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Mathematical Modeling of Mpox with Animal Reservoir Coupling: Stability, Sensitivity, and Calibration to the 2022 US Outbreak

This paper presents a coupled SVEIQR-SEI mathematical model of Mpox transmission between animal reservoirs and humans, which demonstrates that the disease can be eliminated when the basic reproduction number is below one and, upon calibration to the 2022 US outbreak, reveals that targeted quarantine and behavioral interventions were the primary drivers of containment while highlighting the shifting influence of reservoir dynamics across different epidemic phases.

Original authors: Mst. Srabony Akhter, Md. Ahnaf Sadik Inan, Ashrafi Meher Niger

Published 2026-08-11

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Mst. Srabony Akhter, Md. Ahnaf Sadik Inan, Ashrafi Meher Niger

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, invisible game of "hot potato" being played across the globe, but instead of a ball, the object is a virus, and the players are living creatures. This is the world of epidemiology, the science of tracking how diseases move through populations. To understand these games, scientists use mathematical models, which are like digital simulators or video game engines that let researchers test rules without risking real lives. In these simulations, they calculate a crucial number called the basic reproduction number ( $R_0$ ). Think of $R_0$ as a "spread score": if the score is below 1, the game fizzles out because each infected person passes the virus to fewer than one new person; if it's above 1, the game explodes into an outbreak. The big question scientists ask is: How do we stop the game before it gets out of control? Specifically, when a disease jumps from animals to humans (like a squirrel passing a virus to a person), does the animal side keep the fire burning even after we put out the human fires?

This paper, titled "Mathematical Modeling of Mpox with Animal Reservoir Coupling," dives deep into that exact question using the 2022 Mpox outbreak as a real-world test case. The authors built a sophisticated digital twin of the virus's journey, creating a system that links two separate worlds: the human population and the animal reservoir (specifically rodents). They didn't just guess; they wrote a set of equations that track people moving through different stages: Susceptible, Vaccinated, Exposed, Infected, Quarantined, and Recovered (SVEIQR), while simultaneously tracking rodents in their own Susceptible-Exposed-Infected (SEI) loop.

The researchers discovered that the game has two distinct phases. In the beginning, the "spread score" is driven almost entirely by human-to-human contact. It's like a spark in a dry forest; the wind (human behavior) carries the fire. However, as the outbreak peaks and starts to fade, the animal reservoir becomes the hidden engine keeping the embers alive. The paper proves mathematically that if you can get the overall spread score below 1, the disease will vanish completely—there are no tricky "hidden traps" where the disease lingers even if the score looks low.

When they calibrated their model to the actual 2022 outbreak in the United States, the results were strikingly accurate. By adjusting the model to reflect that the virus was mostly spreading within a small, specific group of people (about 5% of the population) rather than the whole country, they achieved a fit so precise that the model's curve matched the real data with an $R^2$ of 0.985. This means the model explained 98.5% of the variation in the real-world cases. The simulation showed that the outbreak peaked exactly when it did in reality (Week 10) and that the combination of rapid quarantine and behavioral changes (like social distancing) was the primary reason the fire was put out. The paper concludes that while humans start the fire, the animals can keep it smoldering, so the best strategy is a phase-specific approach: crush human transmission early, but keep a close watch on the animal side as the outbreak winds down.

Technical Summary: Mathematical Modeling of Mpox with Animal Reservoir Coupling

Problem Statement
Mpox (formerly monkeypox) is a zoonotic disease characterized by complex transmission dynamics involving both animal reservoirs (primarily rodents) and human populations. While previous mathematical models have addressed Mpox using frameworks like SVEIR or fractional-order derivatives, there remains a need for a comprehensive model that explicitly couples animal reservoir dynamics with human transmission while incorporating specific interventions such as vaccination, quarantine, and behavioral changes. The 2022 global outbreak, particularly in the United States, highlighted a shift from traditional zoonotic spillover to sustained human-to-human transmission within specific high-risk networks, necessitating a model capable of capturing these heterogeneous dynamics and evaluating phase-specific control strategies.

Methodology
The authors developed a coupled SVEIQR-SEI mathematical framework, dividing the population into two interacting subsystems:

Human Population (SVEIQR): Susceptible ( $S_h$ ), Vaccinated ( $V_h$ ), Exposed ( $E_h$ ), Infectious ( $I_h$ ), Quarantined/Isolated ( $Q_h$ ), and Recovered ( $R_h$ ).
Animal Reservoir (SEI): Susceptible ( $S_r$ ), Exposed ( $E_r$ ), and Infectious ( $I_r$ ).

The model incorporates two transmission routes: rodent-to-human ( $\theta_1$ ) and human-to-human ( $\theta_2$ ).

Mathematical Analysis: The study established the well-posedness of the model (positivity and boundedness of solutions). Using the next-generation matrix approach, the authors derived the basic reproduction number, $R_0 = \max\{R_r, R_h\}$ , where $R_r$ and $R_h$ represent the reproduction numbers for the rodent and human populations, respectively. Stability analysis of the disease-free equilibrium (DFE) was conducted via Jacobian eigenvalues. Furthermore, the authors applied Castillo-Chavez and Song's center manifold theory to analyze bifurcation behavior at the epidemic threshold ( $R_0 = 1$ ).
Sensitivity Analysis: A global sensitivity analysis was performed using Partial Rank Correlation Coefficients (PRCC) combined with Latin Hypercube Sampling (LHS) over 240 days. This assessed the influence of 15 parameters on the human reproduction number and infected population across four distinct epidemic phases: Early Growth, Peak Dynamics, Decline, and Extended Surveillance.
Model Calibration to 2022 US Outbreak: To address the limitations of homogeneous mixing assumptions in the context of the 2022 US outbreak, the model was adapted with risk stratification. The susceptible human compartment was split into a high-risk subpopulation ( $S_{h,risk} \approx 5\%$ of the population) and a low-risk subpopulation. A time-varying transmission rate, $\theta_2(t)$ , modeled via a logistic function, was introduced to capture behavioral changes and intervention effects. The model was calibrated to weekly case data from "Our World in Data" (May 17 – October 1, 2022) using a three-stage optimization pipeline (Differential Evolution, Dual Annealing, and L-BFGS-B) minimizing a Weighted Sum of Squared Errors (WSSE) with heuristic penalties for peak timing and magnitude.

Key Contributions and Results

Theoretical Stability and Bifurcation: The analysis proves that the disease-free equilibrium is locally asymptotically stable if and only if $R_0 < 1$ . Crucially, the bifurcation analysis confirms a forward (supercritical) bifurcation at $R_0 = 1$ . This finding implies that disease elimination is feasible and sufficient by reducing $R_0$ below unity, with no possibility of backward bifurcation or bistability near the epidemic threshold.
Phase-Specific Sensitivity: The PRCC analysis revealed distinct temporal drivers of the epidemic:
- Early Phase (Days 1–30): Human-to-human transmission ( $\theta_2$ ) and progression rates ( $\tau_1$ ) dominate. Quarantine effectiveness ( $\psi_2$ ) shows the strongest negative correlation.
- Peak Phase (Days 45–90): While human transmission remains critical, animal reservoir dynamics (rodent transmission $\theta_r$ and progression $\xi_r$ ) emerge as highly influential, peaking in sensitivity.
- Decline and Surveillance (Days 105–240): Sensitivity shifts back toward human transmission and quarantine, though reservoir monitoring remains relevant for preventing resurgence.
Calibration Performance: The risk-stratified, time-varying model achieved an exceptional fit to the 2022 US outbreak data ( $R^2 = 0.985$ ). It accurately predicted the peak timing (Week 10) with a magnitude error of only 3.2% and matched total cumulative cases within 0.2%. The model demonstrated that the effective reproduction number ( $R_{eff}$ ) dropped from 2.84 to 0.32 by Week 23, crossing the control threshold ( $R_{eff}=1$ ) at Week 12, coinciding with scaled-up vaccination campaigns.
Intervention Efficacy: Simulations indicated that while vaccination of newborns had negligible impact on the short-term epidemic curve, quarantine and social distancing were the most effective single interventions. Combined strategies, particularly those targeting high-risk networks, were shown to fully control disease spread.

Significance and Claims
The paper claims that its primary significance lies in providing a phase-specific control framework for zoonotic diseases. By demonstrating that animal reservoirs become critical drivers during the peak and decline phases, the authors argue that control strategies must be adaptive: prioritizing rapid quarantine and human transmission reduction in the early response, while maintaining reservoir monitoring and balanced interventions during the peak and decline.

The study emphasizes that the 2022 US outbreak was driven primarily by human-to-human transmission within a small, well-defined network, validating the necessity of risk stratification in modeling. The authors conclude that their quantitative framework offers public health authorities a tool for evidence-based decision-making, specifically highlighting that targeted quarantine and behavioral interventions within high-risk networks were the primary drivers of the 2022 outbreak's containment. The confirmation of a forward bifurcation further reinforces that standard threshold-based elimination strategies are theoretically sound for Mpox, provided $R_0$ is reduced below unity near the epidemic threshold.

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