed control problem is given in (4.2).

subject to initial conditions as in (2.5). The objective functional corresponding to the above control system is described as

While $\mathcal{A}_1 \, ,\mathcal{A}_2 \, ,\mathcal{A}_3 \, ,\mathcal{A}_{4}$ are the associated balancing constants, $\mathcal{B}_1 \, ,\mathcal{B}_2 \, ,\mathcal{B}_3\, ,\mathcal{B}_4\, ,\mathcal{B}_5$, $\mathcal{B}_6$ represents the associated cost factors, and $T$ represents the final step size. Our main objective is to seek optimal controls $u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*}$, and $u_{6}^{*}$, so that

with the control set corresponds to above is given by

5.1.   Existence of the problem

Here, we demonstrate the existence of the optimized control MPX problem (4.2). To obtain the desired outcome, we employ the methodology established in ^[31,32]. The problem in (4.2) comprised bounded equations, and if the subsequent conditions are met, the outcomes provided in ^[31,32] can be utilized to address the existence of the control problem under consideration:

$(1)$ The control set $\Xi$ and model state variables are not empty as defined earlier.

$(2)$ The control variables represented by $\Xi$ form a closed and convex set.

$(3)$ One can notice that every equation situated on the right-hand side of the control system (4.2) represents a continuous function that has an upper bound equivalent to the total of the bounded control and state variables. Additionally, these equations can be written as a linear function of $u$, with coefficients that rely on the time and state variables.

$(4)$ There exist some constants $\zeta_1>0$, $\zeta_2>0$, and $q>1$, so that the integrand involved in the objective functional $\mathcal{J}$ given by (4.3) is convex and fulfills the inequality stated below:

The approach established in ^[31,32] easily satisfies the aforementioned conditions 1 to 4. The first two conditions are evident as as the state variables and controls are both nonempty and bounded, resulting in solutions that are also convex and bounded. Additionally, it can be seen that the third condition is fulfilled due to the fact that the system is bilinear in control variables. Furthermore, the bilinearity of the system in control variables satisfies Condition 3. Finally, to assess Condition 4, one can conveniently write:

Thus, we state the following theorem for the optimal control problem existence.

Theorem 5.1. The optimal control set shown by $u^* = \left(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*}, u_{6}^{*} \right)$ exists if the objective functional $\mathcal{J}$ over the control set $\Xi$ corresponding to (5.1) fulfills the conditions (1–4) stated above. Furthermore, we have

5.1.1.   Solution of control problem

This section analyzes the solution to the optimal control problem for the MPX model dynamics considered in (4.2). The well-known Pontryagin $'$ s maximum principle ^[33] is used for this purpose. The required optimal solution is represented by $\left(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*}, u_{6}^{*} \right)$. Furthermore, for the necessary optimal conditions utilized in the finding the solution, we need Lagrangian and Hamiltonian, which are stated as follows

and

where, $h_i, i = 1, 2, ..., 7$ demonstrates the adjoint variables.

The criteria stated in (5.6) with the theorem given below have been used to attain the solution of the optimum problem.

Theorem 5.2. If the controls $\left(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*}, u_{6}^{*} \right)$ and the solution $\left(S_{h}^{*}, I_{h}^{*}, Q_{h}^{*}, R_{h}^{*}, S_{r}^{*}, I_{r}^{*}, E_{n}^{*}\right)$ of the corresponding control problem (5.1) minimizes the objective function in the problem, then there exist an adjoint variables (co-state variables) $h_i, i = 1, 2, ..., 7$ further the corresponding transversality conditions $h_i \left(T \right) = 0, \, \, \, i = 1, 2, 3.....7,$ such that

Moreover, we obtain the optimum controls $\left(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*}, u_{6}^{*} \right)$ which minimizes the problem over the region $\Xi$ as follows

Proof. By utilizing the conditioned mentioned in (5.6), and the transversilty conditions, the results given in (5.7) are obtained for (5.5) by setting $S_{h} = S_{h}^{*}, I_{h} = I_{h}^{*}, Q_{h} = Q_{h}^{*}, S_{r} = S_{r}^{*}, I_{r} = I_{r}^{*}$ and $E_{n} = E_{n}^{*}.$ Furthermore, using the condition $\frac{\partial }{{\partial k}}H\left({t, \mathord{\buildrel{\lower3pt\hbox{$ \smile $}} \over k} _j, h_i } \right) = 0$ for $j = 1, 2, ...6$ and $i = 1, 2, ...7$ given in (5.6), the optimal control in (5.7) are obtained. □

6.   Simulation and discussion of the control MPX problem

Here, we focus to demonstrate the numerical simulation of the MPX model with and without optimized control interventions to analyze the impact of control variables introduced in the previous section. The numerical solution of the control problem (4.2) and the corresponding adjoint control system is carried out by employing the backward fourth-order Runge-Kutta. The initial conditions for the state variables in section 2 and the values of the model parameters are given in Table 1. Most of the parameters are esteemed from reported cases during the 2022 outbreak and some are taken from literature.

Four different optimized predictive control strategies are developed to illustrate their impact on the infection incidence in a community. The following combination of control variables constructs four control strategies, namely A, B, C, and D:

● Strategy A: consist of a combination of all control measures.

● Strategy B: combination of $u_3$, $u_5$ and $u_6$ control variables only. In this case, we ignore the use of personal protection controls.

● Strategy C: using all control variables except the isolation control $u_3$.

● Strategy D: using all control variables except $u_5$ and $u_6$.

The detailed simulation along with a discussion of the above four interventions are given in the following subparts.

6.1.   Strategy A: using all the control variables

In this case, the MPX control problem (4.2) is simulated by activating all control variables $u_1, u_2, u_3, u_4, u_5, u_6$. This means that all interventions mentioned in the previous section are applied simultaneously. The simulation corresponding to the present strategy for human and animal populations are illustrated in Figure 4(a–d) and Figure 5(a, b) respectively. The profiles of all optimal control variables are shown in Figure 5(c). Figure 4(a) demonstrates the dynamics of susceptible human individuals under the application of all control interventions. The population in this class increases with the application of the optimal controls. Figure 4(b) shows the dynamics of solution trajectories of the infected MPX human population. It can be observed that without optimal controls, the population of infected humans raised quickly. However, with?application of the suggested optimal controls, the infected population vanishes. Figure 4(c) describes the impact of using all control variables on the population of the isolated class. As the controls act quickly from the onset of the application, the isolated population increased significantly. On the other hand, without controls, the isolated population decreased and reached zero. This shows the efficacy of effective contact tracing and isolation the infected population. Figure 4(d) describes the dynamics of the recovered human population with and without optimal control variables. The recovered population decreases with the application of control measures. By applying effective preventive controls, fewer people will acquire infection, and as a result, fewer individuals recover from the infection. The dynamics of susceptible and infected animals are analyzed in Figures 5(a, b), respectively. In Figure 5(a), we observed that the population of susceptible animals was almost the same with and without optimal controls. Howsoever, the number of infected animals significantly decreases under the application of all control interventions and reaches zero after 70 days. Figure 5(c) shows the concentration of the MPXV in the environment. The application of optimal controls effectively decreased the viral concentration level in the environment compared to the case without optimal controls. Figure 6(a–f) illustrates the control profiles of all control variables, showing their level of application over considered time intervals.

6.2.   Strategy B: using $u_3$, $u_5$ and $u_6$ control variables only

This strategy analyzes the combined impact of control measures $u_3$, $u_5$, and $u_6$ and sets the rest as zero (i.e., $u_1 = u_2 = u_4 = 0$). This means that the use of personal protection measures for both human and animal populations is not consideredin this case. Simulation of strategy B with control profiles are shown in Figures 7–9. This strategy is less useful for disease eradication although the infected human population decreases with the application of optimal controls as can be seen in Figure 7(b). The population in the isolated class raises quickly with control compared to without control application, as seen in Figure 7(c). Further, this strategy has no impact on the dynamics of the animal population, as seen in Figure 8(a, b). The concentration of the MPX virus in the environmentsignificantly reduces as the controls act quickly from the onset of the application, as seen in Figure 8(c). Figure 9(a–f) shows the control profiles corresponding to strategy B. Overall, it is clear that without personal protection measures, the infection can not be curtailedeffectively.

6.3.   Strategy C: using all control variables except the isolation control $u_3$

Strategy C describes the impact of all control variables, except for the isolation control $u_3$ on the disease dynamics. The simulation of human and animal populations in this case is depicted in Figure 10(a–d) and Figure 11(a, b), respectively. The concentration of the virus in the environment is depicted in Figure 11(c). The control profiles of all control variables in strategy C are analyzed in Figure 12(a–f). Since no isolation policy is utilized, the population in isolated classes decreases in both with and without cases. However, it is observed that the individuals in the infected human class declined quickly with the application of optimal controls as compared to the without control case. Although, the reduction in infected human individuals is comparatively slower than strategy A, though it can be used for disease eradication.

6.4.   Strategy D: using all control variables except $u_5$ and $u_6$

This strategy presents the impact of personal protection measures and isolation control. The role of this case in infection control is graphically illustrated in Figures 13 and 14. The plots describing control profiles of each optimal control variable in this strategy are depicted in Figure 15. It is observed, that the environmentalviral concentration reduces at a slower rate as compared to previous strategies, as seen in Figure 13(c). Still, it can be used to curtail the infection in a community. Overall, from the simulation of all strategies, we conclude that the best and optimal control intervention that can be used effectively to eradicate the infection is strategy A (i.e., considering all control measures at the same time).

7.   Conclusions

Besides the less severity of the MPX infection, it is alarming that a current outbreak is reported in many non-endemic countries. The early eradication of the infection is necessary before it emerges as a new health issue around the globe. In this study, we developed a mathematical model addressing the dynamics of monkeypox under some optimized predictive control interventions. The model parameters are estimated from the reported infected cases during the 2022 outbreak in the USA in order to make the study more useful. The model is first formulated with constant control measures using a nonlinear differential system. The viral concentration in the environment and the environmental transmission are taken into consideration in the model formulation. The model is reformulated by incorporating personal protection measures, isolation control, and fumigating commercial areas. The control interventions are respectively repressed by six time-dependent control variables namely: $u_1(t)$, $u_2(t)$, $u_3(t)$, $u_4(t)$, $u_5(t)$, and $u_6(t)$. The following outcomes of the present study are observed

● The model exhibits a disease-free and endemic equilibria.

● The model is found to be stable locally and globally at the disease-free equilibrium when, $\mathcal{R}_0 < 1$.

● The basic reproduction number is evaluated based on estimated values of model parameters.

● Normalized sensitivity analysis indicates that personal protection measures, isolation control, and fumigating commercial areas are effective to control interventions.

● The usage of time-dependent controls can reduce the total number of infected individuals in the human and animal populations.

● The most effective control intervention involves all control variable simultaneously.

Fractional order modeling are more reliable than the integer epidemic models. Therefore, the present study could be further extended using fractional case modeling with singular and nonsingular kernels.

Acknowledgments

This research work was funded by Institutional Fund Projects under grant no. (IFPIP: 970-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Conflict of interest

The authors declare no conflicts of interest.