A novel mathematical modeling and optimal control analysis of monkeypox transmission incorporating double-dose vaccination strategies: Insights from the recent outbreak

Abstract

Monkeypox (Mpox) has re-emerged as a serious global public health concern due to its potential human-to-human transmission and persistence in the environment. This study develops a new deterministic model incorporating double-dose vaccination approach to analyze the transmission dynamics and effective interventions for the Mpox outbreak. The model divides the human population in six groups and includes an environmental reservoir, capturing both direct transmission from infectious individuals and indirect transmission through environmental contamination. Global stability results of the equilibria are examined utilizing a standard Lyapunov function approach. The parameters are estimated using cumulative Mpox cases reported during 2022 outbreak in the United States. Further, normalized sensitivity analysis is performed to indicate the key parameters influencing disease transmission and eradication. Pontryagin's maximum principle is applied to formulate an optimal control model using time-dependent variables for vaccination, treatment, and environmental disinfection. Simulation illustrates that double-dose vaccination substantially reduces new infections, particularly when coupled with timely treatment and environmental clearance measures. The findings of the present study highlight the importance of double-dose vaccination combined with complementary controlling measures in managing Mpox incidence and offer practical insights for public health planning.

1. Introduction

Viral infectious diseases have consistently posed serious challenges to global public health. Throughout history, numerous viral epidemics and pandemics have emerged, causing widespread morbidity and mortality. Certain examples of infectious diseases include mpox, measles, influenza, acquired immunodeficiency syndrome (AIDS) caused by Human immunodeficiency virus, Middle East respiratory syndrome (MERS), Ebola virus disease, and the recent coronavirus disease 2019 (COVID-19). An estimated 50,000 people die each day due to various viral and bacterial infections (Shen et al., 2025). Mpox caused by the monkeypox virus (MpoxV), is a zoonotic disease belonging to the orthopoxvirus genus and is transmitted from animals to humans. This zoonotic virus was initially discovered in 1958, when a pox-like illness was observed in monkey colony in a Danish research laboratory. As a result, the disease was given the name ”Monkeypox”. However, the first known case of Mpox was actually found in an animal. In 1970, the first documented case of Mpox in a human was reported in a young child from the Democratic Republic of Congo (Alshehri & Ullah, 2023). According to (Okongo et al., 2024), Mpox primarily occurs in Central and West Africa, with occasional cases reported in other parts of the world. MpoxV may spread through several potential routes including environmental exposure, close contact between people, as well as from animals to humans. Human-to-human transmission may occur through close face-to-face interaction with an infected individual's respiratory droplets, sores, bodily fluids or skin lesions (O'Shea, 2022).

The incubation period of the MpoxV usually lasts between 7 and 14 days, though it can range from 5 to 21 days (Okongo et al., 2024). During this time, an infected person may not show any symptoms but can still spread the virus. The persistence of acute infection typically remains for 1-3 days. The symptoms in this phase include intense headaches, fever, back and muscle pain, swollen lymph nodes, pronounced fatigue etc. The subsequent phase is known as the skin eruption phase which generally disappear between 2 and 4 weeks. In this stage of infection, lesions undergo a characteristic progression: beginning as macules (flat spots), then transforming into papules (raised, firm, and often painful lesions), followed by vesicles (fluid-filled blisters), and later pustules (pus-filled lesions), before ultimately forming crusts (Hraib et al., 2022; Singhal et al., 2022). This infection is reported in many countries including the United States (US), Brazil, Spain and France, where the people are significantly affected by Mpox (Cheema et al., 2024). The global distribution of Mpox is illustrated in Fig. 1 hightailing the alarming situation.

Fig. 1.

Global burden of Mpox disease (CDC).

Fig. 2 demonstrates the per day incidence of Mpox in the US during the 2022 outbreak.

Fig. 2.

Daily reported Mpox cases in the US during the 2022 (CDC).

Vaccination is among the effective ways to prevent the spread of diseases. As reported by the UK Health Security Agency, 2022 (UKHSA 2022) no vaccine has been specifically developed against MpoxV. However, since MpoxV belongs to the same genus as other orthopoxviruses, such as smallpox, the smallpox vaccine has been found approximately 85% effective in protecting against MpoxV (Petersen et al., 2019), and is currently recommended for use in MpoxV prevention (UKHSA 2022). Individuals who received smallpox vaccination in their early years are thought to retain partial immunity that may provide some level of protection against Mpox. This residual immunity is attributed to the close genetic relationship between the smallpox and Mpox viruses, which allows cross protective immune responses to persist even decades after vaccination (Deputy et al., 2023). Recent findings by Adepoju et al. indicate that Mpox infection does not confer complete immunity, as cases of reinfection have been documented within the population (Adepoju & Ibrahim, 2024). Two vaccines are available: Modified Vaccinia Ankara (Jynneos, Imvamune, or Imvanex) and ACAM2000 (by Emergent BioSolutions). The JYNNEOS vaccine uses a weakened, non-replicating form of the vaccinia virus, making it suitable for immuno compromised individuals and pregnant women (Centers for Disease Control and Prevention).

Mathematical modeling plays a vital role in analyzing the dynamics and ultimate mitigation of infectious diseases around the globe (Brauer et al., 2019). Many mathematical models addressing different aspects of Mpox dynamics have been recently presented in the literature. A compartmental model focusing on direct human-to-human contacts with vaccination intervention for minimizing Mpox incidence was presented in (Ahmad et al., 2024). The analysis of a single-dose vaccination strategy impacting the incidence of Mpox was explored through a nonlinear computational model in (Soni & Sinha, 2024). A novel modeling approach capturing the environmental viral transmission was recently proposed in (Madubueze et al., 2022). The dynamics of Mpox under various transmission routes and control measures was recently analyzed in (Bankuru et al., 2020; Peter et al., 2022). In reference (Grant et al., 2020), the authors studied the transmission and control of MpoxV with direct human-to-human contacts. Application of optimization theory to effectively minimize the human-to-human transmission of MpoxV was presented in (Baroudi et al., 2024), where the authors identified that public education and vaccination can be considered as key control interventions. Reference (Awoke et al., 2024) formulated a novel deterministic framework to assess cost-effective strategies for managing Mpox spread incorporating both direct and indirect transmission routes as well as environmental influences. This study focused on developing a strain-specific epidemiological model to study transmission dynamics of the infection capturing statistical patterns and epidemic trends across the globe (Idisi et al., 2025). A recent study focused on the sexual transmission of Mpox infection in the US and highlighting the use of condom as a key control strategy was conducted in (Rabiu et al., 2024). The dynamics and cost-effectiveness analysis of Mpox infection to explore the optimal strategy for disease eradication in Africa were presented (Ochieng, 2025). A recent computational analysis of Mpox transmission dynamics and control through a robust numerical approach was performed in (Khan et al., 2024; Shyamsunder & Meena, 2025). Optimal control theory has been used as a valuable technique for formulating and identifying effective intervention strategies that aims to minimize the infectious disease outbreaks (Hassan et al., 2025; Khan et al., 2022; Li et al., 2025). In recent literature, the application of optimal control strategies to reduce the spread of Mpox and highlight the effectiveness of different interventions was studied in (Peter et al., 2023). Another investigation explored the co-infection dynamics of Mpox and COVID-19, employing optimal control theory to analyze their interactions and potential management strategies (Acheneje et al., 2024). A few recent modeling analyses of Mpox in different regions can be found in (Rabiu et al., 2026; Woldegerima & Ugwu, 2025).

In the present paper, we develop a nonlinear mathematical model to explain transmission through close contact between people of MpoxV, incorporating the effect of a double-doses vaccination strategy. The proposed research intends to understand how both the first and second vaccine doses impact the spread of the disease and to evaluate their effectiveness in reducing infection rates. Furthermore, we estimate the model parameters using reported Mpox cases from outbreaks in the US. Most Mpox cases in the US have occurred among individuals who are either unvaccinated or have received only single JYNNEOS vaccination. For optimal protection, two doses of JYNNEOS are recommended.

Key highlights of the present study are listed as follows:

• •Development of a novel compartmental model for Mpox that incorporates susceptible, single-dose vaccinated, double-dose vaccinated, exposed, infectious, recovered, and environmental reservoir classes.
• •Integration of environmental transmission alongside human-to-human spread to reflect the complex dynamics of Mpox.
• •Assessment of the global stability of the model's equilibria using standard techniques.
• •Estimation of key parameters from reported US Mpox outbreak data, followed by sensitivity analysis to identify the most influential factors driving transmission.
• •The formulation of optimal time-dependent control problem Pontryagin's Maximum Principle (PMP) that combines vaccination, treatment, and environmental disinfection.
• •To perform numerical simulation illustrating that a double-dose vaccination campaign when coupled with treatment of actively infected individuals is effective in reducing disease incidence and mitigating outbreaks.

1.1. Structure of the study

The organization of the whole paper consists of eight sections. Section 2 briefly presents the model's formulation process incorporating six human groups and an environmental reservoir compartments. Section 3 presents the basic theoretical results establishing positivity, boundedness, and the existence of an invariant region for the proposed Mpox model. Section 4 computes the basic reproduction number and the equilibria of the model. This section further analyzes both local and global stability properties using Lyapunov function approach. Section 5 perform parameter estimation using documented Mpox cases during 2022 outbreak in the US and section 6 presents the sensitivity analysis of various parameters. Section 7 formulates the optimal control problem by applying control theory in order to suggest effective interventions incorporating vaccination, treatment, and environmental disinfection strategies. Section 8 illustrates a comprehensive simulation of the optimal control model based on various strategies, highlighting the effectiveness of double-dose vaccination in minimizing transmission and prevalence of the infection. Section 9 concludes the study by summarizing the main findings and discussing their implications for public health policies and vaccination strategies against Mpox.

2. Model formulation

To develop the model, we assume that the total human population represented as P(t) at time t is homogenously mixing. This population is further divided into several compartments: susceptible individuals denoted by S(t), first dose vaccinated represented as V₁(t), second dose vaccinated represented as V₂(t), individuals who have been exposed but are not yet infectious represented as E(t), individuals who are infectious denoted as I(t), those who have recovered denoted as R(t). Thus, we have the relation:

$$P(t)=S(t)+V_1(t)+V_2(t)+E(t)+I(t)+R(t).$$

Similarly, the pathogens in the environment is shown by the class B(t). We represent the birth rate by λ and the natural death rate by μ. The force of infection is represented by ϕ(t), which is given by:

$$\varphi \left(t\right)=\frac{({\beta }_{1}I\left(t\right)+{\beta }_{2}B\left(t\right))}{P\left(t\right)}.$$

The parameter β₁ represents the effectvie contact rate by which susceptible individuals become infected through close contact with infected individuals. The parameter β₂ represents contact rate of susceptible individuals who become infected with the virus from the contaminated environment. The parameter α₁ is the rate of vaccination for susceptible individuals and the parameter α₂ is the rate of second dose vaccination. τ is the rate at which individuals in the first vaccination class after lose immunity and become susceptible again. Since vaccines do not provide complete protection, we account for vaccine efficiency by introducing the parameter η ∈ [0, 1]. Infected individuals transition from the exposed to the infectious compartment at a rate δ while φ is the death rate due to disease at infected class. The parameter γ₁ represent the recovery rate of the second dose vaccinated individual and γ₂ represent the recovery rate of infected individuals. Recent research shows that people who are fully vaccinated with two doses of the vaccine have approximately 85% effectiveness against Mpox and are significantly less chance to be infected compared to unvaccinated individuals (Centers for Disease Control and Prevention). We assume that people who receive two doses acquire lifelong immunity and move directly into the recovered class. The environment is being contaminated by the infected people through the rate ξ, while ω is their clearance rate. Fig. 3 presents the transmission dynamics of the proposed model, showing the movement of individuals through different epidemiological compartments. A detailed summary of the model parameters is provided in Table 1.

Fig. 3.

Schematic diagram illustrating the transmission dynamics of the model.

Table 1.

Biological description and corresponding estimated values of the model parameters.

Parameters	Description	Value [per day]	Source
λ	Birth rate	12109	Estimated
μ	Natural death rate	$\frac{1}{79.11\times 365}$	Estimated
β1	Transmission rate from infectious	0.8716	Fitted
β2	Transmission rate from contaminated environment	0.8016	Fitted
α1	Vaccination rate for first doses	0.1094	Fitted
α2	Vaccination rate for second doses	0.2316	(CDC)
τ	Immunity loss rate from first dose vaccinated	0.1949	Fitted
η	Vaccination efficiency	0.8666	Fitted
δ	Transmission rate from exposed to infected	0.0080	Fitted
γ1	Recovery rate from second dose vaccination	0.2268	Fitted
γ2	Recovery rate from Infected	0.0235	Fitted
ϕ	Death rate due to disease	0.00077	Estimated
ξ	Rate at which the virus is added to the environment	0.9000	Assumed
ω	Clearance rate	0.0125	Alshehri and Ullah (2023)

Based on the assumptions described above, the system of differential equations takes the following form.

$$\left\{\begin{array}{ll}{S}^{\text{'}}& =\lambda -\frac{({\beta }_{1}I+{\beta }_{2}B)S}{P}-({\alpha }_1+\mu )S+\tau V_1,\\ V_1^{\text{'}}& ={\alpha }_{1}S-\frac{\eta ({\beta }_{1}I+{\beta }_{2}B)V_1}{P}-({\alpha }_2+\mu +\tau )V_1,\\ V_2^{\text{'}}& ={\alpha }_2{V}_1-({\gamma }_1+\mu )V_2,\\ E^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)}{P}(S+\eta V_1)-(\delta +\mu )E,\\ I^{\text{'}}& =\delta E-({\gamma }_2+\phi +\mu )I,\\ R^{\text{'}}& ={\gamma }_1{V}_2+{\gamma }_{2}I-\mu R,\\ B^{\text{'}}& =\xi I-\omega B,\end{array}\right)$$ (1)

with non-negative initial conditions, the system guarantees that all solutions remain biologically feasible for every t > 0. The corresponding initial conditions for system 1 are specified as follows:

$$\begin{array}{lll}S(0)& =& S_0>0,V_1(0)={V_1}_0\ge 0,V_2(0)={V_2}_0\ge 0,E(0)=E_0\ge 0,\\ I(0)& =& I_0\ge 0,R(0)=R_0\ge 0,B(0)=B_0\ge 0.\end{array}$$

To simplify notations, let we denote k₁ = (α₁+μ), k₂ = (α₂+μ+τ), k₃ = (γ₁+μ), k₄ = (δ+μ), k₅ = (γ₂+φ+μ); system (1) simplifies to:

$$\{\begin{array}{ll}{S}^{\prime }& =\lambda -\varphi (t)S-k_{1}S+\tau V_1,\\ V_1^{\prime }& ={\alpha }_{1}S-\eta \varphi (t)V_1-k_2{V}_1,\\ V_2^{\prime }& ={\alpha }_2{V}_1-k_3{V}_2,\\ E^{\prime }& =\varphi (t)(S+\eta V_1)-k_{4}E,\\ I^{\prime }& =\delta E-k_{5}I,\\ R^{\prime }& ={\gamma }_1{V}_2+{\gamma }_{2}I-\mu R,\\ B^{\prime }& =\xi I-\omega B.\end{array}$$ (2)

3. Basic theoretical analysis

3.1. Boundedness and positivity

To establish the numerical results of the model (1), it is necessary to confirm the non-negativity of all state variables and for any solution starting with positive initial conditions yields a positive invariant solution. This forms the basis for the following lemma.

Lemma 3.1

All solutions of system (2) with non-negative initial conditions will remain within the invariant region $\mathrm{\Omega }=\{(S,V_1,V_2,E,I,R,B)\in {\mathbb{R}}_{+}^7:0<P\le \frac{\lambda }{\mu },0<B\le \frac{\xi \lambda }{\omega \mu }\}$.

Proof. Adding all equations representing dynamics of human population in the system (2)

$$P^{\prime }(t)=S^{\prime }(t)+V_1^{\prime }(t)+V_2^{\prime }(t)+E^{\prime }(t)+I^{\prime }(t)+R^{\prime }(t)$$

$$P^{\prime }(t)\le \lambda -\mu N(t).$$

For the initial value P(0) = P₀, we obtain that

$$P(t)\le \frac{\lambda }{\mu }-\left(\frac{\lambda }{\mu }-P_0\right)e^{-\mu t}.$$

Hence, $P(t)\le \frac{\lambda }{\mu }$ for all t > 0. Further, we can now express the seventh equation of system (2) as:

$$B(t)\le \frac{\xi \lambda }{\omega \mu }-\left(\frac{\xi \lambda }{\omega \mu }-B_0\right)e^{-\omega t}.$$

By performing a similar analysis as above, we can conclude that $B(t)\le \frac{\xi \mathrm{\Lambda }}{\omega \mu }$ for t > 0. Therefore, the region is positively invariant. Furthermore, the region Ω is biologically feasible and attract all solutions in ${\mathbb{R}}_{+}^7$.

Lemma 3.2

Let P(0) ≥ 0 denotes the initial solution of the model (2) such that P(0) = (S(0), V₁(0), V₂(0), E(0), I(0), R(0), B(0)). Then, all solutions of the model remain non-negative for all time t > 0.

Proof. In the system (2), the functions on the right-hand side are continuous and Lipschitz continuous on ${\mathbb{R}}_{+}^7$, ensuring the existence and uniqueness of solutions.

Further, define the set

$$t_0=\sup \left\{t>0:P(s)\in {\mathbb{R}}_{+}^7\text{for all}s\in [0,t]\right\}.$$

Assume that t₀ < ∞. Then at least one component of the solution becomes zero at t₀, while the remaining components are non-negative.

We examine the derivative of each variable on the boundary:

$$S^{\prime }(t){}_{S=0}=\lambda +\tau V_1\ge \lambda >0,$$

$$V_1^{\prime }(t){}_{V_1=0}={\alpha }_{1}S\ge 0,$$

$$V_2^{\prime }(t){}_{V_2=0}={\alpha }_2{V}_1\ge 0,$$

$$E^{\prime }(t){}_{E=0}=\varphi (S+\eta V_1)\ge 0,$$

$$I^{\prime }(t){}_{I=0}=\delta E\ge 0,$$

$$R^{\prime }(t){}_{R=0}={\gamma }_1{V}_2+{\gamma }_{2}I\ge 0,$$

$$B^{\prime }(t){}_{B=0}=\xi I\ge 0.$$

Thus, all derivatives on the boundary of ${\mathbb{R}}_{+}^7$ are non-negative. Therefore, the solution cannot cross the boundary into the negative region. This contradicts the assumption that t₀ < ∞. Hence, t₀ = ∞, and consequently,

$$P(t)\in {\mathbb{R}}_{+}^7\text{for all}t>0.$$

Moreover, since λ > 0, it follows that S(t) > 0 ∀ t > 0.

4. Model equilibria and basic reproduction number

To derive the threshold number ${\mathcal{R}}_0$ for the model (2), we utilized the well-known next-generation technique described in (Diekmann et al., 2010; Driessche & Watmough, 2002).

In the proposed Mpox model (2), we incorporate a double-dose vaccination approach and assume lifelong vaccine-induced immunity. Consequently, individuals who receive the second dose of vaccine are assumed to fully immunized and move directly into the recovered class R. Therefore, at the disease-free state, the recovered class R remains nonzero, comprising individuals who have acquired immunity through second-dose vaccination, even in the absence of infection.

Within the present modeling framework and assumption, the only infection-related compartments are E, I, and B. By setting these compartments equal to zero, we obtain the corresponding disease-free equilibrium (DFE) for the Mpox model (2) as follows:

$$E_0=\left(\frac{\lambda k_2}{k_1{k}_2-\tau {\alpha }_1},\frac{\lambda {\alpha }_1}{k_1{k}_2-\tau {\alpha }_1},\frac{\lambda {\alpha }_1{\alpha }_2}{k_3(k_1{k}_2-\tau {\alpha }_1)},0,0,\frac{\lambda {\alpha }_1{\alpha }_2{\gamma }_1}{\mu k_3(k_1{k}_2-\tau {\alpha }_1)},0\right).$$

By substituting the corresponding coordinates of E₀, the Jacobian matrix F is obtained as:

$$F=\left(\begin{array}{lll}0& \frac{\mu k_3\left(k_2+\eta {\alpha }_1\right){\beta }_1}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}& \frac{\mu k_3\left(k_2+\eta {\alpha }_1\right){\beta }_2}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$$

and the Jacobian V given by

$$V=\left(\begin{array}{lll}{K}_4& 0& 0\\ -\delta & K_5& 0\\ 0& -\xi & \omega \end{array}\right).$$

According to (Diekmann et al., 2010; Driessche & Watmough, 2002), the basic reproduction number ${\mathcal{R}}_0$ is defined as the spectral radius of the matrix product FV⁻¹. Hence, we have:

$${\mathcal{R}}_0=\frac{\delta \mu k_3\left(k_2+\eta {\alpha }_1\right)\left(\omega {\beta }_1+\xi {\beta }_2\right)}{\omega k_4{k}_5\left(\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)\right)}.$$

We have split ${\mathcal{R}}_0$ into two parts as follows:

$${\mathcal{R}}_{01}=\frac{\delta \mu k_3\left(k_2+\eta {\alpha }_1\right){\beta }_1}{k_4{k}_5\left(\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)\right)},$$

$${\mathcal{R}}_{02}=\frac{\delta \mu k_3\left(k_2+\eta {\alpha }_1\right)\xi {\beta }_2}{\omega k_4{k}_5\left(\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)\right)}.$$

The expression for ${\mathcal{R}}_0$ can be decomposed into two distinct components, each representing a different transmission pathway. The first component, ${\mathcal{R}}_{01}$, corresponds to infections generated through direct contact with infectious individuals in the population. The second part, ${\mathcal{R}}_{02}$, measures the indirect transmission contribution through the contaminated environment, reflecting the role of the pathogen's persistence in the environment. It indicates that minimizing direct human-to-human contact will significantly lower ${\mathcal{R}}_{01}$, while implementing effective environmental sanitation interventions and minimizing exposure to contaminated surfaces will reduce ${\mathcal{R}}_{02}$.

4.1. Existence of endemic equilibrium (EE)

We establish the criteria for the existence of EE of Mpox model (2). The EE is denoted by $E_1=(S^{\ast },V_1^{\ast },V_2^{\ast },E^{\ast },I^{\ast },R^{\ast },B^{\ast })$, and is expressed as follows:

$$\left\{\begin{array}{l}{S}^{\ast }=\frac{\lambda \left(\eta \varphi +k_2\right)}{\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1},\\ V_1^{\ast }=\frac{\lambda {\alpha }_1}{\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1},\\ V_2^{\ast }=\frac{\lambda {\alpha }_1{\alpha }_2}{k_3(\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1)},\\ E^{\ast }=\frac{\lambda \varphi (\eta \varphi +k_2+\eta {\alpha }_1)}{k_4(\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1)},\\ I^{\ast }=\frac{\delta \lambda \varphi (\eta \varphi +k_2+\eta {\alpha }_1)}{k_4{k}_5(\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1)},\\ R^{\ast }=\frac{\lambda (k_4{k}_5{\alpha }_1{\alpha }_2{\gamma }_1+\delta \varphi k_3(\eta \varphi {\gamma }_2+k_2{\gamma }_2+\eta {\alpha }_1{\gamma }_2))}{\mu k_3{k}_4{k}_5(\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1)},\\ B^{\ast }=\frac{\delta \lambda \xi \varphi (\eta \varphi +k_2+\eta {\alpha }_1)}{\omega k_4{k}_5(\eta {\varphi }^2+\eta \varphi k_1+\varphi k_2+k_1{k}_2-\tau {\alpha }_1)}.\end{array}\right)$$ (3)

To evaluate the infection force ϕ∗, the values from (3) are substituted into the following expression:

$${\varphi }^{\ast }=\frac{{\beta }_1{I}^{\ast }+{\beta }_2{B}^{\ast }}{P^{\ast }}.$$ (4)

After simplification, the expression reduces to the following form:

$$C_1{{\varphi }^{\ast }}^2+C_2{\varphi }^{\ast }+C_3=0,$$ (5)

where

$$C_1=\eta \omega k_3(\mu k_5+\delta (\mu +{\gamma }_2)),$$

$$C_2=k_3\left(\mu \omega k_2\left(\delta +k_5\right)+\eta \mu \left(k_4{k}_5\omega +\left(\omega {\alpha }_1\left(\delta +k_5\right)-\delta \left(\omega {\beta }_1+\xi {\beta }_2\right)\right)\right)+\delta \omega {\gamma }_2\left(k_2+\eta {\alpha }_1\right)\right),$$

$$C_3=\omega k_4{k}_5(\mu k_2{k}_3+{\alpha }_1(\mu k_3+{\alpha }_2(\mu +{\gamma }_1)))(1-{\mathcal{R}}_0).$$ (6)

Theorem 4.1

The condition governing the existence of EE for Mpox double-dose vaccine model (2) can be described in the following parts:

• (a)The model exhibits a unique EE if C₃ < 0, or equivalently, when ${\mathcal{R}}_0>1$.
• (b)The model exhibits a unique EE if C₂ < 0, and additionally either C₃ = 0 or the discriminant satisfies $C_2^2-4C_1{C}_3=0$.
• (c)Two distinct EE will exist if C₃ > 0, C₂ < 0, and the condition $C_2^2-4C_1{C}_3>0$ holds.
• (d)There will be no EE otherwise.

Proof. In accordance of condition (a), the model guarantees the existence of a unique MpoxV EE, implying that under the given criteria, the system consistently approaches a single stable endemic state.

4.2. Local stability of the equilibria

Theorem 4.2

The DFE $E_0\left(\frac{k_2\lambda }{k_1{k}_2-{\alpha }_1\tau },\frac{\lambda {\alpha }_1}{k_1{k}_2-\tau {\alpha }_1},\frac{\lambda {\alpha }_2{\alpha }_1}{k_3(k_1{k}_2-{\alpha }_1\tau )},0,0,\frac{\lambda {\alpha }_2{\alpha }_1{\gamma }_1}{k_3\mu (k_1{k}_2-{\alpha }_1\tau )},0\right)$ is locally asymptotically stable (LAS) if ${\mathcal{R}}_0<1$. Conversely, when ${\mathcal{R}}_0>1$, the DFE E₀ becomes unstable.

Proof. To investigate the local stability of the DFE, we compute the Jacobian of (2) at E₀. The Jacobian obtained is expressed as follows:

$$Z=\left(\begin{array}{lllllll}-k_1& \tau & 0& 0& -\frac{\mu k_2{k}_3{\beta }_1}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}& 0& -\frac{\mu k_2{k}_3{\beta }_2}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}\\ {\alpha }_1& -k_2& 0& 0& -\frac{\eta \mu k_3{\alpha }_1{\beta }_1}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}& 0& -\frac{\eta \mu k_3{\alpha }_1{\beta }_2}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}\\ 0& {\alpha }_2& -k_3& 0& 0& 0& 0\\ 0& 0& 0& -k_4& \frac{\mu k_3(k_2+\eta {\alpha }_1){\beta }_1}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}& 0& \frac{\mu k_3(k_2+\eta {\alpha }_1){\beta }_2}{\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right)}\\ 0& 0& 0& \delta & -k_5& 0& 0\\ 0& 0& {\gamma }_1& 0& {\gamma }_2& -\mu & 0\\ 0& 0& 0& 0& \xi & 0& -\omega \end{array}\right).$$

From the Jacobian matrix Z above, it is evident that the eigenvalues −μ, and −k₃ have negative real parts. The remaining eigenvalues are obtained from the sub matrix corresponding to the infectious classes. When ${\mathcal{R}}_0<1$, all eigenvalues of the matrix Z possess negative real parts, ensuring that the DFE is LAS. Conversely, if ${\mathcal{R}}_0>1$, at least one eigenvalue has a positive real part, leading to the instability of the DFE.

$$b_0{\mathrm{\Lambda }}^5+b_1{\mathrm{\Lambda }}^4+b_2{\mathrm{\Lambda }}^3+b_3{\mathrm{\Lambda }}^2+b_4\mathrm{\Lambda }+b_5=0.$$ (7)

The expression for all coefficients in equation (7) are as follows:

$$b_0=a^3,$$

$$b_1=a^3\left(\omega +k_1+k_2+k_4+k_5\right),$$

$$b_2=a^3\left(\left(\omega \left(k_1+k_2+k_4+k_5\right)+k_2\left(k_1+k_4+k_5\right)+k_1\left(k_4+k_5\right)-\tau {\alpha }_1\right)k_4{k}_5\left(1-{\mathcal{R}}_{01}\right)\right),$$

$$\begin{array}{ll}{b}_3& =a^3\left(\omega \left(k_1{k}_2+k_1{k}_4+k_2{k}_4+k_1{k}_5+k_2{k}_5\right)+k_1{k}_2\left(k_4+k_5\right)-\tau {\alpha }_1\left(\omega +k_4+k_5\right)\right)\\ & \left(+\omega k_4{k}_5\left(1-{\mathcal{R}}_0\right)+k_4{k}_5\left(1-{\mathcal{R}}_{01}\right)\left(k_2+k_1\right)\right).\end{array}$$

$$\begin{array}{ll}{b}_4& =a^3\left(k_1{k}_2(\omega (k_4+k_5)+k_4{k}_5)-\tau {\alpha }_1(\omega (k_4+k_5)+k_4{k}_5(1-{\mathcal{R}}_{01}))+k_1{k}_2{k}_4{k}_5(1-{\mathcal{R}}_{01})\right)\\ & \left(+(\omega k_4{k}_5(1-{\mathcal{R}}_0))(k_1+k_2)\right).\end{array}$$

$$b_5=a^2\left(k_1{k}_2-\tau {\alpha }_1\right)+a\omega k_4{k}_5\left(1-{\mathcal{R}}_0\right),$$

where

$$a=\mu k_2{k}_3+{\alpha }_1\left(\mu k_3+{\alpha }_2\left(\mu +{\gamma }_1\right)\right).$$

It follows that all coefficients b_j (for j = 0, 1, 2, 3, 4, 5) remain strictly positive whenever the basic reproduction number satisfies ${\mathcal{R}}_0<1$. Under this condition, the subsequent Routh–Hurwitz criterion for the fifth-degree characteristic polynomial in equation (7) can be verified. Hence, we deduce that the DFE is LAS whenever ${\mathcal{R}}_0<1$.

4.3. Global stability of DFE

Theorem 4.3

The DFE $E_0\left(\frac{\lambda k_2}{k_1{k}_2-\tau {\alpha }_1},\frac{\lambda {\alpha }_1}{k_1{k}_2-\tau {\alpha }_1},\frac{\lambda {\alpha }_1{\alpha }_2}{k_3(k_1{k}_2-\tau {\alpha }_1)},0,0,\frac{\lambda {\alpha }_2{\alpha }_1{\gamma }_1}{k_3\mu (k_1{k}_2-{\alpha }_1\tau )},0\right)$ is globally asymptotically stable (GAS) in $\mathrm{\Omega }=\{(S,V_1,V_2,E,I,R,B)\in {\mathbb{R}}_{+}^7\}$ if ${\mathcal{R}}_0<1$.

Proof. To investigate the global stability of the DFE, we construct the following Lyapunov function (Ullah & Khan, 2020):

$$F(t)={\varrho }_{1}E+{\varrho }_{2}I+{\varrho }_{3}B.$$ (8)

Here, the constants ϱ_i for i = 1, 2, 3 are chosen to be positive. Computing the derivative of F(t) w.r.t to t over the solution of system (2), we get the following form:

$$\begin{array}{ll}{F}^{\text{'}}=& {\varrho }_1\left(\frac{{\beta }_{1}I+{\beta }_{2}B}{P}(S+\eta V_1)-k_{4}E\right)\\ & +{\varrho }_2(\delta E-k_{5}I)+{\varrho }_3(\xi I-\omega B),\end{array}$$

$$\begin{array}{ll}{F}^{\text{'}}=& \left(\frac{{\varrho }_1{\beta }_1}{P}(S+\eta V_1)-{\varrho }_2{k}_5+{\varrho }_3\xi \right)I\\ & +\left(\frac{{\varrho }_1{\beta }_2(S+\eta V_1)}{P}-{\varrho }_3\omega \right)B\\ & +\left({\varrho }_2\delta -{\varrho }_1{k}_4\right)E,\end{array}$$

$$\begin{array}{ll}{F}^{\prime }\le & \left({\varrho }_1\left(\frac{k_2{k}_3{\beta }_1}{k_2{k}_3+{\alpha }_1(k_3+{\alpha }_2)}+\frac{k_3{\alpha }_1{\beta }_1\eta }{k_2{k}_3+{\alpha }_1(k_3+{\alpha }_2)}\right)-{\varrho }_2{K}_5+{\varrho }_3\xi \right)I\\ & +\left({\varrho }_1\left(\frac{k_2{k}_3{\beta }_2}{k_2{k}_3+{\alpha }_1(\mu k_3+{\alpha }_2)}+\frac{k_3{\alpha }_1{\beta }_2\eta }{k_2{k}_3+{\alpha }_1(k_3+{\alpha }_2)}\right)-{\varrho }_3\omega \right)B\\ & +({\varrho }_2\delta -{\varrho }_1{K}_4)E.\end{array}$$

Choosing the value of constants ϱ₁, ϱ₂, ϱ₃ as follows:

$${\varrho }_1=\delta ,{\varrho }_2=k_4,{\varrho }_3=\frac{\delta k_3{\beta }_2(k_2+{\alpha }_1\eta )}{\omega \left(k_2{k}_3+{\alpha }_1\left(k_3+{\alpha }_2\right)\right)},$$

we obtained after some simplification

$$F^{\prime }\le k_4{k}_5\left({\mathcal{R}}_0-1\right)I.$$ (9)

Hence, when ${\mathcal{R}}_0\le 1$, it follows that F′(t) ≤ 0. As a result, the largest compact invariant set within Ω is the singleton set E₀. Therefore, by applying LaSalle’s Invariance Principle (Wei et al., 2011), we conclude that E₀ is GAS in Ω.

4.4. Global stability of EE

From the from system (2) we have

$$\left\{\begin{array}{ll}\lambda & ={\varphi }^{\ast }{S}^{\ast }+k_1{S}^{\ast }-\tau V_1^{\ast },\\ {\alpha }_1{S}^{\ast }& =\eta {\varphi }^{\ast }{V}_1^{\ast }+k_2{V}_1^{\ast },\\ k_3{V}_2^{\ast }& ={\alpha }_2{V}_1^{\ast },\\ k_4{E}^{\ast }& ={\varphi }^{\ast }(S^{\ast }+\eta V_1^{\ast }),\\ k_5{I}^{\ast }& =\delta E^{\ast },\\ \xi I^{\ast }& =\omega B^{\ast },\end{array}\right)$$ (10)

and specifically considering

$${\varphi }^{\ast }=({\beta }_1{I}^{\ast }+{\beta }_2{B}^{\ast }),$$ (11)

then we state the following theorem.

Theorem 4.4

The EE $E_1=(S^{\ast },V_1^{\ast },V_2^{\ast },E^{\ast },I^{\ast },R^{\ast },B^{\ast })$ is GAS in Ω if ${\mathcal{R}}_0>1$.

Proof. To investigate the global stability of the EE, we construct a Lyapunov function V(t) defined in the following form.

$$\begin{array}{ll}V(t)=& \left(S(t)-S^{\ast }-S^{\ast }ln\frac{S}{S^{\ast }}\right)+\left(V_1(t)-V_1^{\ast }-V_1^{\ast }ln\frac{V_1}{V_1^{\ast }}\right)+\frac{k_2}{{\alpha }_2}\left(V_2(t)-V_2^{\ast }-V_2^{\ast }ln\frac{V_2}{V_2^{\ast }}\right)\\ & +\left(E(t)-E^{\ast }-E^{\ast }ln\frac{E}{E^{\ast }}\right)+\frac{{\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)}{k_5}\left(I(t)-I^{\ast }-I^{\ast }ln\frac{I}{I^{\ast }}\right)\\ & +\frac{{\beta }_2{B}^{\ast }\left(S^{\ast }+\eta V_1^{\ast }\right)}{\xi (I^{\ast })}\left(B(t)-B^{\ast }-B^{\ast }ln\frac{B}{B^{\ast }}\right).\end{array}$$

Taking time derivative along the model solution, we get

$$\begin{array}{ll}{V}^{\prime }(t)=& \left(1-\frac{S^{\ast }}{S}\right)S^{\prime }+\left(1-\frac{V_1^{\ast }}{V_1}\right)V_1^{\prime }+\frac{k_2}{{\alpha }_2}\left(1-\frac{V_2^{\ast }}{V_2}\right)V_2^{\prime }\\ & +\left(1-\frac{E^{\ast }}{E}\right)E^{\prime }+\frac{{\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)}{k_5}\left(1-\frac{I^{\ast }}{I}\right)I^{\prime }+\frac{{\beta }_2{B}^{\ast }\left(S^{\ast }+\eta V_1^{\ast }\right)}{\xi (I^{\ast })}\left(1-\frac{B^{\ast }}{B}\right)B^{\prime }.\end{array}$$

First we find

$$\begin{array}{ll}\left(1-\frac{S^{\ast }}{S}\right)S^{\prime }& =\left(1-\frac{S^{\ast }}{S}\right)\left[\lambda -\varphi S-k_{1}S+\tau V_1\right]\\ & =\left(1-\frac{S^{\ast }}{S}\right)\left[{\varphi }^{\ast }{S}^{\ast }+k_1{S}^{\ast }-\tau V_1^{\ast }-\varphi S-k_{1}S+\tau V_1\right]\\ & =\left(1-\frac{S^{\ast }}{S}\right)\left[{\beta }_1(I^{\ast }{S}^{\ast }-IS)+{\beta }_2(B^{\ast }{S}^{\ast }-BS)\right]\\ & +k_1{S}^{\ast }\left(2-\frac{S}{S^{\ast }}-\frac{S^{\ast }}{S}\right)-\tau V_1^{\ast }\left(1-\frac{V_1}{V_1^{\ast }}-\frac{S^{\ast }}{S}+\frac{V_1}{V_1^{\ast }}\frac{S^{\ast }}{S}\right)\\ & ={\beta }_1{I}^{\ast }{S}^{\ast }\left(1-\frac{IS}{I^{\ast }{S}^{\ast }}-\frac{S^{\ast }}{S}+\frac{I}{I^{\ast }}\right)+{\beta }_2{B}^{\ast }{S}^{\ast }\left(1-\frac{BS}{B^{\ast }{S}^{\ast }}-\frac{S^{\ast }}{S}+\frac{B}{B^{\ast }}\right)\\ & +k_1{S}^{\ast }\left(2-\frac{S}{S^{\ast }}-\frac{S^{\ast }}{S}\right)-\tau V_1^{\ast }\left(1-\frac{V_1}{V_1^{\ast }}-\frac{S^{\ast }}{S}+\frac{V_1{S}^{\ast }}{V_1^{\ast }S}\right).\end{array}$$ (12)

$$\begin{array}{ll}\left(1-\frac{V_1^{\ast }}{V_1}\right)V_1^{\prime }& =\left(1-\frac{V_1^{\ast }}{V_1}\right)\left[{\alpha }_{1}S-\eta \varphi V_1-k_2{V}_1\right]\\ & =\left(1-\frac{V_1^{\ast }}{V_1}\right)\left[{\alpha }_1(S-S^{\ast })-\eta (\varphi V_1-{\varphi }^{\ast }{V}_1^{\ast })-k_2(V_1-V_1^{\ast })\right]\\ & ={\alpha }_1{S}^{\ast }\left(1-\frac{V_1^{\ast }}{V_1}\right)\left(\frac{S}{S^{\ast }}-1\right)-\eta \left(1-\frac{V_1^{\ast }}{V_1}\right)\left[{\beta }_1(I{V}_1-I^{\ast }{V}_1^{\ast })+{\beta }_2(B{V}_1-B^{\ast }{V}_1^{\ast })\right]\\ & -k_2{V}_1^{\ast }\left(1-\frac{V_1^{\ast }}{V_1}\right)\left(\frac{V_1}{V_1^{\ast }}-1\right)\\ & =\eta {\beta }_1{I}^{\ast }{V}_1^{\ast }\left(\frac{S}{S^{\ast }}-\frac{I{V}_1}{I^{\ast }{V}_1^{\ast }}-\frac{V_1^{\ast }S}{V_1{S}^{\ast }}+\frac{I}{I^{\ast }}\right)\\ & +\eta {\beta }_2{B}^{\ast }{V}_1^{\ast }\left(\frac{S}{S^{\ast }}-\frac{B{V}_1}{B^{\ast }{V}_1^{\ast }}-\frac{V_1^{\ast }S}{V_1{S}^{\ast }}+\frac{B}{B^{\ast }}\right)+k_2{V}_1^{\ast }\left(1+\frac{S}{S^{\ast }}-\frac{V_1}{V_1^{\ast }}-\frac{V_1^{\ast }S}{V_1{S}^{\ast }}\right).\end{array}$$ (13)

We find

$$\begin{array}{ll}\frac{k_2}{{\alpha }_2}\left(1-\frac{V_2^{\ast }}{V_2}\right)V_2^{\prime }& =\frac{k_2}{{\alpha }_2}\left(1-\frac{V_2^{\ast }}{V_2}\right)({\alpha }_2{V}_1-k_3{V}_2)\\ & =k_2\left(1-\frac{V_2^{\ast }}{V_2}\right)V_1-\frac{k_2{k}_3}{{\alpha }_2}\left(1-\frac{V_2^{\ast }}{V_2}\right)V_2,\left(\frac{k_2{k}_3}{{\alpha }_2}=\frac{k_2{V}_1^{\ast }}{V_2^{\ast }}\right)\\ & =k_2{V}_1\left(1-\frac{V_2^{\ast }}{V_2}\right)-\frac{k_2{V}_1^{\ast }}{V_2^{\ast }}\left(V_2-V_2^{\ast }\right)\\ & =k_2{V}_1\left(1-\frac{V_2^{\ast }}{V_2}\right)+k_2{V}_1^{\ast }\left(1-\frac{V_2}{V_2^{\ast }}\right)\\ & =k_2{V}_1^{\ast }\left(1+\frac{V_1}{V_1^{\ast }}-\frac{V_2}{V_2^{\ast }}-\frac{V_1{V}_2^{\ast }}{V_2{V}_1^{\ast }}\right).\end{array}$$ (14)

Further,

$$\begin{array}{ll}\left(1-\frac{E^{\ast }}{E}\right)E^{\prime }& =\left(1-\frac{E^{\ast }}{E}\right)\left[\varphi (S+\eta V_1)-k_{4}E\right]\\ & =\left(1-\frac{E^{\ast }}{E}\right)\left[\varphi (S+\eta V_1)-{\varphi }^{\ast }(S^{\ast }+\eta V_1^{\ast })\right]\\ & =\left(1-\frac{E^{\ast }}{E}\right)\left[{\beta }_1\left(I(S+\eta V_1)-I^{\ast }(S^{\ast }+\eta V_1^{\ast })\right)\right)\\ & \left(+{\beta }_2\left(B(S+\eta V_1)-B^{\ast }(S^{\ast }+\eta V_1^{\ast })\right)\right]\\ & ={\beta }_1{I}^{\ast }{S}^{\ast }\left(\frac{IS}{I^{\ast }{S}^{\ast }}-\frac{E}{E^{\ast }}-\frac{IS{E}^{\ast }}{I^{\ast }{S}^{\ast }E}+1\right)+{\beta }_2{B}^{\ast }{S}^{\ast }\left(\frac{BS}{B^{\ast }{S}^{\ast }}-\frac{E}{E^{\ast }}-\frac{BS{E}^{\ast }}{B^{\ast }{S}^{\ast }E}+1\right)\\ & +\eta {\beta }_1{I}^{\ast }{V}_1^{\ast }\left(\frac{I{V}_1}{I^{\ast }{V}_1^{\ast }}-\frac{E}{E^{\ast }}-\frac{I{V}_1{E}^{\ast }}{I^{\ast }{V}_1^{\ast }E}+1\right)+\eta {\beta }_2{B}^{\ast }{V}_1^{\ast }\left(\frac{B{V}_1}{B^{\ast }{V}_1^{\ast }}-\frac{E}{E^{\ast }}-\frac{B{V}_1{E}^{\ast }}{B^{\ast }{V}_1^{\ast }E}+1\right).\end{array}$$ (15)

$$\begin{array}{ll}\frac{{\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)}{k_5}\left(1-\frac{I^{\ast }}{I}\right)I^{\prime }& =\frac{{\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)}{k_5}\left(1-\frac{I^{\ast }}{I}\right)I^{\prime }\\ & =\frac{{\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)}{k_5}\left(1-\frac{I^{\ast }}{I}\right)\left[\delta E-k_{5}I\right]\\ & ={\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)\left[\frac{I^{\ast }E}{E^{\ast }}-I-\frac{I^{\ast 2}E}{E^{\ast }I}+I^{\ast }\right]\\ & ={\beta }_1\left(S^{\ast }+\eta V_1^{\ast }\right)I^{\ast }\left[1-\frac{I}{I^{\ast }}+\frac{E}{E^{\ast }}-\frac{I^{\ast }E}{I{E}^{\ast }}\right].\end{array}$$ (16)

$$\begin{array}{ll}\frac{{\beta }_2{B}^{\ast }(S^{\ast }+\eta V_1^{\ast })}{\xi I^{\ast }}\left(1-\frac{B^{\ast }}{B}\right)B^{\prime }& =\frac{{\beta }_2{B}^{\ast }(S^{\ast }+\eta V_1^{\ast })}{\xi I^{\ast }}\left(1-\frac{B^{\ast }}{B}\right)(\xi I-\omega B)\\ & =\frac{{\beta }_2{B}^{\ast }(S^{\ast }+\eta V_1^{\ast })}{\xi I^{\ast }}\xi I^{\ast }\left(\frac{I}{I^{\ast }}-\frac{B}{B^{\ast }}-\frac{I{B}^{\ast }}{I^{\ast }B}+1\right),(\xi I^{\ast }=\omega B^{\ast })\\ & ={\beta }_2{B}^{\ast }(S^{\ast }+\eta V_1^{\ast })\left(\frac{I}{I^{\ast }}-\frac{B}{B^{\ast }}-\frac{I{B}^{\ast }}{I^{\ast }B}+1\right).\end{array}$$ (17)

Combining all we get,

$$\begin{array}{ll}{V}^{\prime }(t)=& k_1{S}^{\ast }\left(2-\frac{S}{S^{\ast }}-\frac{S^{\ast }}{S}\right)+{\beta }_1{S}^{\ast }{I}^{\ast }\left(3-\frac{S^{\ast }}{S}-\frac{S{E}^{\ast }I}{S^{\ast }{I}^{\ast }E}-\frac{I^{\ast }E}{I{E}^{\ast }}\right)\\ & +\eta {\beta }_1{I}^{\ast }{V}_1^{\ast }\left(2+\frac{S^{\ast }}{S}-\frac{S{V}_1^{\ast }}{S^{\ast }{V}_1}-\frac{I^{\ast }E}{I{E}^{\ast }}-\frac{I{V}_1{E}^{\ast }}{I^{\ast }{V}_1^{\ast }E}\right)\\ & +{\beta }_2{B}^{\ast }{S}^{\ast }\left(3-\frac{S^{\ast }}{S}-\frac{E}{E^{\ast }}+\frac{I}{I^{\ast }}-\frac{I{B}^{\ast }}{B{I}^{\ast }}-\frac{SB{E}^{\ast }}{S^{\ast }{B}^{\ast }E}\right)\\ & +\eta {\beta }_2{B}^{\ast }{V}_1^{\ast }\left(2+\frac{S}{S^{\ast }}-\frac{E}{E^{\ast }}+\frac{I}{I^{\ast }}-\frac{I{B}^{\ast }}{I^{\ast }B}-\frac{S{V}_1^{\ast }}{S^{\ast }{V}_1}-\frac{V_{1}B{E}^{\ast }}{V_1^{\ast }{B}^{\ast }E}\right)\\ & -\tau V_1^{\ast }\left(1-\frac{V_1}{V_1^{\ast }}-\frac{S^{\ast }}{S}+\frac{V_1{S}^{\ast }}{S{V}_1^{\ast }}\right)+k_2{V}_1^{\ast }\left(2+\frac{S}{S^{\ast }}-\frac{V_2}{V_2^{\ast }}-\frac{V_1^{\ast }S}{V_1{S}^{\ast }}-\frac{V_1{V}_2^{\ast }}{V_1^{\ast }{V}_2}\right).\end{array}$$ (18)

Using properties of the arithmetic-geometric means, it is obvious that

$$\left\{\begin{array}{l}\left(2-\frac{S}{S^{\ast }}-\frac{S^{\ast }}{S}\right)\le 0,\\ \left(3-\frac{S^{\ast }}{S}-\frac{S{E}^{\ast }I}{S^{\ast }{I}^{\ast }E}-\frac{I^{\ast }E}{I{E}^{\ast }}\right)\le 0.\end{array}\right)$$ (19)

Hence, it follows that $\frac{dV}{dt}\le 0$ under the conditions when the remaining parts in (18) are negative. Moreover, the equality holds only when the system variables take their equilibrium values, $S=S^{\ast },V_1=V_1^{\ast },V_2=V_2^{\ast },E=E^{\ast },I=I^{\ast },R=R^{\ast },B=B^{\ast }$. Hence, by applying LaSalle’s invariance principle, we establish that the EE E₁ is GAS in the feasible region Ω whenever ${\mathcal{R}}_0>1$.

5. Parameter estimation

This section presents the procedure for data fitting and parameter estimation of the proposed Mpox epidemic model. Integrating epidemic modeling with real-world data is crucial for enhancing the realism and relevance of the study. A number of well-know statistical methods can be used to perform the estimation of parameters. In this study, a standard least squares approach is utilized for the estimation procedure based on the documented Mpox infection cases reported during the recent outbreaks in the US. Specifically, for parametrizing the Mpox model, two complementary approaches were adopted. Some parameters such as recruitment rate, natural death rate, and related quantities were estimated using available in the literature (Worldometer, 2022). For example, considering the average life expectancy in the US, which is approximately 79.11 years, the corresponding value of μ per day is computed as:

$$\mu =\frac{1}{79.11\times 365}.$$

The remaining biological parameters were estimated using statistical data obtained from relevant sources (CDC). The objective function is defined as follows:

$$\mathrm{\Phi }(\phi )=\sum _{i=1}^n{l}_i^2,\text{where}{l}_i=x(t_i,\phi )-\overline{x}(t_i),$$ (20)

where x(t_i, φ) denotes the model's outcomes at time t_i with parameter set φ, and $\overline{x}(t_i)$ denotes the reported data set points. The term l_i know as residual which is the difference between the model solutions and the corresponding observed data. Further, n is used for the total number of available data points in the selected time period. We aimed to minimize the sum of squared residuals, Φ(φ), such that the model's outcomes will remain close as possible with the observed data points. This optimization process is performed in Matlab via the least_squares fit function. The procedure uses the residuals l_i, along with guess initial parameter estimates φ, and applies the Trust Region Reflective algorithm to compute the optimal parameter values. The initial conditions and relevant details used for the data fitting are assumed as: S(0) = 349648579.4, V₁(0) = 500, V₂(0) = 10, E(0) = 100, I(0) = 2, and B(0) = 100. The total human population is therefore expressed as P(t) = S(t) +V₁(t) +V₂(t) +E(t) +I(t) +R(t) +B(t). The estimated parameter values obtain through this optimization framework are tabulated in Table 1. Fig. 4 depicts the cumulative reported cases of the 2022 Mpox outbreak in the US, while the proposed model best predication is displayed in Fig. 5. Based on the parameters values mentioned in Table 1, the basic reproduction number is computed as ${\mathcal{R}}_0=1.3346$.

Fig. 4.

Cumulative confirmed cases in the 2022 Mpox outbreak. The data set is taken from CDC official website (CDC).

Fig. 5.

The model best fitted curve versus the cumulative reported cases.

6. Sensitivity analysis

Sensitivity analysis plays a significant role across multiple disciplines, including engineering, economics and epidemiology, where it helps in identifying the most influential factors in a given system. This approach further provides a deeper insights into a model's behavior, particularly by examining how variations in input parameters influence the resulting outputs. We compute the sensitivity indices (SI) of the basic reproduction number, ${\mathcal{R}}_0$, with respect to model embedded parameters by employing the normalized forward sensitivity method (Chitnis et al., 2008). The resulting SI of different parameters are indicated in Table 2. Parameters associated with a positive sensitivity index show a direct relationship with ${\mathcal{R}}_0$, which means that increasing their values leads to an enhancement in the value of ${\mathcal{R}}_0$, thereby favoring disease persistence. Conversely, parameters with negative indices exhibit an inverse relationship with ${\mathcal{R}}_0$, suggesting that enhancing such parameters can help to reduce disease incidence.

Table 2.

SI of the model parameters.

Parameter	Sensitivity Index
β1	0.0148771
β2	0.985123
α1	−0.817544
α2	−0.55533
η	0.18185
τ	0.373566
δ	0.00447984
γ1	0.0
γ2	−0.752337
ϕ	−0.24651
ξ	0.985123
ω	−0.985123
μ	0.993676

Fig. 6 demonstrate the SI of various parameters with bar plot. The figure illustrates that the parameters β₁, β₂, η, τ, δ, γ₁, ξ, and μ exhibit a direct relationship with ${\mathcal{R}}_0$, meaning that increasing these parameters leads to a corresponding rise in ${\mathcal{R}}_0$. On the other hand, the parameters α₁, α₂, γ₂, φ, and ω show an inverse relationship with ${\mathcal{R}}_0$. This shows that higher values of these parameters contribute to minimizing the ${\mathcal{R}}_0$. This analysis helps in formulating the optimal control problem.

Fig. 6.

Normalized sensitivity indices of ${\mathcal{R}}_0$ with respect to the parameters considered in system 1.

Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11 demonstrate the impact of various model parameters on the individuals in exposed, infected and environmental viral concentration classes.

Fig. 7.

Simulation of the infectious compartments across a range of transmission rates β₁.

Fig. 8.

Simulation of the infectious compartments across a range of transmission rates β₂.

Fig. 9.

Simulation of the infectious compartments across a range of first-dose vaccination rates α₁.

Fig. 10.

Simulation of the infectious compartments across different values of the second-dose vaccination rate α₂.

Fig. 11.

Simulation of the infectious compartments across a range of recovery rates γ₂.

7. Formulation of the optimal control problem

The central aim of mathematical modeling of epidemics coupled with real data implementation is to explore effective ways of prevention and ultimate elimination of an infection in a community (Sargent, 2000). Theory of control optimization provides a powerful framework for designing preventive strategies that facilitate disease eradication while reducing the overall impact of infection. To address the spread of MpoxV, we incorporate four time dependent control functions, denoted by u₁(t), u₂(t), u₃(t), and u₄(t). These control functions are selected on the basis of the sensitivity analysis and are explained as:

• •u₁(t): Protective measures for the susceptible and partially vaccinated individuals

This control accounts interventions aimed at reducing exposure of the susceptible and first-dose vaccinated groups. These interventions may include awareness campaigns, use of protective equipment, limiting social contacts, and avoiding interaction with infected people or contaminated surfaces.

• •u₂(t): Promotion of second-dose vaccination

This control enhances the administration of the second vaccine dose to individuals in the first-dose class. It reflects efforts to improve immunity through vaccination campaigns, accessibility of vaccination services, and awareness programs that encourage full immunization.

• •u₃(t): Improved treatment of infectious individuals

This control denotes the timely and effective clinical management of infectious individuals. Its purpose is to reduce the infectious period, mitigate disease severity, and limit further transmission.

• •u₄(t): Environmental decontamination measures

This control captures actions to reduce virus survival in the environment by adopting proper sanitation and disinfection practices resulting in indirect transmission.

The aforementioned controls are assumed to be Lebesgue measurable and bounded on the closed interval [0,1]. The corresponding optimal time-variable profiles are evaluated by applying the PMP. This approach provides a systematic framework to suggest effective and feasible control strategies under available resources. Incorporating these control variables, the Mpox controlled model is formulated as follows.

$$\left\{\begin{array}{ll}{S}^{\text{'}}& =\lambda -\frac{({\beta }_{1}I+{\beta }_{2}B)S(1-u_1)}{P}-({\alpha }_1+\mu )S+\tau V_1,\\ V_1^{\text{'}}& ={\alpha }_{1}S-\frac{\eta ({\beta }_{1}I+{\beta }_{2}B)V_1(1-u_1)}{P}-({\alpha }_2+u_2+\mu +\tau )V_1,\\ V_2^{\text{'}}& =({\alpha }_2+u_2)V_1-({\gamma }_1+\mu )V_2,\\ E^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)}{P}(S+\eta V_1)(1-u_1)-(\delta +\mu )E,\\ I^{\text{'}}& =\delta E-({\gamma }_2+u_3+\phi +\mu )I,\\ R^{\text{'}}& ={\gamma }_1{V}_2+({\gamma }_2+u_3)I-\mu R,\\ B^{\text{'}}& =\xi I-(1+u_4)\omega B.\end{array}\right)$$ (21)

The system is considered subject to nonnegative initial conditions for all state variables. The aim is to minimize not only the overall disease burden but also the economic and logistical costs associated with applying control measures. Accordingly, the objective functional is formulated as follows:

$$J(u_1,u_2,u_3,u_4)={\int }_0^{T_f}\left[A_{1}E(t)+A_{2}I(t)+A_{3}B(t)+\frac{1}{2}\left(A_4{u}_1^2(t)+A_5{u}_2^2(t)+A_6{u}_3^2(t)+A_7{u}_4^2(t)\right)\right]dt.$$ (22)

Here, the terms A_i for i = 1, 2, ⋯, 7 denote constant weighting factors that balance the relative costs of implementing control measures, while T_f represents the final. A quadratic form of the objective functional is adopted, as it effectively captures the nonlinear nature of the applied interventions.

The main objective is to determine the optimal control strategies $u_1^{\ast }$, $u_2^{\ast }$, $u_3^{\ast }$, and $u_4^{\ast }$ such that:

$$J(u_1^{\ast },u_2^{\ast },u_3^{\ast },u_4^{\ast })={\min }_{\mathrm{\Omega }}\{J(u_1,u_2,u_3,u_4)\}.$$ (23)

The corresponding set of admissible control functions is defined as:

$$\mathrm{\Omega }=\{(u_1,u_2,u_3,u_4):[0,T_f]\to [0,1],(u_1,u_2,u_3,u_4)\text{is Lebesgue measurable}\}.$$ (24)

The Lagrangian corresponding to the formulated optimal control system is given as:

$$L=A_{1}E+A_{2}I+A_{3}B+\frac{1}{2}\left(A_4{u}_1^2+A_5{u}_2^2+A_6{u}_3^2+A_7{u}_4^2\right).$$ (25)

Subsequently, the corresponding Hamiltonian represented by $\mathcal{H}$ is formulated in the following equation

$$\mathcal{H}=L+{\lambda }_1{S}^{\prime }+{\lambda }_2{V}_1^{\prime }+{\lambda }_3{V}_2^{\prime }+{\lambda }_4{E}^{\prime }+{\lambda }_5{I}^{\prime }+{\lambda }_6{R}^{\prime }+{\lambda }_7{B}^{\prime }.$$ (26)

Here, λ_j for j = 1, 2, ⋯, 7 denote the adjoint variables associated with the corresponding state variables. The Hamiltonian $\mathcal{H}$ for the formulated optimal control problem is then given by:

$$\left\{\begin{array}{ll}\mathcal{H}& =A_{1}E+A_{2}I+A_{3}B+\frac{1}{2}\left(A_4{u}_1^2+A_5{u}_2^2+A_6{u}_3^2+A_7{u}_4^2\right)\\ & +{\lambda }_1\left[\lambda -\frac{({\beta }_{1}I+{\beta }_{2}B)S(1-u_1)}{P}-({\alpha }_1+\mu )S+\tau V_1\right]\\ & +{\lambda }_2\left[{\alpha }_{1}S-\frac{\left(\eta \right)({\beta }_{1}I+{\beta }_{2}B)V_1(1-u_1)}{P}-({\alpha }_2+u_2+\mu +\tau )V_1\right]\\ & +{\lambda }_3\left[({\alpha }_2+u_2)V_1-({\gamma }_1+\mu )V_2\right]\\ & +{\lambda }_4\left[\frac{({\beta }_{1}I+{\beta }_{2}B)}{P}(S+\eta V_1)(1-u_1)-(\delta +\mu )E\right]\\ & +{\lambda }_5\left[\delta E-({\gamma }_2+u_3+\phi +\mu )I\right]\\ & +{\lambda }_6\left[{\gamma }_1{V}_2+({\gamma }_2+u_3)I-\mu R\right]\\ & +{\lambda }_7\left[\xi I-(1+u_4)\omega B\right].\end{array}\right)$$ (27)

7.1. Derivation of the optimal control strategy

By applying PMP (Pontryagin et al., 1962), we establish the necessary conditions for optimality. These conditions comprise the system of adjoint equations, the transversality conditions imposed at the terminal time T_f, and the explicit expressions for the optimal controls, which are obtained by minimizing the Hamiltonian with respect to each control variable.

$$\left\{\begin{array}{l}\frac{dz}{dt}=-\frac{\partial \mathcal{H}(t,u_j^{\ast },{\lambda }_j)}{\partial {\lambda }_j},\\ \frac{\partial \mathcal{H}(t,u_j^{\ast },{\lambda }_j)}{\partial u}=0,\\ \frac{d{\lambda }_j(t)}{dt}=-\frac{\partial \mathcal{H}(t,u_j^{\ast },{\lambda }_j)}{\partial z}.\end{array}\right)$$ (28)

The resulting adjoint equations is given by:

$$\begin{array}{ll}{\lambda }_1^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)(1-u_1)}{P^2}\left[({\lambda }_1-{\lambda }_4)(P-S)+({\lambda }_4-{\lambda }_2)\eta V_1\right]+({\lambda }_1-{\lambda }_2){\alpha }_1+{\lambda }_1\mu ,\\ {\lambda }_2^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)(1-u_1)}{P^2}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_2-{\lambda }_4)(P-V_1)\right]+({\lambda }_2-{\lambda }_3)({\alpha }_2+u_2)-{\lambda }_1\tau +{\lambda }_2(\mu +\tau ),\\ {\lambda }_3^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)(1-u_1)}{P^2}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_4-{\lambda }_2)V_1\right]+({\lambda }_3-{\lambda }_6){\gamma }_1+{\lambda }_3\mu ,\\ {\lambda }_4^{\text{'}}& =-A_1-\delta {\lambda }_5+\frac{({\beta }_{1}I+{\beta }_{2}B)(1-u_1)}{P^2}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_4-{\lambda }_2)V_1\right]+{\lambda }_4(\delta +\mu ),\\ {\lambda }_5^{\text{'}}& =-A_2-{\lambda }_6({\gamma }_2+u_3)-{\lambda }_7\xi +{\lambda }_5({\gamma }_2+\mu +\phi +u_3)-\frac{\left({\beta }_1(P-I)-{\beta }_{2}B\right)(1-u_1)}{P^2}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_4-{\lambda }_2)V_1\right],\\ {\lambda }_6^{\text{'}}& =\frac{({\beta }_{1}I+{\beta }_{2}B)(1-u_1)}{P^2}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_4-{\lambda }_2)V_1\right]+{\lambda }_6\mu ,\\ {\lambda }_7^{\text{'}}& =-A_3+{\lambda }_7\omega (1+u_4)+\frac{{\beta }_2(1-u_1)}{P}\left[({\lambda }_1-{\lambda }_4)S+\eta ({\lambda }_2-{\lambda }_4)V_1\right].\end{array}$$ (29)

To obtain the expressions for the optimal control functions, we compute the partial derivatives of the Hamiltonian with respect to each control variable.

$$\frac{\partial \mathcal{H}}{\partial u_1}=A_4{u}_1+\frac{S{\lambda }_1({\beta }_{1}I+{\beta }_{2}B)}{P}-\frac{{\lambda }_4({\beta }_{1}I+{\beta }_{2}B)(S+\eta V_1)}{P}+\frac{\eta V_1{\lambda }_2({\beta }_{1}I+{\beta }_{2}B)}{P},$$ (30)

$$\frac{\partial \mathcal{H}}{\partial u_2}=A_5{u}_2-V_1{\lambda }_2+V_1{\lambda }_3,$$ (31)

$$\frac{\partial \mathcal{H}}{\partial u_3}=A_6{u}_3+I{\lambda }_6-I{\lambda }_5,$$ (32)

$$\frac{\partial \mathcal{H}}{\partial u_4}=A_7{u}_4-\omega B{\lambda }_7.$$ (33)

The optimal controls are obtain as follows

$$u_1^{\ast }=\frac{{\beta }_{1}I+{\beta }_{2}B}{A_{4}P}\left[({\lambda }_4-{\lambda }_1)S+\eta ({\lambda }_4-{\lambda }_2)V_1\right],$$ (34)

$$u_2^{\ast }=\frac{({\lambda }_2-{\lambda }_3)V_1}{A_5},$$ (35)

$$u_3^{\ast }=\frac{({\lambda }_5-{\lambda }_6)I}{A_6},$$ (36)

$$u_4^{\ast }=\frac{{\lambda }_7\omega B}{A_7}.$$ (37)

8. Numerical simulations and discussion

The simulation of the Mpox model considering different control strategies are discussed in this section. For the solution of nonlinear optimal control problem, we use the forward–backward sweep method. Parameter values and initial conditions are derived from relevant epidemiological data as listed in the previous section. The core aim of the simulation is to assess the effectiveness of the proposed control measures in combating the infection levels, improving vaccination coverage, and lowering the disease burden in the population. This section further presents a comprehensive analysis of the simulation results corresponding to various combinations of the control functions: u₁(t), u₂(t), u₃(t), and u₄(t). The impact of these interventions are visualized in Fig. 12, Fig. 13, Fig. 14.

Fig. 12.

Simulation of the Mpox model under the simultaneous implementation of all four control strategies (u₁, u₂, u₃, u₄) with corresponding control profile.

Fig. 13.

Simulation for Case II: Implementing u₂ ≠ 0 and u₄ ≠ 0, while the remaining controls are not implemented i.e., (u₁ = 0) and (u₃ = 0) with the corresponding control profile.

Fig. 14.

Simulation for Case II: Implementing u₂ ≠ 0 and u₃ ≠ 0, while the protective measures and environmental decontamination are inactive i.e., (u₁ = 0, u₄ = 0); with corresponding control profile.

8.1. Simultaneous implementation of all control measures

The impact of implementing all four controls u₁(t), u₂(t), u₃(t), and u₄(t) on the dynamics of Mpox transmission is presented. As shown in Fig. 12, the simultaneous implementation of all control leads to a substantial decrease in the number of individuals in the infected classes compared with the scenario without controls. Specifically, the control u₁(t) reduces exposure among susceptible and partially vaccinated individuals, resulting in delaying and lowering the infection peaks. The control u₂(t) enhances the rate of second-dose vaccination, strengthening immunity in the people and limiting the infection outbreak. The control u₃(t) improves and enhance treatment efforts for infectious people, leading to a significant reduction in the duration of infection and the overall active cases. Finally, the control variable u₄(t) targets decontamination of environment, aiming to minimize indirect transmission and reduce the risk of resurgence. Thus, the combined application of all controls effectively decline the epidemic curve and highlights the importance of coordinated interventions in mitigating Mpox outbreaks.

8.2. Effectiveness of combined dual control approaches

Case I

Absence of u₁ and u₃

In this scenario, protective measures and treatment of infectious individuals are not considered (i.e., u₁ = u₃ = 0), while second-dose vaccination (u₂) and environmental decontamination (u₄) remain active. The resulting simulation is depicted in Fig. 13. Without preventive interventions to reduce exposure and the clinical management of cases, the outbreak reaches a higher and more prolonged peak. The inclusion of u₂ strengthens immunity through improved vaccination coverage, which moderates the rise of new infections to some extent. Similarly, u₄ lowers indirect transmission by reducing environmental contamination, thereby limiting secondary spread. However, these measures alone are insufficient to suppress the timely and complete eradication of epidemic. These results emphasize that the use of only vaccination and decontamination interventions are beneficial, but cannot effectively control the infection in the absence of direct prevention strategies and timely treatment of infectious individuals.

Case II

Absence of u₁ and u₄

This case considers the absence of protective measures and environmental decontamination (u₁ = u₄ = 0), while second-dose vaccination (u₂) and treatment of infectious individuals (u₃) are active as shown in Fig. 14. The lack of preventive interventions to reduce direct exposure, combined with the absence of efforts to eliminate environmental reservoirs, leads to widespread and sustained transmission. Although u₂ increases immunity through improved vaccine coverage and u₃ shortens the infectious period by enhancing treatment, these measures are not sufficient to offset the continuous spread of infection. These results indicate that with no implementation of behavioral protection and environmental control measures, the infection remains at high level despite vaccination and treatment efforts, emphasizing the importance of integrated strategies in combating Mpox outbreak.

9. Conclusion

We developed and analyzed a compartmental model of Mpox transmission dynamics that integrates vaccination and various pathways of disease spread. The model is constructed with seven compartments, incorporating both direct and indirect pathways of the infection transmission. By considering waning immunity, progression through vaccination stages and the role of environmental contamination, the model provides a realistic framework for understanding the dynamics and control of Mpox. The biological and threshold parameters of the model were estimated using reported Mpox cases from the 2022 outbreak in the US. The mathematical analysis began with establishing the basic properties of the model, including positivity, boundedness, and uniqueness of the solution. Local stability of the disease-free equilibrium was proved through the Routh-Hurwitz criteria, confirming that the infection dies out when ${\mathcal{R}}_0<1$. Furthermore, suitable Lyapunov functions were constructed, and the necessary criteria were derived for the global stability of both steady states of the model. A normalized sensitivity analysis was carried out to identify the parameters with the greatest influence on ${\mathcal{R}}_0$. These results demonstrate that transmission rates and parameters linked to vaccination coverage and waning immunity play particularly important role in the disease control. In addition to this, an optimal control framework was developed incorporating four time-dependent control variables. These controls were formulated to reduce exposure, enhance vaccination efficacy, mitigate waning immunity, and minimize environmental contamination. The necessary conditions for optimality were established using Pontryagin maximum principle and the resulting system was numerically solved using the forward–backward sweep method. The illustrated simulation confirmed that simultaneous implementation of four controls produced the most significant and timely reduction in the infections. Comparisons between with and without time-depended control scenarios further highlighted the substantial advantages of coordinated, multi-faceted intervention strategies in reducing the burden of Mpox. These results aligned with epidemiological evidence, underscoring that vaccination, timely treatment, and control of environmental contamination are critical components in curbing Mpox outbreaks. These findings not only improve understanding of the epidemiological drivers of Mpox but also highlight priority areas for targeted interventions.

CRediT authorship contribution statement

Shuo Li: Writing – review & editing, Supervision, Software, Methodology, Conceptualization. Nimra Sada: Writing – original draft, Software, Methodology, Investigation, Data curation, Conceptualization. Saif Ullah: Writing – original draft, Software, Conceptualization. Muhammad Bilal Riaz: Writing – review & editing, Visualization, Validation, Supervision, Resources, Investigation, Funding acquisition, Formal analysis, Data curation.

Data availability statement

The associated data will be available on request.

Declaration of competing interest

We declare that there does not exist any conflict of interest regarding this paper.

Handling Editor: Dr Daihai He