Modeling the transmission dynamics of Mpox virus

Abstract

The global emergence of human monkeypox (Mpox) necessitates stage-structured transmission models. We develop a compartmental framework with Prodromal, Rash, and Complication stages, identifying critical thresholds through bifurcation analysis. A transcritical bifurcation at day separates stable disease-free equilibrium () from endemic spread. Normalized sensitivity analysis establishes transmission rate () as the dominant epidemic driver, with mortality () and progression rate () as key modulators. Intervention analysis reveals: (1) 22.7% outbreak reduction (95% CI: 19.4–25.1%) through prodromal case isolation (days 0-5) requiring 92% diagnostic accuracy; (2) Linear response to transmission controls (0.0398 reduction per 10% decrease); (3) Phase-adaptive resource allocation (60% to transmission reduction) sustains subcritical operation. The framework advocates real-time monitoring via wastewater surveillance and -optimized diagnostics.

Introduction

Monkeypox (Mpox) is an emerging zoonotic viral disease caused by the Mpox virus, belonging to the Orthopoxvirus genus, which also includes the variola virus responsible for smallpox¹. First identified in 1958 in laboratory monkeys, Mpox has garnered increased global attention due to recent outbreaks beyond its endemic regions in Africa, primarily driven by human-to-human transmission². The disease is transmitted through close contact with bodily fluids, skin lesions, or contaminated surfaces, with transmission dynamics similar to those of other orthopoxviruses³. The clinical manifestation of Mpox includes fever, malaise, lymphadenopathy, and a characteristic rash that progresses through several stages before recovery or death in severe cases^4,5. Monkeypox can occur in individuals of any age, progressing through three distinct stages: incubation, prodrome, and the eruptive phase^6–9. The initial infection may not always be easy to pinpoint, particularly in cases of zoonotic transmission. On average, the incubation period lasts around 13 days but can range from 3 to 34 days. Following this, the prodromal stage typically persists for 1 to 4 days and is marked by symptoms such as fever, headaches, exhaustion, and frequently swollen lymph nodes, particularly in the neck, jaw areas and infectious¹⁰. Recent outbreaks have renewed interest in understanding the transmission dynamics and potential control strategies through mathematical modeling. Mathematical models, especially compartmental models, have been widely used to understand and predict the spread of infectious diseases. Compartmental models such as the SEIR (Susceptible-Exposed-Infectious-Recovered) framework have proven effective in capturing key epidemiological features and guiding public health interventions^11,12. Mpox, compartmental models can be extended to include additional stages such as Prodromal, Rash, Complications, and Death to more accurately reflect the progression of the disease¹³. This study develops a compartmental model for Mpox, incorporating key epidemiological compartments that represent different stages of infection and outcomes. The model is constructed to account for the zoonotic nature of Mpox, human-to-human transmission, and the potential for severe complications.

Literature review

In recent years, mathematical modeling has proven to be a valuable tool in understanding the spread of infectious diseases, including Mpox. The use of compartmental models to simulate the dynamics of infectious diseases has a long history, dating back to the early 20th century with Kermack and McKendrick’s seminal work on the SIR model^14,15. These models have been extensively used in epidemiology to capture the dynamics of various infectious diseases by dividing the population into different classes based on their health status^16,17. Since then, these models have evolved to accommodate more complex disease dynamics, such as latency periods, varying transmission rates, and multi-host systems, making them essential tools in epidemiology¹⁸. The approach helps in estimating key parameters such as transmission rates, recovery rates, and the impact of interventions on the epidemic’s progression. Extension of these model, such as the SEIR (Susceptible-Exposed-Infectious-Recovered) and SEIRS models, account for additional stages in disease progression, such as the latent or incubation period¹⁹. The models have been instrumental in shaping public health policies by simulating potential outcomes under various scenarios of disease transmission and intervention strategies. Mpox, caused by the monkeypox virus, has recently gained attention due to its re-emergence in certain parts of the world. The disease shares similarities with smallpox, yet it has distinct epidemiological features that require tailored modeling approaches²⁰. Various studies have employed compartmental models to describe the dynamics of Mpox. Several studies have extended the basic SEIR framework to include multiple compartments^21–23, reflecting the progression of symptoms and severity in viral diseases such as Ebola, SARS, and Mpox itself²⁴. Early models of Mpox outbreaks in Africa, where the virus is endemic, primarily used the SEIR framework to simulate transmission dynamics²⁵. These models typically included compartments for susceptible, exposed, infectious, and recovered individuals, capturing the basic epidemiological dynamics²⁶. However, as more data on Mpox became available, particularly from outbreaks in non-endemic regions, the need for more detailed models that include additional stages of disease progression became apparent. The 2022 Mpox outbreak, which spread to multiple countries outside of Africa, spurred a wave of new research into the transmission dynamics and potential interventions for Mpox. Several studies have used real-time data from this outbreak to refine existing models and provide projections on the outbreak’s trajectory under different intervention scenarios^27,28. Models incorporating quarantine, isolation, and vaccination strategies have provided valuable insights for public health authorities in managing these outbreaks. Additionally, researchers have explored the role of asymptomatic transmission and super-spreading events in driving Mpox outbreaks, which have implications for modeling efforts^29,30. In the context of understanding Mpox virus transmission dynamics, the study by²⁵ provides valuable insights through the development of an SEIR-based deterministic model, which incorporates the prodromal stage, differential infectivity, and hospitalization. This model is instrumental in enhancing control interventions, as it computes key epidemiological metrics like the basic reproduction number () to estimate the potential spread of infections in a fully susceptible population. The study further investigates the stability of equilibrium states to assess the virus’s transmission potential and utilizes sensitivity analysis, specifically the partial rank correlation coefficient, to identify the most influential parameters for disease control. Numerical simulations and model predictions offer additional evaluation of how critical parameters affect the prevention and containment of Mpox outbreaks, strengthening public health strategies³¹. investigated the impact of imperfect vaccination, primarily derived from the Smallpox vaccine, on the spread of the Mpox virus. Their results suggest that while these vaccines do not provide complete immunity, they still play a crucial role in reducing transmission. Lowering the rate of new infections, even imperfect vaccines can significantly slow the spread of Mpox within a population, thereby contributing to more effective control strategies³². developed a mathematical deterministic model that captures the dynamics of Monkeypox by considering both human and rodent populations. While recent studies advance understanding of Mpox control strategies^33,32, they often overlook the stage-specific transmission dynamics intrinsic to Mpox progression. Clinically, Mpox evolves through three phases: incubation, prodromal (high temperature, headache, fatigue, and often, lymphadenopathy, especially in the cervical and maxillary regions), and eruptive (rash/lesions)^34,35, with transmission risks escalating dramatically during the rash phase due to high viral loads in pustular fluids³⁶. Existing models frequently aggregate transmission into a single infectious compartment³⁷, obscuring the distinct epidemiological roles of prodromal () and rash () stages. To address this gap, we propose a SEPRRvC model (Susceptible-Exposed-Prodromal-Rash-Recovered-Complications) with two key innovations:

1. Stage-separated force of infection: The total force of infection explicitly captures contributions from both Prodromal () and Rash () compartments. While the transmission rate is constant, the model acknowledges that contributes disproportionately to transmission due to higher viral shedding in lesions³⁸.

2. Phase-dependent interventions: Isolating and compartments, the model evaluates targeted strategies (e.g., early detection during prodromal vs. strict isolation during rash).

The model further incorporates a Complications (C) compartment (Eq. 1) to quantify severe outcomes³⁹, aligning with WHO guidelines for stage-specific case management⁴⁰. This structure enables precise evaluation of public health measures, such as:

• Optimizing testing during the prodromal window (low infectivity but critical for containment).

• Scaling isolation capacity during the eruptive surge (high infectivity).

The force of infection in Eq. 1 reflects empirical evidence that:

• Prodromal transmission () usually 1–4 days^41,42,39,43.

• Rash-stage transmission () dominates () due to lesion-driven viral shedding and cover 14 to 28 days³⁶.

Separating and , the model avoids conflating their transmission roles a limitation of single-infectious-compartment frameworks²⁵. The structure of the paper is as follows: Section 1 provides the introduction, followed by Section 2, which presents the literature review. Section 3 details the methodology, while Section 4 discusses the results and analysis. Finally, Section 5 concludes the paper.

Methodology

The methodology employs a compartmental modeling approach to investigate the transmission dynamics of Monkeypox (Mpox). This model extends the SEIR framework by stratifying the infectious phase into Prodromal (P) and Rash (R) compartments, capturing distinct clinical stages with differing transmission potentials as shown in Fig. 1. The inclusion of Complications (C) further enables analysis of severe outcomes, providing a holistic representation of Mpox progression⁴⁴.

Fig. 1.

Flowchart of Mpox Dynamics.

Model formulation

The population is divided into six compartments:

• Susceptible (S): Individuals at risk of Mpox. Infection occurs via contact with Prodromal (P) or Rash (R) individuals at rate . Dynamics include birth () and natural death ().

• Exposed (E): Infected but not yet infectious. Transition to P occurs at rate .

• Prodromal (P): Early symptomatic stage (fever, lymphadenopathy) with low viral shedding. While individuals in are mildly infectious, clinical studies suggest their contribution to transmission is minimal (2–5% of total cases)⁴⁵. Transition to R (), recover (), or die ().

• Rash (R): High-transmission stage characterized by rash/lesions, contributing 90–98% of infections due to elevated viral shedding³⁸. Recover (), develop complications (), or die ().

• Recovered (): Lifelong immunity is assumed post-recovery. Inflows originate from (), (), and ().

• Complications (C): Severe cases (e.g., secondary infections, sepsis) with prolonged hospitalization. Recovery occurs at rate ; death occurs at rate .

1

Parameter descriptions

The epidemiological parameters in the model are defined as follows:

• : Recruitment rate into the susceptible population (persons/day)

• : Transmission rate of Mpox disease (day)

• : Natural mortality rate (day)

• : Progression rate from exposed to prodromal stage (day)

• : Disease-induced mortality rate in complications (day)

• : Progression rate from prodromal to rash stage (day)

• : Recovery rate directly from prodromal stage (day)

• : Recovery rate from rash stage (day)

• : Recovery rate from complications (day)

• : Complication development rate from rash stage (day)

Model dynamics and biological assumptions

The model incorporates the following biologically-motivated assumptions:

1. Transmission Mechanism: The force of infection accounts only for transmission from prodromal (P) and rash (R) compartments, excluding complications (C) based on clinical evidence:

   1. Individuals in the prodromal stage (P) exhibit early symptoms (fever, lymphadenopathy) while maintaining community mobility, enabling transmission through close contact.
   2. Rash-stage individuals (R) develop characteristic skin lesions containing high viral loads, representing peak infectiousness through direct contact and fomites.
   3. Complications (C) represent severe cases (hemorrhagic manifestations, sepsis, encephalitis) typically requiring hospitalization and isolation, substantially reducing transmission risk. This aligns with clinical management protocols that isolate severe cases within 48 hours of complication onset. This exclusion is supported by epidemiological studies of orthopoxviruses indicating <2% secondary transmission from hospitalized cases compared to community cases^45,46.
2. Compartment-Specific Mortality:

   1. Natural mortality (): Affects all compartments equally, representing background mortality.
   2. Disease-induced mortality (): Exclusive to complications (C) compartment, capturing elevated mortality risk from severe manifestations like secondary infections and sepsis.
3. Transition Dynamics:

   1. Stage progression: Follows exponential distributions with mean durations:

      
   2. Recovery pathways:

      • i.
        Direct recovery from prodromal stage (γ₂)
      • ii.
        Recovery from rash stage (δ₁)
      • iii.
        Recovery from complications (δ₂)
4. Population Dynamics:

   1. Non-negative populations: All compartments satisfy for given non-negative initial conditions.
   2. Total population:

      2

      evolves according to:
      
      reflecting both natural and disease-induced mortality.
   3. Lifelong immunity: Recovered individuals () retain permanent immunity against reinfection, consistent with orthopoxvirus immunology⁴⁷.
5. Justification for Exclusion of C in Calculation: The basic reproduction number quantifies secondary infections generated by a single typical infectious individual in a fully susceptible population. Our exclusion of the complications compartment (C) from calculation is based on three principal considerations:

   1. Transmission potential: Clinical studies indicate viral loads in severe cases are comparable to earlier stages, but infection control measures in healthcare settings reduce transmission opportunities by 85-95% compared to community settings.
   2. Epidemiological significance: During the initial invasion phase when is calculated, complications develop after mean duration days (Table 1), while serial intervals for Mpox are 8-12 days. Thus, complication-derived transmission occurs too late to significantly influence .
   3. Compartmental contribution: Sensitivity analysis shows C contributes <2% to overall transmission force in endemic equilibrium (Fig. 3), justifying its exclusion from calculation without significantly altering threshold estimates. The next-generation matrix approach accordingly considers E, P, R as infected compartments for derivation, while C is excluded from transmission dynamics but included in disease progression and mortality calculations.

Fig. 3.

Time-series simulation under disease-free equilibrium conditions (). Initial conditions: , , .

Theorem 1

(Boundedness of the Total Population) Under the dynamics described by the system of differential equations in (1), the total population , as defined in (2), remains bounded for all .

Proof

To establish the boundedness of , note that the total population at any time is given as:

Taking the time derivative of :

Substituting the updated system of equations (1):

Simplifying by canceling terms:

Since , , , and , we have:

3

Thus, the solution is bounded for , and the total population is bounded above by .

Computation of the basic reproduction number

The infectious compartments are , , and , with the state vector:

System of equations for infectious compartments

From the model equations, the dynamics of the infectious compartments are:

At the Disease-Free Equilibrium (DFE), , and , so the total population is .

Next-generation matrix

Using the next-generation matrix method, we decompose the system into new infections () and transitions ().

Infection Matrix : The new infection terms are:

The Jacobian of with respect to is:

Transition Matrix : The transition terms are:

The Jacobian of with respect to is:

The next-generation matrix is . The inverse is:

Multiplying by :

The basic reproduction number is the spectral radius (dominant eigenvalue) of . Simplifying the dominant eigenvalue:

This accounts for transmission from both the prodromal () and rash () stages, weighted by their respective residence times and infectivity.

Theorem 2

(Existence and Characterization of Equilibrium Points) The system of differential equations in (1) admits two types of equilibrium points:

• Disease-Free Equilibrium (DFE):

  
• Endemic Equilibrium (EE):

  

Proof

To determine the equilibrium points, set all time derivatives in (1) to zero:

Disease-Free Equilibrium (DFE): Set . Solving gives:

Thus, , and the endemic Equilibrium (EE) exist in (4) for .

4

Theorem 3

(Local Stability of Equilibrium Points) Define:

where is:

The stability of equilibrium points is characterized as follows:

1. Disease-Free Equilibrium (DFE): The DFE is locally asymptotically stable if and unstable if .

2. Endemic Equilibrium (EE): If , the EE is locally asymptotically stable.

Proof

1. Disease-Free Equilibrium (DFE): The Jacobian at is:

   

   The characteristic polynomial is found by solving . Through block matrix decomposition and determinant properties, we obtain:

   

   where:

   

   with coefficients:

   

   where , , , and .

Trivial eigenvalues: (double root),

Routh-Hurwitz Criteria For a cubic polynomial , the necessary and sufficient conditions for all roots to have negative real parts are:

• (i)
• (ii)
• (iii)

1. Coefficient Positivity:

   • (all )
   • 
   • 
2. Determinant Condition:

   

3. All Routh-Hurwitz conditions satisfied. DFE is locally asymptotically stable If .

• 2.
  Endemic Equilibrium (EE): The Jacobian at is:

  

where:

The eigenvalues of are roots of the characteristic polynomial:

The characteristic polynomial is:

where:

and

By Routh-Hurwitz Criteria since coefficients and the Hurwitz determinants are satisfied.

Theorem 4

(Global Stability of Equilibria in the SEPRRvC Model) Consider the SEPRRvC model governed by the system in (1) with positive parameters , and total population (2) bounded as per Theorem 1.

• Disease-Free Equilibrium (DFE): If , the DFE is globally asymptotically stable (GAS) in the feasible region.

• Endemic Equilibrium (EE): If , the unique endemic equilibrium defined in (4) is GAS in the interior of the feasible region.

Proof

Global Stability of the DFE ()

Define the Lyapunov function:

This function is positive definite in the infected compartments and vanishes only at the DFE. The time derivative is:

Simplifying the expression:

Since (from Theorem 1), we have , and:

Thus:

Substituting the expression for :

When , with equality if and only if . When , the system dynamics imply:

By LaSalle’s Invariance Principle, all trajectories converge to . Thus, the DFE is GAS when .

Global Stability of the EE ()

We prove GAS of the endemic equilibrium using Lyapunov function. Define the Lyapunov function:

The time derivative is:

5

6

7

8

9

10

11

12

13

Using the system equations and equilibrium conditions eqs. (5) to (12), we compute each term:

Step 1: Compute Each Term

Term 1:

From the system, . Using (5), , we get:

Thus:

Simplify:

For the terms:

Thus:

The term , since , , and .

Term 2:

From the system, . Using (6), , so:

Thus:

So:

Term 3:

From the system, . Using (7), , so:

Thus:

So:

Term 4:

From the system, . Using (8), , so:

Thus:

So:

Term 5:

From the system, . Using (10), , so:

Thus:

So:

Term 6:

From the system, . Using (9), , so:

Since , we have:

The term involving :

Step 2: Combine Terms

Summing all terms:

Step 3: Simplify Infection Terms

Since from from 3, so . Define , . Using (6), . The infection terms are:

Rewrite:

Step 4: Rewrite Transition Terms

For the transition terms, rewrite each to form :

- :

Add and subtract :

- :

- :

- :

Thus:

Since , and all coefficients are positive, , with equality only when , , , , , .

Table 1.

Epidemiological Parameters and Sources.

Parameter	Description (Units)	Value	Source
	Recruitment rate (persons/day)	9832.97	48
	Natural mortality rate (day)	0.018	49
	Transmission rate (day)	0.06	49
	ExposedProdromal rate (day)	0.02	50
	ProdromalRash rate (day)	0.08	50
	Prodromal recovery rate (day)	0.014	49
	Rash recovery rate (day)	0.126	49
	Complications recovery (day)	0.02	39
	Complication rate (day)	0.05	39
	Disease mortality (day)	0.1	39

Results and discussion

Epidemiological thresholds and bifurcation analysis

The fundamental dynamics of Mpox transmission are governed by the basic reproduction number , which serves as a critical epidemiological threshold determining disease persistence or elimination. Our analysis establishes that when , the disease-free equilibrium (DFE) is locally asymptotically stable, leading to eventual disease extinction. Conversely, when , the endemic equilibrium (EE) emerges and becomes stable, indicating sustained transmission within the population. This transition at represents a transcritical bifurcation, mathematically characterized through stability analysis. The critical transmission threshold was calculated as day using the formula: Fig. 2 demonstrates this fundamental relationship, revealing a clear epidemiological phase transition. The bifurcation diagram shows the disease-free equilibrium for where , and the endemic equilibrium emerging supercritically for . The absence of hysteresis indicates that reducing transmission below the critical threshold reliably eliminates endemic transmission.

Fig. 2.

Bifurcation diagram with demographic recalibration. Transcritical bifurcation at (data1) with DFE stability for (, others zero). Endemic equilibrium emerges supercritically for . Phase arrows show attraction basins. Parameters: persons/day, , , , , . DFE: .

Disease-free equilibrium dynamics

Under subcritical conditions (), Mpox transmission exhibits self-limiting dynamics characterized by rapid outbreak extinction.The compartmental progression follows clinically expected timelines with minimal population impact. The susceptible population experiences negligible reduction (104 individuals over 100 days), confirming limited secondary transmission. Figure 3 visualizes these dynamics, showing the sequential peaking of compartments: exposed cases peaking at day 44 (103 cases), rash cases at day 9 (11 cases), and complications at day 19 (3 cases). This compartmental decay without renewal confirms outbreak extinction within 60 days despite continuous importation risk (Fig. 7), demonstrating that prevents sustained community transmission.

Fig. 7.

Self-limiting outbreak from imported case despite .

Endemic transmission dynamics

In supercritical transmission scenarios (), Mpox exhibits explosive epidemic growth followed by endemic stabilization. Table 2 quantifies the dramatic scale of endemic transmission, with the susceptible population experiencing rapid depletion of 60.41% (323,426 individuals) within the first 100 days. Figure 4 visualizes the progression from initial outbreak to endemic equilibrium. Infectious compartments demonstrate sequential peaking driven by disease progression kinetics: exposed cases peak at day 64 (193,451 cases), prodromal at day 74 (34,289 cases), rash at day 80 (14,108 cases), and complications at day 87 (5,089 cases). Figure 5 provides deeper insight into endemic equilibrium characteristics. The bifurcation analysis shows how equilibrium values vary with transmission rate , with the critical threshold separating disease-free and endemic states. The forward bifurcation indicates no hysteresis - reducing below will eliminate the disease. The system stabilizes into endemic equilibrium, characterized by persistent circulation with approximately 46% susceptibles, 1.2% prodromal cases, 0.8% rash cases, 0.3% complications, and 49% recovered individuals. This final seroprevalence of 51% provides substantial herd immunity, yet insufficient to interrupt transmission due to the high force of infection.

Table 2.

Quantitative results: Endemic equilibrium ().

Metric	Value	Time (days)	Description
Susceptible decrease	323,426	100	60.41% reduction
Exposed peak	193,451	64	Maximum exposed cases
Prodromal peak	34,289	74	Maximum prodromal cases
Rash peak	14,108	80	Maximum rash cases
Complications peak	5,089	87	Maximum severe cases
Recovered at day 500	103,952	500	Cumulative recovered

Fig. 4.

Endemic transmission dynamics (). Initial conditions: , , .

Fig. 5.

Bifurcation analysis of infectious compartments. Variation of equilibrium values with transmission rate . Critical threshold (vertical dashed line) separates disease-free () and endemic () states. Prodromal cases () dominate, showing steep increase above . Rash cases () follow similar pattern while complication cases () remain low due to low complication rates.

Parameter sensitivity and intervention efficacy

Comprehensive sensitivity analysis (Table 3) reveals the hierarchical influence of parameters on transmission dynamics. Transmission rate () dominates the sensitivity spectrum (), where a 10% reduction decreases by 0.0398 - equivalent to 10% vaccine coverage under perfect efficacy. Figure 6 demonstrates the mortality reduction achievable through interventions. Reducing transmission by 50% (from to 0.15) crosses the critical threshold , decreasing peak complication burden by 78% (from 5,089 to 1,120 cases). The linear relationship between -reduction and -decrease enables precise intervention calibration. Figure 7 quantifies the outbreak potential despite , showing that imported cases can trigger limited outbreaks peaking at 27 prodromal cases. This demonstrates why interventions remain valuable even in subcritical regions.

Table 3.

Parameter sensitivity analysis ( baseline).

Parameter	Sensitivity index	(10% Change)	Impact
(Transmission)	1.0000	+0.0398	High
(Mortality)	0.6615	0.0264	High
(PR progression)	0.4223	0.0168	Moderate
(Incubation)	0.4737	+0.0189	Moderate
(R recovery)	0.1896	0.0076	Low
(P recovery)	0.1250	0.0050	Low
(Complications)	0.0753	0.0030	Negligible

Fig. 6.

Cumulative mortality under baseline vs. intervention scenarios. Gray band shows 95% CI from 10,000 LHS simulations. Transmission reduction decreases mortality by 78% compared to baseline.

Public health implementation framework

Our analysis supports three evidence-based control strategies optimized through sensitivity findings. First, transmission-focused resource allocation prioritizes -sensitive interventions (60% of resources), including contact tracing and PPE distribution. Second, diagnostic-targeted vaccination concentrates on prodromal cases and their contacts, requiring 92% diagnostic sensitivity. Third, adaptive surveillance integrates wastewater monitoring and genomic sequencing for early outbreak detection. Region-specific implementation varies: in DFE regions (), targeted surveillance with 60-day monitoring windows suffices; in endemic regions (), aggressive transmission reduction and complication management are essential. Healthcare planning must account for the substantial complication burden shown in Fig. 4, requiring approximately 20 ICU bed-days per severe case.

Modeling innovations and contributions

This study advances Mpox modeling through four significant contributions derived from our analysis. First, the granular compartmentalization of infection stages (Figs. 3 and 4) captures differential transmission dynamics between prodromal, rash, and complication phases, enabling stage-specific intervention analysis. Second, stability analysis (Fig. 2) formally establishes the transcritical bifurcation at with critical threshold , providing a mathematically framework for epidemic prediction. Third, clinically parameterization (Table 2) incorporates progression rates and mortality risks based on empirical data, enhancing model biological fidelity. Fourth, the intervention optimization framework (Table 3, Fig. 6) translates sensitivity indices into actionable public health strategies, demonstrating that 10% transmission reduction decreases by 0.0398. These elements collectively bridge theoretical epidemiology with practical disease management, offering both analytical foundations and implementable control targets.

Limitations and research directions

Several methodological limitations warrant consideration in interpreting results. The homogeneous mixing assumption likely overestimates rural transmission intensity by 12-15% due to unaccounted spatial heterogeneity in contact patterns. Exclusion of healthcare capacity constraints may underestimate mortality rates during epidemic peaks (Fig. 6), particularly in resource-limited settings where complication management could be compromised. The static parameterization doesn’t capture potential behavioral adaptations during outbreaks, such as voluntary contact reduction. Additionally, waning immunity is not incorporated, potentially overestimating long-term protection in recovered cohorts. Future research should address these limitations through: 1) Spatial-explicit modeling incorporating mobility networks and heterogeneous mixing; 2) Healthcare system integration with resource-dependent mortality functions; 3) Dynamic behavioral modules capturing risk perception and intervention adherence; and 4) Multi-strain frameworks accounting for viral evolution. Validation against emerging phylodynamic data would further strengthen parameter estimation. These refinements would enhance model applicability to real-world outbreak scenarios where spatial, behavioral, and healthcare constraints significantly influence transmission trajectories.

Conclusion

This study has systematically investigated the transmission dynamics of Mpox through a compartmental modeling framework, yielding critical insights into epidemic thresholds, intervention efficacy, and public health implications. Our analysis establishes day as the fundamental threshold governing transmission behavior, demarcating two distinct epidemiological regimes: disease extinction below this threshold and endemic persistence above it. The transcritical bifurcation at (Fig. 2) provides a mathematically foundation for predicting epidemic outcomes, with stability analysis confirming the disease-free equilibrium (DFE) is stable when and the endemic equilibrium (EE) stable when . Under subcritical conditions (), our simulations demonstrate self-limiting outbreaks characterized by negligible susceptible depletion (0.0% over 100 days), sequential compartmental peaking aligning with progression rates, and complete outbreak extinction within 60 days despite importation risk (Fig. 3. In supercritical scenarios (), we observe explosive transmission featuring rapid susceptible depletion (60.41% within 100 days), substantial complication burden peaking at 5,089 cases, and endemic stabilization with persistent low-level circulation (Fig. 4, Table 2). Parameter sensitivity analysis reveals a hierarchical intervention efficacy structure where transmission reduction dominates (), with a quantifiable relationship showing each 10% reduction in transmission rate decreases by 0.0398 (Table 3). This enables precise calibration of control measures, demonstrating that 50% transmission reduction crosses the critical threshold , potentially decreasing complications by 78% (Fig. 6). The study contributes four key innovations: 1) Granular compartmentalization capturing differential stage infectivity; 2) Formal stability analysis establishing bifurcation properties; 3) Clinically parameterization based on progression kinetics; and 4) Sensitivity-derived intervention optimization framework. While the model has limitations in spatial resolution and healthcare constraints, it provides operational targets for control programs (), resource allocation guidance (60:40 transmission-to-clinical focus), and quantitative metrics for surveillance systems. These findings collectively advance our capacity to predict, prevent, and manage Mpox transmission through evidence-based public health strategies grounded in mathematical epidemiology. The established thresholds, sensitivity relationships, and compartmental framework offer both theoretical foundations for epidemic forecasting and practical tools for outbreak response, creating a transferable methodology for orthopoxvirus threat management.

Author contributions

S.R. conceptualized the mathematical model, implemented computational analyses (bifurcation, sensitivity), and drafted the manuscript. M.K.M.A. supervised the research design, validated analytical methods, and critically revised epidemiological interpretations. Both authors contributed to intervention strategy development, approved the final manuscript, and agree to be accountable for all aspects of the work.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

All data that support the findings of this study are included in the article as cited in table 1.

Declarations

Competing interests

The authors declare no competing interests.