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Modeling the Complex Dynamics of Monkeypox and Typhoid Fever Co-Infection Using a Novel Fractal-Fractional Approach

This study introduces a novel fractal-fractional mathematical model to analyze the complex co-infection dynamics of monkeypox and typhoid fever, demonstrating that this advanced framework effectively captures memory effects and multi-scale heterogeneities to reveal how dual pathogen interactions accelerate disease progression and inform integrated control strategies.

Original authors: Hawah Oyiza Rabiu, Jeremiah Amos, Godwin Onuche Acheneje, Benedict Celestine Agbata, A.K. Awasthi, Hambeer Singh, Aseel Smerat, Bolarinwa Bolaji

Published 2026-07-10

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CC BY 4.0

Original authors: Hawah Oyiza Rabiu, Jeremiah Amos, Godwin Onuche Acheneje, Benedict Celestine Agbata, A.K. Awasthi, Hambeer Singh, Aseel Smerat, Bolarinwa Bolaji

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine the world of disease spread not as a simple line on a graph, but as a chaotic, bouncy trampoline where every jump leaves a lasting ripple. That's the world this paper explores. The authors, a team of mathematicians from Nigeria, India, Jordan, and the UK, built a brand-new kind of digital simulation to understand what happens when two very different diseases—Monkeypox and Typhoid fever—decide to crash the same party.

The Old Way vs. The New Way
Usually, scientists use "integer-order" models to predict how diseases spread. Think of these like a standard, flat map. They tell you where you are right now, but they don't remember where you've been. They assume that once you move, the past doesn't matter.

The authors argue that this flat map isn't enough for real life. In the real world, history matters. If you got sick last week, it changes how you act today. If a community has poor sanitation for years, that history shapes how Typhoid spreads right now. To fix this, the team introduced a "Fractal-Fractional" approach.

Here is the analogy: Imagine a fractal is like a coastline. From far away, it looks smooth, but if you zoom in, it's jagged and complex, with smaller jagged bits inside the big jagged bits. This represents how people mix in a community—it's not a smooth flow; it's messy and self-similar at every level. Now, add "fractional" math, which acts like a memory bank. It remembers every step the disease took in the past. By combining these two, the authors created a model that captures both the messy, jagged reality of human contact and the long-term memory of how the disease has behaved over time.

The Double Trouble: Monkeypox and Typhoid
The paper focuses on a scary scenario where Monkeypox and Typhoid fever infect the same people at the same time.

Monkeypox is like a sneaky guest that jumps from animals (like rodents) to humans and then spreads between people through close contact.
Typhoid is like a dirty water guest that spreads through contaminated food and water, sticking around in the environment.

The authors found that when these two "guests" meet, they make things much worse. The simulation suggests that having Monkeypox can weaken a person's immune system, making the Typhoid infection more severe. Conversely, the stress of fighting Typhoid might make a person more likely to catch Monkeypox or spread it faster. It's a vicious cycle where one disease helps the other win.

What the Math Says
The team didn't just guess; they ran complex computer simulations using their new fractal-fractional equations.

Memory Matters: They discovered that the "fractional order" (a number representing how much memory the system has) changes the outcome. When they lowered this number to simulate stronger memory effects, the number of infected people didn't just go up or down smoothly; it started to oscillate, or bounce up and down, in a way that standard models missed. This suggests that real-world outbreaks might be more unpredictable and "bouncy" than we thought.
The Danger Zone: The model calculated a "Basic Reproduction Number" (a score that tells us if a disease will spread). If this score is above 1, the disease keeps spreading. The authors found that specific factors—like how easily Typhoid spreads through water, how easily Monkeypox jumps between people, and how well treatments work—are the main drivers of this score.
The Solution: The simulations suggest that to stop the spread, we can't just treat one disease. We need a combined attack: better sanitation to stop Typhoid, vaccination and isolation to stop Monkeypox, and treating co-infected people quickly.

What They Didn't Find
It's important to note what this paper doesn't say. The authors did not claim to have found a cure or a magic bullet. They didn't say this model is perfect for every single village on Earth. They explicitly ruled out the idea that simple, old-fashioned math (the flat maps) is enough to understand these complex, dual-disease outbreaks. They also didn't provide real-world data proving these exact numbers happen in a specific city right now; instead, they used their model to simulate what would happen under these conditions.

The Bottom Line
This paper suggests that to understand the messy, overlapping world of Monkeypox and Typhoid, we need a new kind of math that remembers the past and respects the complexity of how people actually live. Their simulations show that ignoring the "memory" of the disease or the "jagged" nature of human contact could lead us to underestimate how dangerous these co-infections really are. The authors propose that by using this advanced, memory-rich approach, public health officials might be able to design better strategies to keep both diseases in check.

Technical Summary: Modeling the Complex Dynamics of Monkeypox and Typhoid Fever Co-Infection Using a Novel Fractal-Fractional Approach

Problem Statement
The paper addresses the epidemiological challenge of the simultaneous circulation of Monkeypox (MPX) and Typhoid fever, particularly in resource-limited regions where sanitation is poor and human-animal contact is frequent. While both diseases share overlapping initial symptoms (fever, fatigue, muscle aches) and transmission pathways (fecal-oral for typhoid; zoonotic and human-to-human for monkeypox), existing mathematical models have largely treated them as single-pathogen systems or used classical integer-order differential equations for co-infections. The authors argue that integer-order models fail to capture critical real-world complexities, specifically the "memory effects" (long-term dependencies in disease progression) and "multi-scale heterogeneities" (self-similar patterns in population mixing) inherent in dual-pathogen transmission. There is a noted gap in the literature regarding the application of fractal-fractional derivative frameworks to this specific co-infection scenario.

Methodology
The study proposes a novel 18-compartment fractal-fractional order mathematical model. The human population is stratified into susceptible, exposed, quarantined, infected, treated, recovered, and vaccinated classes for both Monkeypox and Typhoid, including specific compartments for co-infected individuals. The model also incorporates a rodent population (reservoir for Monkeypox) and a bacterial population (environmental reservoir for Typhoid).

The core methodological innovation is the use of the Fractal-Fractional (FF) derivative with a Mittag-Leffler kernel in the Caputo sense. This approach integrates:

Fractional Calculus (Order $\alpha$ ): To account for non-local dynamics and memory effects, where the current state of the system depends on its historical trajectory.
Fractal Geometry (Dimension $\sigma$ ): To represent the self-similar, heterogeneous nature of contact patterns and transmission pathways.

The analytical framework includes:

Existence and Uniqueness: Proven using fixed-point theorems (Banach and Krasnoselskii) within a defined Banach space.
Positivity and Boundedness: Demonstrated to ensure the model is biologically feasible (populations remain non-negative and within realistic limits).
Equilibrium Analysis: Derivation of Disease-Free Equilibrium (DFE) and Endemic Equilibrium points.
Basic Reproduction Numbers ( $R_0$ ): Calculation of $R_{0m}$ (Monkeypox) and $R_{0T}$ (Typhoid) using the Next-Generation Matrix method.
Sensitivity Analysis: Performed to identify parameters that most significantly influence the transmission dynamics.
Numerical Simulations: Implemented using fractal-fractional computational schemes to visualize the impact of varying $\alpha$ and $\sigma$ .

Key Contributions

First Integrated Model: This work presents the first comprehensive fractal-fractional model specifically for Monkeypox-Typhoid co-infection, bridging the gap between single-pathogen fractional models and integer-order co-infection models.
Dual-Pathogen Dynamics: The model explicitly captures the interaction between zoonotic transmission (rodent-to-human for MPX) and environmental persistence (bacteria in water/food for Typhoid), as well as the bidirectional impact of co-infection on disease progression and mortality.
Enhanced Mathematical Framework: By employing the FF-derivative, the model offers a more flexible tool to simulate complex epidemic behaviors that integer-order models cannot, specifically regarding memory retention and fractal contact structures.

Results

Stability: The analysis confirms that the system is globally asymptotically stable at the endemic equilibrium when the basic reproduction numbers exceed unity.
Sensitivity: Sensitivity analysis identifies the transmission rates for both diseases, relative infectiousness of co-infected individuals, and recovery/treatment rates as primary drivers of epidemic dynamics.
Impact of Fractional Parameters: Numerical simulations reveal that varying the fractional order ( $\alpha$ ) and fractal dimension ( $\sigma$ ) significantly alters population oscillatory behaviors. Lower fractional orders produce more pronounced memory effects, leading to sustained fluctuations in co-infected compartments compared to integer-order models.
Co-infection Severity: The model demonstrates that co-infection substantially increases morbidity and mortality rates. Monkeypox co-infection exacerbates Typhoid progression, while Typhoid episodes accelerate Monkeypox viral shedding and disease severity.
Control Implications: The results suggest that integrated strategies—combining zoonotic control, sanitation improvements (WASH), and coordinated vaccination—are critical for managing dual-pathogen epidemics.

Significance and Claims
The paper claims that the fractal-fractional framework provides superior performance over conventional integer-order models in capturing the complexities of real-world epidemics. By accounting for non-local interactions and long-term dependencies, the model offers enhanced predictive capabilities for intervention strategies. The authors posit that this approach is particularly vital for regions with high Typhoid endemicity and emerging Monkeypox outbreaks, where traditional models may underestimate the persistence and volatility of the disease burden. The study concludes that understanding the memory effects and fractal nature of transmission is essential for designing effective, coordinated public health interventions for co-infection control.

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