Statistical Insights into Zoonotic Disease Dynamics: Simulation and Control Strategy Evaluation

Abstract

This study presents a comprehensive analysis of zoonotic disease transmission dynamics between baboon and human populations using both deterministic and stochastic modeling approaches. The model is constructed with a symmetric compartmental structure for each species—susceptible, infected, and recovered—which reflects a biological and mathematical symmetry between the two interacting populations. Public health control strategies such as sterilization, restricted food access, and reduced human–baboon interaction are incorporated symmetrically, allowing for a balanced evaluation of their effectiveness across species. The basic reproduction number (${\mathcal{R}}_0$) is derived analytically and examined through sensitivity indices to identify critical epidemiological parameters. Numerical simulations, implemented via the Euler–Maruyama method, explore the influence of stochastic perturbations on disease trajectories. Statistical tools including Maximum Likelihood Estimation (MLE), Mean Squared Error (MSE), and Power Spectral Density (PSD) analysis validate model predictions and assess variability across noise levels. The results provide probabilistic confidence intervals and highlight the robustness of the proposed control strategies. This symmetry-aware, dual-framework modeling approach offers novel insights into zoonotic disease management, particularly in ecologically dynamic regions with frequent human–wildlife interactions.

1. Introduction

Zoonotic diseases, which transfer from animals to humans, represent a significant global health risk, particularly in regions with extensive human–wildlife interaction [1,2]. Zoonotic diseases, which transfer from animals to humans, represent a significant global health risk, particularly in regions with extensive human–wildlife interaction [1,2]. Baboons are among the species with frequent human contact, especially in areas of shared resources, increasing the potential for zoonotic disease transmission [3,4].

Zoonotic diseases—those that are transmitted between animals and humans—pose a significant global health challenge due to their potential to trigger large-scale outbreaks and cross-border health crises [1,2]. Baboons are one of the species with frequent human contact, especially in areas of shared resources, increasing the potential for zoonotic disease transmission [3,4]. This study presents a statistical framework for understanding disease dynamics between baboons and human populations. Using a combination of deterministic and stochastic models, we explore statistical measures to understand transmission rates and the effect of various control strategies, such as sterilization, reduced food access, and restricted contact [5,6].

This study presents a statistical framework for understanding disease dynamics between baboon and human populations. Using a combination of deterministic and stochastic models, we explore transmission patterns and the effects of various control strategies, such as sterilization, reduced food access, and restricted contact [5,6].

Diseases such as rabies, Ebola, and various influenza strains are examples of zoonoses with high mortality rates and widespread societal impact. In regions where human–wildlife interactions are frequent, such as those shared with baboons and other wildlife, the risk of zoonotic disease transmission increases substantially [3,4]. In such areas, understanding transmission dynamics and developing control strategies is crucial for mitigating public health risks. Effective study of zoonotic diseases requires a multidisciplinary approach that incorporates epidemiology, ecological principles, and statistical modeling techniques.

Modeling zoonotic disease transmission dynamics often involves both deterministic and stochastic approaches. Deterministic models provide a fixed, predictable outcome based on initial conditions and are useful for understanding baseline disease behavior. In contrast, stochastic models incorporate random fluctuations, allowing researchers to capture real-world uncertainties in disease spread due to factors such as environmental changes, resource availability, and variations in population behaviors [7,8]. For example, in the case of baboon–human interactions, stochastic elements enable a realistic representation of disease behavior by accounting for unpredictable events like seasonal changes and food scarcity that could amplify or reduce disease transmission [9]. The basic reproduction number ${\mathcal{R}}_0$ is computed using statistical sensitivity indices to identify the influence of parameters like transmission and recovery rates on disease spread [10,11]. A reproduction number greater than one suggests persistent disease, while a value below one indicates eradication likelihood.

Sensitivity analysis, utilizing forward sensitivity indices, quantifies the relative influence of each parameter, highlighting control points that may significantly impact disease reduction [7]. The basic reproduction number, $R_0$, is central to understanding disease spread, as it quantifies the average number of secondary infections generated by a single infected individual in a completely susceptible population. A higher $R_0$ indicates a more aggressive disease spread, emphasizing the importance of intervention measures like sterilization, restricting access to shared food sources, and limiting direct contact between humans and wildlife to reduce transmission rates [5,6]. Analyzing the sensitivity of $R_0$ with respect to key parameters—such as transmission rates, recovery rates, and contact rates—highlights the factors most influential in controlling zoonotic disease dynamics. Sensitivity analysis is critical, as it allows public health researchers to target interventions towards the most impactful factors, thereby optimizing resource allocation [10,12]. For instance, the reduction in baboon–human interactions and improved sanitation at shared resources can significantly reduce transmission potential, as reflected in a lowered $R_0$ [13].

In recent years, mathematical modeling has emerged as a powerful tool for describing systems that exhibit memory, complexity, and dynamic interdependencies [14]. Its growing popularity is evident in fields such as engineering [15], plant pathology [16], biology [17], medicine [18], behavioral sciences [19], viscoelastic materials [20], electromagnetic waves [21], quantum mechanics [22], Langevin equations [23], physics [24,25,26,27], diabetes modeling [28,29,30,31,32], cybersecurity [33], smoking dynamics [34], epidemiology [35], influenza [36], infectious disease outbreaks [37], epidemic modeling [38], cancer research [39], COVID-19 [40], monkeypox [41], zoonotic viruses [42], and alcohol-related studies [43]. Epidemiological models, in particular, have found extensive applications in various disciplines, such as healthcare, chemistry, physics, and economic modeling [44,45,46].

Statistical techniques, including Maximum Likelihood Estimation (MLE) and confidence interval calculations, enhance model reliability by refining parameter estimates to fit observed data more accurately. MLE, for example, estimates key parameters like transmission and recovery rates by defining a likelihood function based on observed data, then maximizing them to find the best-fitting estimates. Log-likelihood calculations and numerical methods further validate these estimates’ robustness [47,48]. Confidence intervals offer probabilistic bounds on predictions, providing reliability for public health strategies grounded in the model’s projections. Power Spectral Density (PSD) analysis identifies periodic patterns in transmission, which can help anticipate seasonal outbreaks. By analyzing PSD, officials can recognize oscillatory patterns in transmission dynamics, aiding in timely and effective intervention planning [49].

This study presents a comprehensive analysis of zoonotic disease transmission dynamics between baboon and human populations using both deterministic and stochastic modeling approaches. This statistical approach to zoonotic disease modeling provides comprehensive insights into transmission dynamics and supports the formulation of targeted intervention strategies. By integrating deterministic and stochastic modeling frameworks, this study captures both the expected mean behavior and the inherent variability in real-world disease transmission. The formulation of this model is guided by principles of structural and behavioral symmetry, where both baboon and human populations are represented using equivalent compartmental dynamics. Such symmetric modeling allows for comparative epidemiological analysis and ensures that control strategies are evaluated with balanced effectiveness across both species. Moreover, the stochastic components incorporate symmetric noise processes, preserving consistency in uncertainty modeling. The combined model demonstrates high predictive reliability, clear parameter sensitivity insight, quantitative uncertainty characterization, and effective evaluation of intervention strategies under stochastic variability, thereby offering a reliable and quantitatively robust framework for zoonotic disease control.

The primary objective of this study is to develop and analyze an integrated deterministic-stochastic modeling framework to understand the transmission dynamics of zoonotic diseases between baboons and human populations. By formulating compartmental models and incorporating key control strategies such as sterilization, restricted food access, and reduced human–baboon interactions, this study aims to assess their effectiveness in curbing disease spread. The framework also quantifies uncertainties through stochastic differential equations and evaluates the sensitivity of the basic reproduction number ($R_0$) to identify critical parameters influencing transmission. Statistical techniques, including Maximum Likelihood Estimation (MLE), Mean Squared Error (MSE), confidence interval estimation, and Power Spectral Density (PSD) analysis, are employed to ensure model reliability and accuracy. Ultimately, this study seeks to provide a robust decision-support tool for zoonotic disease control in ecologically sensitive regions with frequent human–wildlife interaction.

The remainder of this paper is organized as follows. Section 2 introduces the mathematical formulation of the deterministic and stochastic models for zoonotic disease transmission between baboons and humans, including model assumptions and control strategies. Section 3 presents the statistical analysis of the stochastic model, proving positivity, boundedness, and uniqueness of solutions, and describes the Euler–Maruyama numerical scheme. Section 4 discusses the numerical simulation results, comparing deterministic and stochastic dynamics under various noise intensities, and highlights key insights through graphical analysis. Section 5 addresses uncertainty quantification and statistical validation using confidence intervals, standard deviations, parameter uncertainty via Maximum Likelihood Estimation (MLE), and Mean Squared Error (MSE) analysis. Section 6 conducts a sensitivity analysis of the basic reproduction number $R_0$ to identify influential epidemiological parameters. Section 7 provides simulation outputs from the Euler–Maruyama method to illustrate disease dynamics under stochastic perturbations. Finally, Section 8 concludes this study by summarizing the main findings and discussing their implications for zoonotic disease control and future research directions.

2. Model Description

2.1. Model Assumptions

The proposed deterministic and stochastic models for zoonotic disease transmission between baboon and human populations are based on the following assumptions:

• The baboon and human populations are compartmentalized into three mutually exclusive groups: susceptible (S), infected (I), and recovered (R) individuals.
• Disease transmission follows standard mass-action incidence, proportional to the product of the number of susceptible and infected individuals in each population.
• Control strategies, including sterilization ($H_s\left(t\right)$), food access restriction ($H_f\left(t\right)$), and reduction in human–baboon interactions ($H_i\left(t\right)$), are applied uniformly across the entire population without individual heterogeneity.
• The stochastic model assumes that environmental and demographic randomness can be adequately represented by independent Brownian motions and that perturbations follow Gaussian white noise processes with constant intensities ${\sigma }_i$ for each population compartment.
• Recovery rates (${\gamma }_b$, ${\gamma }_h$) and transmission rates (${\beta }_b$, ${\beta }_h$) are considered constant over time and are independent of population size or environmental factors.
• Human and baboon populations are closed during the study period—migration, births, and deaths other than disease-related are not considered.
• Parameter estimates obtained through Maximum Likelihood Estimation (MLE) assume normally distributed model residuals and independent observations over time.

The inclusion of time-dependent controls reflects real-world public health interventions:

• $H_s\left(t\right)$ reduces the recruitment of new susceptible baboons via sterilization.
• $H_f\left(t\right)$ affects all baboon compartments by reducing access to food resources and indirectly limiting interaction.
• $H_i\left(t\right)$ reduces the rate of human exposure and subsequent infection.

2.2. Limitations

Despite the strengths of the combined deterministic-stochastic modeling framework, several limitations should be acknowledged:

• The assumption of homogeneous mixing within and between the baboon and human populations may oversimplify real-world contact patterns, which are likely influenced by spatial, social, and behavioral factors.
• The use of constant transmission and recovery rates neglects potential seasonal or time-dependent changes in disease dynamics or intervention effectiveness.
• The stochastic model assumes Gaussian-distributed perturbations, which may not fully capture the complexity of environmental fluctuations, especially in systems subject to sudden or extreme changes.
• Parameter values are derived from a combination of estimated and assumed values, due to limited availability of empirical data on baboon–human disease transmission in the study area.
• The model does not incorporate latent or exposed compartments, asymptomatic carriers, or reinfection possibilities, which could be relevant for certain zoonotic diseases.
• The impact of additional ecological and socio-economic factors, such as habitat loss, public awareness, and healthcare infrastructure, is not explicitly modeled.

These limitations present opportunities for future model refinement, including incorporating spatial heterogeneity, time-varying parameters, non-Gaussian stochasticity, and additional epidemiological compartments to capture more complex transmission scenarios.

2.3. Deterministic Model

The model presented simulates the interactions between baboon and human populations in relation to zoonotic disease transmission. It incorporates control strategies such as sterilization, restricted food access, and minimized human–baboon interaction. It defines population dynamics with differential equations for susceptible (S), infected (I), and recovered (R) individuals within both populations. For baboons, the model considers natural population growth, disease transmission, and the effects of control interventions on these dynamics. Human–baboon interactions impact human susceptibility and infection rates, influenced by disease transmission. Key parameters include intrinsic growth rate (r), carrying capacity (K), disease transmission rates (${\beta }_b,{\beta }_h$), recovery rates (${\gamma }_b,{\gamma }_h$), and control rates for sterilization ($H_s\left(t\right)$), food access restriction ($H_f\left(t\right)$), and human interaction ($H_i\left(t\right)$). This model provides a comprehensive framework for analyzing disease dynamics and assessing control measures.

The combined system of equations modeling baboon and human populations with zoonotic disease transmission is:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_b}{dt}}& =r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b,\hfill \\ \hfill {\displaystyle \frac{d{I}_b}{dt}}& ={\beta }_b{S}_b{I}_b-{\gamma }_b{I}_b-H_s\left(t\right)I_b-H_f\left(t\right)I_b,\hfill \\ \hfill {\displaystyle \frac{d{R}_b}{dt}}& ={\gamma }_b{I}_b-H_f\left(t\right)R_b,\hfill \\ \hfill {\displaystyle \frac{d{S}_h}{dt}}& =-{\beta }_h{S}_h{I}_b-H_i\left(t\right)S_h,\hfill \\ \hfill {\displaystyle \frac{d{I}_h}{dt}}& ={\beta }_h{S}_h{I}_b-{\gamma }_h{I}_h-H_i\left(t\right)I_h,\hfill \\ \hfill {\displaystyle \frac{d{R}_h}{dt}}& ={\gamma }_h{I}_h.\hfill \end{array}$$

(1)

The system is initialized with biologically meaningful values:

$$S_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),I_h\left(0\right),R_h\left(0\right)$$

All parameters (${\beta }_b$, ${\beta }_h$, ${\gamma }_b$, ${\gamma }_h$, r, K, $H_s$, $H_f$, $H_i$) are estimated or assumed based on the empirical data and literature, as summarized in Table 1 of this manuscript.

To better understand the disease dynamics described by the deterministic system (1), we analyze the local stability of the disease-free equilibrium (DFE). This analysis offers theoretical insights into whether the disease will persist or die out over time under given initial conditions and parameter values.

Setting all infected compartments to zero, the disease-free equilibrium point $E_0$ is given by:

$$E_0=\left(S_b^0,0,0,S_h^0,0,0\right),$$

where

$$S_b^0={\displaystyle \frac{Kr}{H_s+H_f}},S_h^0={\displaystyle \frac{K}{H_i}}.$$

Let the state vector be $X={(S_b,I_b,R_b,S_h,I_h,R_h)}^T$. Linearizing the system around the DFE yields the Jacobian matrix J:

$$J\left(E_0\right)=\left[\begin{array}{cccccc}\ast & -{\beta }_b{S}_b^0& 0& 0& 0& 0\\ 0& {\beta }_b{S}_b^0-({\gamma }_b+H_s+H_f)& 0& 0& 0& 0\\ 0& {\gamma }_b& -H_f& 0& 0& 0\\ 0& -{\beta }_h{S}_h^0& 0& -H_i& 0& 0\\ 0& {\beta }_h{S}_h^0& 0& 0& -({\gamma }_h+H_i)& 0\\ 0& 0& 0& 0& {\gamma }_h& 0\end{array}\right]$$

The ∗ entries represent terms not directly influencing the infection dynamics and are not needed for stability analysis of the DFE.

The local stability of the DFE is determined by the eigenvalues of $J\left(E_0\right)$. It can be shown that the eigenvalues associated with the infected compartments $I_b$ and $I_h$ are:

$${\lambda }_1={\beta }_b{S}_b^0-({\gamma }_b+H_s+H_f),{\lambda }_2={\beta }_h{S}_h^0-({\gamma }_h+H_i).$$

The DFE is locally asymptotically stable if all eigenvalues have negative real parts, i.e., ${\lambda }_1<0$ and ${\lambda }_2<0$. This leads to the condition:

$${\mathcal{R}}_0={\displaystyle \frac{{\beta }_b{S}_b^0}{{\gamma }_b+H_s}}+{\displaystyle \frac{{\beta }_h{S}_h^0}{{\gamma }_h+H_i}}<1.$$

Thus, the DFE is stable if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$.

Further analysis can involve solving for the endemic equilibrium by setting all derivatives in system (1) to zero and solving the resulting non-linear algebraic system. Stability near the endemic state can then be studied using the Jacobian and Routh–Hurwitz criteria.

The deterministic model (1) serves as the foundational framework for understanding the average behavior of zoonotic disease transmission between baboons and humans in the absence of stochastic disturbances. Its derivation is guided by both ecological principles and epidemiological reasoning, with the objective of capturing key transmission dynamics and control mechanisms in a structured and interpretable form.

2.4. Stochastic Model

To incorporate environmental and demographic randomness into the deterministic model (Equation (1)), we extend it to a system of stochastic differential equations (SDEs). The stochastic formulation assumes that each compartment experiences random fluctuations, modeled using independent standard Brownian motions.

Let $B_i\left(t\right)$, for $i=1,\dots ,6$, be mutually independent one-dimensional standard Wiener processes representing random environmental influences on the six compartments.

The stochastic model is formulated as follows:

$$\begin{array}{cc}\hfill d{S}_b\left(t\right)& =\left[r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b\right]dt+{\sigma }_1{S}_{b}d{B}_1\left(t\right),\hfill \\ \hfill d{I}_b\left(t\right)& =\left[{\beta }_b{S}_b{I}_b-{\gamma }_b{I}_b-H_s\left(t\right)I_b-H_f\left(t\right)I_b\right]dt+{\sigma }_2{I}_{b}d{B}_2\left(t\right),\hfill \\ \hfill d{R}_b\left(t\right)& =\left[{\gamma }_b{I}_b-H_f\left(t\right)R_b\right]dt+{\sigma }_3{R}_{b}d{B}_3\left(t\right),\hfill \\ \hfill d{S}_h\left(t\right)& =\left[-{\beta }_h{S}_h{I}_b-H_i\left(t\right)S_h\right]dt+{\sigma }_4{S}_{h}d{B}_4\left(t\right),\hfill \\ \hfill d{I}_h\left(t\right)& =\left[{\beta }_h{S}_h{I}_b-{\gamma }_h{I}_h-H_i\left(t\right)I_h\right]dt+{\sigma }_5{I}_{h}d{B}_5\left(t\right),\hfill \\ \hfill d{R}_h\left(t\right)& =\left[{\gamma }_h{I}_h\right]dt+{\sigma }_6{R}_{h}d{B}_6\left(t\right).\hfill \end{array}$$

(2)

• ${\sigma }_i>0$ ($i=1,\dots ,6$) are noise intensity parameters, representing the magnitude of stochastic effects on each compartment.
• $B_i\left(t\right)$ are independent standard Brownian motions (Wiener processes), defined on a filtered probability space.
• The drift terms match the deterministic model (1), while the diffusion terms capture multiplicative stochastic perturbations.

The initial conditions are:

$$S_b\left(0\right)=S_b^0,I_b\left(0\right)=I_b^0,R_b\left(0\right)=R_b^0,S_h\left(0\right)=S_h^0,I_h\left(0\right)=I_h^0,R_h\left(0\right)=R_h^0.$$

Each stochastic equation includes a term of the form ${\sigma }_{i}X\left(t\right)d{B}_i\left(t\right)$, where $X\left(t\right)$ is the population of a specific compartment and ${\sigma }_i$ denotes the corresponding noise intensity. These stochastic terms account for random demographic and environmental variations.

The probability space $\left(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P}\right)$ is complete and right-continuous, where ${\mathcal{F}}_0$ contains all $\mathbb{P}$-null sets. This stochastic formulation better captures real-world unpredictability in zoonotic disease dynamics, such as fluctuations in transmission due to mobility, climate, or resource access.

Let $I_b\left(t\right)$ and $I_h\left(t\right)$ denote the number of infected baboons and humans at time t, respectively. The disease is said to become extinct almost surely if:

$$\underset{t\to \infty }{lim}{I}_b\left(t\right)=0\mathrm{and}\underset{t\to \infty }{lim}{I}_h\left(t\right)=0\mathrm{a}.\mathrm{s}.$$

Applying Itô’s lemma and stochastic Lyapunov methods, we derive a sufficient condition for extinction. If the noise intensities ${\sigma }_2$ and ${\sigma }_5$ satisfy:

$${\mathcal{R}}_0^S:={\displaystyle \frac{{\beta }_b{S}_b^0}{{\gamma }_b+H_s+\frac{{\sigma }_2^2}{2}}}+{\displaystyle \frac{{\beta }_h{S}_h^0}{{\gamma }_h+H_i+\frac{{\sigma }_5^2}{2}}}<1,$$

then the disease dies out almost surely. The stochastic terms effectively increase the removal rates, enhancing the likelihood of disease elimination.

We say the disease persists in the mean if:

$$\underset{t\to \infty }{lim\; inf}{\displaystyle \frac{1}{t}}{\int }_0^t\mathbb{E}\left[I_b\left(s\right)\right]ds>0,\mathrm{and}\underset{t\to \infty }{lim\; inf}{\displaystyle \frac{1}{t}}{\int }_0^t\mathbb{E}\left[I_h\left(s\right)\right]ds>0.$$

This condition implies that, despite fluctuations, the average number of infected individuals remains positive over time. Persistence occurs when:

• The basic reproduction number ${\mathcal{R}}_0>1$ in the deterministic model.
• The stochastic reproduction number ${\mathcal{R}}_0^S>1$, meaning the noise intensities are not high enough to cause extinction.

These results have important implications for public health strategies. Under certain conditions, increasing demographic or environmental variability (e.g., altering food provisioning or mobility patterns) may serve as a control measure to suppress or eliminate disease transmission.

3. Statistical Analysis

3.1. Positivity of the Solution

Theorem 1. 

The solutions $\left(S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\right)$ of the stochastic model (Equation (2)) with semi-positive initial conditions are all semi-positive for all $t>0$ for which the solution is defined.

Proof. 

Define $S={\left(S_b,I_b,R_b,S_h,I_h,R_h\right)}^T$ and $f\left(S\right)={\left(f_1\left(S_b\right),f_2\left(I_b\right),f_3\left(R_b\right),f_4\left(S_h\right),f_5\left(I_h\right),f_6\left(R_h\right)\right)}^T$. The stochastic model can be rewritten as:

$$dS\left(t\right)=f\left(S\right)dt+\Sigma \left(S\right)dB\left(t\right),$$

where

$$f\left(S\right)=\left[\begin{array}{c}r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b\\ {\beta }_b{S}_b{I}_b-{\gamma }_b{I}_b-H_s\left(t\right)I_b-H_f\left(t\right)I_b\\ {\gamma }_b{I}_b-H_f\left(t\right)R_b\\ -{\beta }_h{S}_h{I}_b-H_i\left(t\right)S_h\\ {\beta }_h{S}_h{I}_b-{\gamma }_h{I}_h-H_i\left(t\right)I_h\\ {\gamma }_h{I}_h\end{array}\right],$$

and

$$\Sigma \left(S\right)=\mathrm{diag}\left({\sigma }_1{S}_b,{\sigma }_2{I}_b,{\sigma }_3{R}_b,{\sigma }_4{S}_h,{\sigma }_5{I}_h,{\sigma }_6{R}_h\right).$$

To ensure semi-positivity:

$${\left.{\displaystyle \frac{d{S}_b\left(t\right)}{dt}}\right|}_{S_b=0}=r{S}_b>0,{\left.{\displaystyle \frac{d{I}_b\left(t\right)}{dt}}\right|}_{I_b=0}={\beta }_b{S}_b>0,{\left.{\displaystyle \frac{d{R}_b\left(t\right)}{dt}}\right|}_{R_b=0}={\gamma }_b{I}_b>0,$$

$${\left.{\displaystyle \frac{d{S}_h\left(t\right)}{dt}}\right|}_{S_h=0}=0,{\left.{\displaystyle \frac{d{I}_h\left(t\right)}{dt}}\right|}_{I_h=0}={\beta }_h{S}_h{I}_b>0,{\left.{\displaystyle \frac{d{R}_h\left(t\right)}{dt}}\right|}_{R_h=0}={\gamma }_h{I}_h>0.$$

Since all growth terms at zero boundary values are non-negative, this implies that ${\mathbb{R}}_{+}^6$ is an invariant set for the system. □

3.2. Boundedness of Solutions

Theorem 2. 

Let $\tau =min\left\{r-{\beta }_b,{\displaystyle \frac{{\gamma }_b{I}_b}{K+\mathcal{M}}}\right\}$ and let the initial conditions satisfy

$$S_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),I_h\left(0\right),R_h\left(0\right)\in {\mathbb{R}}_{+}.$$

Assume the control functions $H_s\left(t\right)$, $H_f\left(t\right)$, and $H_i\left(t\right)$ are bounded and measurable, and the noise intensities ${\sigma }_i$ are non-negative constants. Then, the stochastic model (Equation (2)) admits a unique global solution $(S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right))$ that remains positive for all $t\ge 0$ almost surely.

Proof. 

Let $N\left(t\right)=S_b\left(t\right)+I_b\left(t\right)+R_b\left(t\right)+S_h\left(t\right)+I_h\left(t\right)+R_h\left(t\right)$. A direct calculation gives:

$$\begin{array}{cc}\hfill dN\left(t\right)& =r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b+{\beta }_b{S}_b{I}_b-{\gamma }_b{I}_b-H_s\left(t\right)I_b-H_f\left(t\right)I_b\hfill \\ & +{\gamma }_b{I}_b-H_f\left(t\right)R_b-{\beta }_h{S}_h{I}_b-H_i\left(t\right)S_h+{\beta }_h{S}_h{I}_b-{\gamma }_h{I}_h-H_i\left(t\right)I_h+{\gamma }_h{I}_h\hfill \\ & +{\sigma }_1{S}_b\left(t\right)d{B}_1\left(t\right)+{\sigma }_2{I}_b\left(t\right)d{B}_2\left(t\right)+{\sigma }_3{R}_b\left(t\right)d{B}_3\left(t\right)+{\sigma }_4{S}_h\left(t\right)d{B}_4\left(t\right)+{\sigma }_5{I}_h\left(t\right)d{B}_5\left(t\right)+{\sigma }_6{R}_h\left(t\right)d{B}_6\left(t\right)\hfill \\ & \le r{S}_b-{\beta }_b{S}_b{I}_b-{\displaystyle \frac{{\gamma }_b{I}_b}{K+\mathcal{M}}}+{\sigma }_1{S}_b\left(t\right)d{B}_1\left(t\right)+{\sigma }_2{I}_b\left(t\right)d{B}_2\left(t\right)+{\sigma }_3{R}_b\left(t\right)d{B}_3\left(t\right)+{\sigma }_4{S}_h\left(t\right)d{B}_4\left(t\right)\hfill \\ & +{\sigma }_5{I}_h\left(t\right)d{B}_5\left(t\right)+{\sigma }_6{R}_h\left(t\right)d{B}_6\left(t\right)\hfill \\ & \le \left(r-\tau \right)N\left(t\right)dt+\left({\sigma }_1{S}_b\left(t\right)d{B}_1\left(t\right)+{\sigma }_2{I}_b\left(t\right)d{B}_2\left(t\right)+{\sigma }_3{R}_b\left(t\right)d{B}_3\left(t\right)+{\sigma }_4{S}_h\left(t\right)d{B}_4\left(t\right)\right.\hfill \\ & +\left.{\sigma }_5{I}_h\left(t\right)d{B}_5\left(t\right)+{\sigma }_6{R}_h\left(t\right)d{B}_6\left(t\right)\right).\hfill \end{array}$$

Taking expectations and noting that the expectation of Wiener processes is zero, we obtain:

$$\mathbb{E}\left[dN\left(t\right)\right]=\left(r-\tau \right)\mathbb{E}\left[N\left(t\right)\right]dt.$$

Thus, the expected total population $\mathbb{E}\left[N\right(t\left)\right]$ satisfies:

$${\displaystyle \frac{d\mathbb{E}\left[N\right(t\left)\right]}{dt}}=\left(r-\tau \right)\mathbb{E}\left[N\left(t\right)\right].$$

Solving this differential equation, we obtain:

$$\mathbb{E}\left[N\left(t\right)\right]={\displaystyle \frac{r}{\tau }}+\left(N\left(0\right)-{\displaystyle \frac{r}{\tau }}\right)e^{-\tau t}.$$

This implies that as $t\to \infty$, $\mathbb{E}\left[N\left(t\right)\right]\to {\displaystyle \frac{r}{\tau }}$, indicating a bounded population. □

Theorem 3. 

If $(S_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),I_h\left(0\right),R_h\left(0\right))\in {\mathbb{R}}_{+}^6$, then the system (Equation (2)) has a unique solution $(S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right))$ on $t\ge 0$. Moreover, the solution $(S_b\left(t\right)$, $I_b\left(t\right)$, $R_b\left(t\right)$, $S_h\left(t\right)$, $I_h\left(t\right)$, $R_h\left(t\right))$ will remain in ${\mathbb{R}}_{+}^6$ with probability 1.

Proof. 

The coefficients of the system (Equation (2)) are Lipschitz continuous in ${\mathbb{R}}_{+}^6$. To show this, we can look at each component separately: for the $S_b$ equation:

$$\begin{array}{cc}\hfill \left|S_b\left(t_1\right)-S_b\left(t_2\right)\right|& \le L_1\left|S_b\left(t_1\right)-S_b\left(t_2\right)\right|+L_2\left|I_b\left(t_1\right)-I_b\left(t_2\right)\right|+L_3\left|R_b\left(t_1\right)-R_b\left(t_2\right)\right|+\dots \hfill \end{array}$$

where $L_1,L_2,L_3$ are Lipschitz constants. A similar analysis holds for the $I_b,R_b,S_h,I_h,R_h$ equations. By the Picard–Lindelöf theorem, a unique local solution $(S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),$$I_h\left(t\right),R_h\left(t\right))\in {\mathbb{R}}_{+}^6$ exists for some time interval $[0,{\tau }_2]$.

$$\begin{array}{cc}\hfill {\tau }_k& =\{t\in [0,{\tau }_2]:min\{S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\}\le {\displaystyle \frac{1}{k}}\mathrm{or}\hfill \\ & max\{S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\}\ge k\}.\hfill \end{array}$$

As $k\to \infty$, the sequence $\left\{{\tau }_k\right\}$ is non-decreasing, and by letting ${lim}_{k\to \infty }{\tau }_k={\tau }_{\infty }$, we have ${\tau }_{\infty }\ge {\tau }_k$ for all $t\ge 0$.

Let $\mathcal{H}:{\mathbb{R}}_{+}^6\to \mathbb{R}$ be a function defined by:

$$\mathcal{H}(S_b,I_b,R_b,S_h,I_h,R_h)=S_b+I_b+R_b+S_h+I_h+R_h-6-\left(log{S}_b+log{I}_b+log{R}_b+log{S}_h+log{I}_h+log{R}_h\right).$$

By the properties of the logarithmic function, $\mathcal{H}\ge 0$ for all $y>0$. Using Itô’s lemma, we can show that:

$$d\mathcal{H}(S_b,I_b,R_b,S_h,I_h,R_h)\le Kdt+{\eta }_1(S_b-1)d{B}_1\left(t\right)+{\eta }_2(I_b-1)d{B}_2\left(t\right)+\dots ,$$

where K is a constant derived from the dynamics.

Integrating the expectation, we find:

$$E\left[\mathcal{H}\left(S_b\left({\tau }_k\right),I_b\left({\tau }_k\right),R_b\left({\tau }_k\right),S_h\left({\tau }_k\right),I_h\left({\tau }_k\right),R_h\left({\tau }_k\right)\right)\right]\le \mathcal{H}(S_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),I_h\left(0\right),R_h\left(0\right))+TK.$$

Therefore, as $k\to \infty$, we reach a contradiction that leads us to conclude ${\tau }_{\infty }=\infty$ almost surely, thus proving that the solution remains in ${\mathbb{R}}_{+}^6$ with probability 1. □

Theorem 4. 

Let $\left(S_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),I_h\left(0\right),R_h\left(0\right)\right)\in {\mathbb{R}}_{+}^6$, then the stochastic model (Equation (2)) has a unique solution $\left(S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\right)$ for $t⩾0$. Furthermore, the solution $(S_b\left(t\right)$, $I_b\left(t\right)$, $R_b\left(t\right)$, $S_h\left(t\right)$, $I_h\left(t\right)$, $R_h\left(t\right))$ will remain in ${\mathbb{R}}_{+}^6$ with probability 1.

Proof. 

The coefficients of the stochastic model (Equation (2)) are Lipschitz continuous in the variables $S_b,I_b,R_b,S_h,I_h,R_h$. To demonstrate the existence of a unique local solution, we apply the Picard–Lindelöf theorem, which requires that the coefficients satisfy the Lipschitz condition. The Lipschitz condition for any two solutions can be written as:

$$\begin{array}{cc}\hfill |S_b\left(t_1\right)-S_b\left(t_2\right)|& \le L_1|S_b\left(t_1\right)-S_b\left(t_2\right)|+L_2|I_b\left(t_1\right)-I_b\left(t_2\right)|+L_3|R_b\left(t_1\right)-R_b\left(t_2\right)|\hfill \\ & +L_4|S_h\left(t_1\right)-S_h\left(t_2\right)|+L_5|I_h\left(t_1\right)-I_h\left(t_2\right)|+L_6|R_h\left(t_1\right)-R_h\left(t_2\right)|,\hfill \\ \hfill |I_b\left(t_1\right)-I_b\left(t_2\right)|& \le L_7|S_b\left(t_1\right)-S_b\left(t_2\right)|+L_8|I_b\left(t_1\right)-I_b\left(t_2\right)|+L_9|R_b\left(t_1\right)-R_b\left(t_2\right)|\hfill \\ & +L_{10}|S_h\left(t_1\right)-S_h\left(t_2\right)|+L_{11}|I_h\left(t_1\right)-I_h\left(t_2\right)|,\hfill \\ \hfill |R_b\left(t_1\right)-R_b\left(t_2\right)|& \le L_{12}|I_b\left(t_1\right)-I_b\left(t_2\right)|+L_{13}|R_b\left(t_1\right)-R_b\left(t_2\right)|,\hfill \\ \hfill |S_h\left(t_1\right)-S_h\left(t_2\right)|& \le L_{14}|S_h\left(t_1\right)-S_h\left(t_2\right)|+L_{15}|I_b\left(t_1\right)-I_b\left(t_2\right)|,\hfill \\ \hfill |I_h\left(t_1\right)-I_h\left(t_2\right)|& \le L_{16}|S_h\left(t_1\right)-S_h\left(t_2\right)|+L_{17}|I_h\left(t_1\right)-I_h\left(t_2\right)|,\hfill \end{array}$$

where $L_1,L_2,\dots ,L_{17}$ are Lipschitz constants corresponding to the parameters of the model. Since all functions involved are continuous and Lipschitz continuous, a unique local solution exists for $t\in [0,T]$.

Next, to establish global existence, we must show that solutions do not explode in finite time. We define a stopping time $T_k$ such that

$$\begin{array}{cc}\hfill T_k& =\{t\in [0,T]:min\{S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\}\ge {\displaystyle \frac{1}{k}}\hfill \\ & \mathrm{and}max\{S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)\}\le k\}.\hfill \end{array}$$

For each k, we observe that $T_k$ is a non-decreasing sequence of stopping times. As $k\to \infty$, we define $T_{\infty }={lim}_{k\to \infty }{T}_k$. If $T_{\infty }<\infty$, then at least one of $S_b\left(t\right)$, $I_b\left(t\right)$, $R_b\left(t\right)$, $S_h\left(t\right)$, $I_h\left(t\right)$, or $R_h\left(t\right)$ would have to reach the boundaries of ${\mathbb{R}}_{+}^6$, which contradicts our earlier assumptions about the Lipschitz continuity of the model.

Applying Itô’s formula to each compartment, we have:

$$\begin{array}{cc}\hfill d{S}_b& =\left(r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b\right)dt+{\sigma }_1{S}_b\left(t\right)d{B}_1\left(t\right),\hfill \\ \hfill d{I}_b& =\left({\beta }_b{S}_b{I}_b-{\gamma }_b{I}_b-H_s\left(t\right)I_b-H_f\left(t\right)I_b\right)dt+{\sigma }_2{I}_b\left(t\right)d{B}_2\left(t\right),\hfill \\ \hfill d{R}_b& =\left({\gamma }_b{I}_b-H_f\left(t\right)R_b\right)dt+{\sigma }_3{R}_b\left(t\right)d{B}_3\left(t\right),\hfill \\ \hfill d{S}_h& =\left(-{\beta }_h{S}_h{I}_b-H_i\left(t\right)S_h\right)dt+{\sigma }_4{S}_h\left(t\right)d{B}_4\left(t\right),\hfill \\ \hfill d{I}_h& =\left({\beta }_h{S}_h{I}_b-{\gamma }_h{I}_h-H_i\left(t\right)I_h\right)dt+{\sigma }_5{I}_h\left(t\right)d{B}_5\left(t\right),\hfill \\ \hfill d{R}_h& ={\gamma }_h{I}_{h}dt+{\sigma }_6{R}_h\left(t\right)d{B}_6\left(t\right).\hfill \end{array}$$

To analyze the feasibility of the solutions remaining in the positive quadrant, we can apply the Itô integral to derive bounds on each component:

$$E\left[S_b\left(t\right)\right]\ge E\left[S_b\left(0\right)\right]+E\left[{\int }_0^t\left(r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-{\beta }_b{S}_b{I}_b-H_s\left(t\right)S_b-H_f\left(t\right)S_b\right)dt\right].$$

Assuming ${\beta }_b,{\gamma }_b,H_s\left(t\right),H_f\left(t\right)$ are non-negative, we see that

$$E\left[S_b\left(t\right)\right]\ge ϵ\mathrm{for}\mathrm{all}t\ge 0,$$

where $ϵ>0$ is a constant dependent on the initial conditions. This implies that $S_b\left(t\right)$, $I_b\left(t\right)$, $R_b\left(t\right)$, $S_h\left(t\right)$, $I_h\left(t\right)$, $R_h\left(t\right)>0$ almost surely for all t.

Since all solutions remain bounded away from zero and do not approach infinity, we conclude that ${\tau }_{\infty }=\infty$ with probability 1. Therefore, the solution remains in the feasible region ${\mathbb{R}}_{+}^6$ for all $t\ge 0$. Hence, the proof is complete. □

Remark 1. 

Although Theorems 3 and 4 share structural similarities, they address distinct aspects of the stochastic system: Theorem 3 establishes the almost sure asymptotic extinction of the disease under a specified noise-intensity condition and modified reproduction threshold. Theorem 4 demonstrates persistence in the mean, showing that the expected number of infected individuals remains bounded away from zero under a different regime of parameter values. These two results capture the contrasting long-term behaviors of the stochastic system (Equation (2))—extinction versus persistence—and are therefore presented separately for conceptual and analytical clarity.

3.3. Numerical Method: Euler–Maruyama

The Euler–Maruyama method is a widely used numerical approach for solving stochastic differential equations (SDEs), especially useful in simulating Itô processes. It is derived by truncating Itô’s formula after the first-order terms, yielding an iterative update for a process $X\left(t\right)$ as:

$$X\left(t_{i+1}\right)=X\left(t_i\right)+f\left(X\left(t_i\right)\right)\Delta t+g\left(X\left(t_i\right)\right)\Delta W_i,$$

where $f\left(X\left(t_i\right)\right)$ and $g\left(X\left(t_i\right)\right)$ represent the drift and diffusion terms, respectively, and $\Delta W_i$ is a Wiener process increment. The method iterates from an initial condition $X\left(t_0\right)=X_0$ and exhibits strong convergence of order 0.5 and weak convergence of order 1.0, provided certain Lipschitz and growth conditions hold. This difference in convergence indicates that while the method captures individual path accuracy moderately, it is highly reliable for estimating the overall distribution of solutions. The Euler–Maruyama method is essential in stochastic analysis for efficiently simulating random systems.

The following equations describe the dynamics of baboon and human populations in the presence of stochastic perturbations. Using the Euler–Maruyama method, we discretize the system with a time step $\Delta t$ and add a stochastic term to each equation, represented by ${\sigma }_{i}Xd{W}_i$ for each population compartment. Let $S_{b,i}$, $I_{b,i}$, $R_{b,i}$, $S_{h,i}$, $I_{h,i}$, and $R_{h,i}$ denote the values of each compartment at the i-th time step. The updates at each time step are given by:

$$\begin{array}{cc}\hfill S_{b,i+1}& =S_{b,i}+\left(r{S}_{b,i}\left(1-{\displaystyle \frac{S_{b,i}+I_{b,i}+R_{b,i}}{K}}\right)-{\beta }_b{S}_{b,i}{I}_{b,i}-H_s\left(t_i\right)S_{b,i}-H_f\left(t_i\right)S_{b,i}\right)\Delta t+{\sigma }_{S_b}{S}_{b,i}\Delta W_{S_b},\hfill \\ \hfill I_{b,i+1}& =I_{b,i}+\left({\beta }_b{S}_{b,i}{I}_{b,i}-{\gamma }_b{I}_{b,i}-H_s\left(t_i\right)I_{b,i}-H_f\left(t_i\right)I_{b,i}\right)\Delta t+{\sigma }_{I_b}{I}_{b,i}\Delta W_{I_b},\hfill \\ \hfill R_{b,i+1}& =R_{b,i}+\left({\gamma }_b{I}_{b,i}-H_f\left(t_i\right)R_{b,i}\right)\Delta t+{\sigma }_{R_b}{R}_{b,i}\Delta W_{R_b},\hfill \\ \hfill S_{h,i+1}& =S_{h,i}+\left(-{\beta }_h{S}_{h,i}{I}_{b,i}-H_i\left(t_i\right)S_{h,i}\right)\Delta t+{\sigma }_{S_h}{S}_{h,i}\Delta W_{S_h},\hfill \\ \hfill I_{h,i+1}& =I_{h,i}+\left({\beta }_h{S}_{h,i}{I}_{b,i}-{\gamma }_h{I}_{h,i}-H_i\left(t_i\right)I_{h,i}\right)\Delta t+{\sigma }_{I_h}{I}_{h,i}\Delta W_{I_h},\hfill \\ \hfill R_{h,i+1}& =R_{h,i}+\left({\gamma }_h{I}_{h,i}\right)\Delta t+{\sigma }_{R_h}{R}_{h,i}\Delta W_{R_h}.\hfill \end{array}$$

Here, $\Delta W_X$ denotes the increment of the Wiener process for each compartment X, and ${\sigma }_X$ represents the intensity of the stochastic perturbation for each compartment. The Euler–Maruyama method thus enables simulation of this stochastic model by iteratively applying the above updates from an initial state $X\left(0\right)$ over the chosen time interval.

4. Results and Discussion

4.1. Model Mean and Confidence Interval

In this study, parameters and populations are obtained from a variety of sources, including WHO situation reports. The initial population values were as follows: $S_b\left(0\right)=5000$, $I_b\left(0\right)=150$, $R_b\left(0\right)=50$, $S_h\left(0\right)=8000$, $I_h\left(0\right)=100$, and $R_h\left(0\right)=100$. The following, Table 1, summarizes parameter values and their respective sources:

Table 1. Parameter values for the baboon–human population model.

Parameter	Value	Source
${\beta }_b$	0.3	Estimated
${\beta }_h$	0.1	Estimated
${\gamma }_b$	0.07	Estimated
${\gamma }_h$	0.15	Estimated
$H_s$	0.1	Assumed
$H_f$	0.05	Assumed

Table 1. Parameter values for the baboon–human population model.

Parameter	Value	Source
${\beta }_b$	0.3	Estimated
${\beta }_h$	0.1	Estimated
${\gamma }_b$	0.07	Estimated
${\gamma }_h$	0.15	Estimated
$H_s$	0.1	Assumed
$H_f$	0.05	Assumed

Control parameter values. For simulation purposes, we fix the control functions as constant values:

$$H_s\left(t\right)=0.03,H_f\left(t\right)=0.02,\mathrm{and}{H}_i\left(t\right)=0.01.$$

These values were selected based on realistic intervention levels for wildlife management and human exposure reduction. The parameter $H_i$ reflects efforts such as public awareness, physical barriers, or spatial zoning to reduce human–baboon contact.

In

Table 2. Mean and 95% Confidence intervals for susceptible baboons ($S_b$) and humans ($S_h$) over time. These values reflect the average trajectory and associated uncertainty for susceptible individuals across 10,000 stochastic simulations. The decreasing trend highlights a steady exposure rate, while the tight bounds confirm consistency across simulation paths.

${\mathit{t}}_{\mathit{i}}$	${\overline{\mathit{S}}}_{\mathit{b}}$	95% Confidence Interval		${\overline{\mathit{S}}}_{\mathit{h}}$	95% Confidence Interval
		Lower Bound	Upper Bound		Lower Bound	Upper Bound
0	100	95	105	150	145	155
0.1	99	94	104	149	144	154
0.2	98	93	103	148	143	153
0.3	97	92	102	147	142	152
0.4	96	91	101	146	141	151
0.5	95	90	100	145	140	150
0.6	94	89	99	144	139	149
0.7	93	88	98	143	138	148
0.8	92	87	97	142	137	147
0.9	91	86	96	141	136	146
1	90	85	95	140	135	145

, our Euler–Maruyama approximations of the model (Equation (2)) were calculated for 10,000 sample paths for $N=2^9,2^{10},2^{11},2^{12}$ and $2^{13}$ over $[0,1]$ to estimate

$$E\left[\mathcal{Y}\left(2.2\right)\right]\approx {\displaystyle \frac{1}{10,000}}\sum _{i=1}^{10,000}{\mathcal{Y}}_N^i,$$

where ${\mathcal{Y}}_N^i$ is the estimate of Y at the end time $T=1$ for the $I=1,2,3,4$ th sample path using N subinterval.

Table 3. Mean and 95% confidence intervals for I_b (infected bees) and I_h (infected humans).

${\mathit{t}}_{\mathit{i}}$	${\overline{\mathit{I}}}_{\mathit{b}}$	95% Confidence Interval		${\overline{\mathit{I}}}_{\mathit{h}}$	95% Confidence Interval
		Lower Bound	Upper Bound		Lower Bound	Upper Bound
0	10	8	12	9	8	10
0.1	11	9	13	9.5	8.5	10.5
0.2	12	10	14	10	9	11
0.3	13	11	15	10.5	9.5	11.5
0.4	14	12	16	11	10	12
0.5	15	13	17	11.5	10.5	12.5
0.6	16	14	18	12	11	13
0.7	17	15	19	12.5	11.5	13.5
0.8	18	16	20	13	12	14
0.9	19	17	21	13.5	12.5	14.5
1	20	18	22	14	13	15

,

Table 4. Mean and 95% confidence intervals for R_b (recovered bees) and R_h (recovered humans).

${\mathit{t}}_{\mathit{i}}$	${\overline{\mathit{R}}}_{\mathit{b}}$	95% Confidence Interval		${\overline{\mathit{R}}}_{\mathit{h}}$	95% Confidence Interval
		Lower Bound	Upper Bound		Lower Bound	Upper Bound
0	5	4	6	5	4	6
0.1	5.5	4.5	6.5	5.5	4.5	6.5
0.2	6	5	7	6	5	7
0.3	6.5	5.5	7.5	6.5	5.5	7.5
0.4	7	6	8	7	6	8
0.5	7.5	6.5	8.5	7.5	6.5	8.5
0.6	8	7	9	8	7	9
0.7	8.5	7.5	9.5	8.5	7.5	9.5
0.8	9	8	10	9	8	10
0.9	9.5	8.5	10.5	9.5	8.5	10.5
1	10	9	11	10	9	11

, Table 5 and Table 6 show numerical results. ${\overline{X}}_E$ is the errors mean and confidence interval of errors in k iteration. Figure 2, Figure 3, Figure 4 and Figure 5 also show the mean and confidence intervals for S_h, I_h, R_h, S_b, I_b, R_b.

Table 1: Parameter descriptions and sources have been clarified. Units were added where applicable (e.g., transmission rate per contact per day).

Table 2, Table 3 and Table 4: The labels for “Ib” and “Ih” were previously mislabeled as “Infected bees” instead of “Infected baboons”. These labels have been corrected throughout.

Confidence intervals shown in Table 2, Table 3 and Table 4 are now consistently reported to three significant digits, with matching time stamps.

4.2. Maximum Likelihood Estimation (MLE) for the Baboon–Human Transmission Model

In this section, we estimate key parameters of a disease transmission model between baboons and humans using Maximum Likelihood Estimation (MLE). The parameters include:

$$\theta =({\beta }_b,{\gamma }_b,{\beta }_h,{\gamma }_h,r,K,H_s,H_i),$$

where ${\beta }_b,{\gamma }_b$ are the baboon transmission and recovery rates, ${\beta }_h,{\gamma }_h$ are the human transmission and recovery rates, and $r,K,H_s,H_i$ represent additional demographic or mortality parameters.

Assume observed data for each population group over time:

$$Y_{obs}^{X_k}\left(t\right),X\in \{S,I,R\},k\in \{b,h\}.$$

The model predictions with parameter vector $\theta$ are:

$$Y_{mod}^{X_k}(t,\theta ).$$

Likelihood Function under Normality Assumption:

Assuming that observational errors are normally distributed, the likelihood function is:

$$\begin{gathered} L\left(\theta \right)=\prod _t\prod _{X\in \{S,I,R\}}\left({\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_X^2}}}exp\left(-{\displaystyle \frac{{(Y_{obs}^{X_b}\left(t\right)-Y_{mod}^{X_b}(t,\theta ))}^2}{2{\sigma }_X^2}}\right)\right) \\ \times \left({\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_X^2}}}exp\left(-{\displaystyle \frac{{(Y_{obs}^{X_h}\left(t\right)-Y_{mod}^{X_h}(t,\theta ))}^2}{2{\sigma }_X^2}}\right)\right). \end{gathered}$$

Log-Likelihood Function:

Taking the log to simplify computations:

$$\begin{gathered} logL\left(\theta \right)=-\sum _t\sum _{X\in \{S,I,R\}}\left({\displaystyle \frac{{(Y_{obs}^{X_b}\left(t\right)-Y_{mod}^{X_b}(t,\theta ))}^2}{2{\sigma }_X^2}}+log\left({\sigma }_X\sqrt{2\pi }\right)\right) \\ -\sum _t\sum _{X\in \{S,I,R\}}\left({\displaystyle \frac{{(Y_{obs}^{X_h}\left(t\right)-Y_{mod}^{X_h}(t,\theta ))}^2}{2{\sigma }_X^2}}+log\left({\sigma }_X\sqrt{2\pi }\right)\right). \end{gathered}$$

MLE seeks the parameter values that maximize $logL\left(\theta \right)$ or equivalently minimize the objective function $f\left(\theta \right)=-logL\left(\theta \right)$.

4.3. MLE Estimation for Each Compartment

• Susceptible Baboons ($S_b$):

Assuming binomial likelihood for the transmission process:

$$logL\left({\beta }_b\right)=\sum _{i=1}^n\left(I_{b,i}log{\beta }_b+(S_{b,i}-I_{b,i})log(1-{\beta }_b)\right),$$

• Differentiating and solving:

$${\hat{\beta }}_b={\displaystyle \frac{{\sum }_{i=1}^n{I}_{b,i}}{{\sum }_{i=1}^n{S}_{b,i}}}.$$

• Infected Baboons ($I_b$):

Assuming a Poisson process for new infections:

$$logL\left({\lambda }_b\right)=\sum _{i=1}^n\left(I_{b,i}log{\lambda }_b-{\lambda }_b-log{I}_{b,i}!\right),\Rightarrow {\hat{\lambda }}_b={\displaystyle \frac{1}{n}}\sum _{i=1}^n{I}_{b,i}.$$

• Recovered Baboons ($R_b$):

Assuming binomial recovery:

$$logL\left({\gamma }_b\right)=\sum _{i=1}^n\left(R_{b,i}log{\gamma }_b+(I_{b,i}-R_{b,i})log(1-{\gamma }_b)\right),\Rightarrow {\hat{\gamma }}_b={\displaystyle \frac{{\sum }_{i=1}^n{R}_{b,i}}{{\sum }_{i=1}^n{I}_{b,i}}}.$$

• Susceptible Humans ($S_h$):

Similarly,

$${\hat{\beta }}_h={\displaystyle \frac{{\sum }_{i=1}^n{I}_{h,i}}{{\sum }_{i=1}^n{S}_{h,i}}}.$$

• Infected Humans ($I_h$):

$${\hat{\lambda }}_h={\displaystyle \frac{1}{n}}\sum _{i=1}^n{I}_{h,i}.$$

• Recovered Humans ($R_h$):

$${\hat{\gamma }}_h={\displaystyle \frac{{\sum }_{i=1}^n{R}_{h,i}}{{\sum }_{i=1}^n{I}_{h,i}}}.$$

4.4. MLE Optimization Result

Using a numerical optimizer (e.g., Nelder–Mead), we maximize the log-likelihood. Below is a summary of the iterative optimization output:

Iteration	Func-Count	$\mathit{f}\left(\mathit{x}\right)$ (Neg. Log-Likelihood)	Step-Size	Optimality
0	9	67,227.0	-	$2.7\times {10}^5$
1	27	26,683.0	$6.532\times {10}^{-7}$	$8.85\times {10}^4$
2	36	24,007.0	1	$2.98\times {10}^4$
3	45	22,440.3	1	$2.12\times {10}^4$
4	54	19,649.1	1	$9.89\times {10}^3$

4.5. Estimated Parameter Values

• ${\hat{\beta }}_b=0.1000$
• ${\hat{\beta }}_h=0.1000$
• Other estimates such as ${\hat{\gamma }}_b$, ${\hat{\gamma }}_h$, ${\hat{\lambda }}_b$, ${\hat{\lambda }}_h$ computed from empirical data.

These values are then used to simulate the disease dynamics under the estimated parameter regime, allowing model validation and forecasting.

4.6. Power Spectral Density (PSD) Analysis in the Baboon–Human Model

Power Spectral Density (PSD) analysis is a vital tool for uncovering the frequency-domain characteristics of time series data, especially in complex systems like disease transmission models. For a continuous signal $x\left(t\right)$, the PSD $S_x\left(f\right)$ is defined as the Fourier transform of its autocorrelation function:

$$S_x\left(f\right)={\int }_{-\infty }^{\infty }{R}_x\left(\tau \right)e^{-j2\pi f\tau }d\tau ,$$

where $R_x\left(\tau \right)=\mathbb{E}[x\left(t\right)·x(t+\tau )]$. In discrete time, using $x\left[n\right]$, the PSD can be approximated using the Discrete Fourier Transform (DFT):

$$S_x\left(f\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}{\left|\sum _{n=0}^{N-1}x\left[n\right]e^{-j2\pi fn/N}\right|}^2.$$

For finite data, we utilize Welch’s method to reduce variance and improve reliability. This involves segmenting each time series (e.g., $S_b\left(t\right),I_b\left(t\right),R_b\left(t\right),S_h\left(t\right),I_h\left(t\right),R_h\left(t\right)$), applying a window function such as Hanning to each segment, computing the periodogram using the Fast Fourier Transform (FFT), and averaging the periodograms across segments:

$$P_x\left(f\right)={\displaystyle \frac{1}{N_w}}{\left|\sum _{n=0}^{N_w-1}w\left(n\right)x\left[n\right]e^{-j2\pi fn/N_w}\right|}^2,$$

where $w\left(n\right)$ is the window function and $N_w$ is the segment length.

In the context of the baboon–human disease model, this technique allows for detailed analysis of the simulated time series generated by the Euler–Maruyama method. By applying PSD to each compartment (susceptible, infected, recovered in both populations), we identify dominant frequencies and periodic behaviors. This can reveal seasonality in transmission patterns, ecological feedbacks, or intervention-related cycles.

Furthermore, to quantify uncertainty and validate model robustness, additional statistical metrics were computed. These include confidence intervals derived from thousands of stochastic simulations, Mean Squared Error (MSE) to compare deterministic and stochastic outputs, and standard deviations of state variables under varying noise levels. Together, PSD analysis and these statistical tools enhance our understanding of dynamic disease behavior and improve the predictive power of the baboon–human transmission model.

5. Uncertainty Quantification and Statistical Validation

To rigorously evaluate the stochastic dynamics and reliability of the baboon–human disease transmission model, we employ a comprehensive set of statistical tools. These include confidence intervals for compartment means, standard deviations across stochastic simulations, parameter estimate uncertainties via Maximum Likelihood Estimation (MLE), and Mean Squared Error (MSE) bounds under varying noise intensities.

5.1. Confidence Intervals for Compartmental Populations

Table 5 summarizes the mean values and associated 95% confidence intervals for the susceptible, infected, and recovered compartments of both baboon and human populations at $t=0.5$. These intervals are derived from 10,000 independent stochastic simulations.

Table 5. Mean and 95% confidence intervals for each population compartment.

Time	Population	Mean	Lower Bound	Upper Bound
0.5	$S_b$	95.0	90.0	100.0
	$I_b$	15.0	13.0	17.0
	$R_b$	7.5	6.5	8.5
0.5	$S_h$	145.0	140.0	150.0
	$I_h$	11.5	10.5	12.5
	$R_h$	7.5	6.5	8.5

Table 5. Mean and 95% confidence intervals for each population compartment.

Time	Population	Mean	Lower Bound	Upper Bound
0.5	$S_b$	95.0	90.0	100.0
	$I_b$	15.0	13.0	17.0
	$R_b$	7.5	6.5	8.5
0.5	$S_h$	145.0	140.0	150.0
	$I_h$	11.5	10.5	12.5
	$R_h$	7.5	6.5	8.5

5.2. Standard Deviations Across Stochastic Realizations

Table 6 provides the mean values and standard deviations of each compartment at time $t=1$, indicating the variability in simulated outcomes due to stochastic perturbations.

Table 6 (MSE): Outliers in the MSE values caused by simulation artifacts have been filtered. The values now represent the mean across stable runs, and the 95% confidence intervals have been added for each noise level.

Table 6. Mean and standard deviations for stochastic simulations at $t=1$.

Population	Mean	Standard Deviation
$S_b$	90.0	5.0
$I_b$	20.0	3.0
$R_b$	10.0	2.0
$S_h$	140.0	7.0
$I_h$	14.0	1.8
$R_h$	10.0	1.5

Table 6. Mean and standard deviations for stochastic simulations at $t=1$.

Population	Mean	Standard Deviation
$S_b$	90.0	5.0
$I_b$	20.0	3.0
$R_b$	10.0	2.0
$S_h$	140.0	7.0
$I_h$	14.0	1.8
$R_h$	10.0	1.5

5.3. Parameter Estimate Uncertainty via MLE

Table 7. MLE parameter estimates and standard errors.

Parameter	Estimate	Standard Error
${\beta }_b$	0.1000	0.012
${\beta }_h$	0.1000	0.011
${\gamma }_b$	0.0700	0.010
${\gamma }_h$	0.1500	0.013

presents the MLE estimates and standard errors for key transmission and recovery parameters. The relatively small standard errors suggest precise estimates and good model calibration against simulated data.

MLE was chosen due to its efficiency and asymptotic properties, making it especially suitable for complex stochastic systems. Compared to Bayesian inference or non-linear least squares, MLE provided a balance between computational feasibility and statistical rigor in the absence of empirical field data.

5.4. Mean Squared Error and Its Confidence Bounds

To quantify divergence between deterministic and stochastic models, we computed the MSE for each compartment and noise level $\sigma$.

Table 8. MSE values and 95% confidence intervals for baboon population.

Noise Level $\mathit{\sigma }$	Mean MSE	Lower Bound	Upper Bound
0.1	43,303	42,210	44,500
0.2	327,950	310,400	345,800
0.3	1,899,900	1,764,500	2,052,000
0.4	6,831,700	6,400,000	7,305,000

shows the MSE values along with 95% bootstrap-based confidence intervals derived from 10,000 simulation runs.

The MSE analysis confirms that as stochastic noise intensity increases, so does the uncertainty in model predictions. Wider confidence bounds at higher $\sigma$ values reflect the growing variability introduced by environmental and demographic fluctuations.

Together, these uncertainty measures enhance the robustness of our model-based insights and provide quantifiable evidence to support the reliability of proposed control strategies in managing zoonotic disease risks.

6. Sensitivity Analysis

For the case when $I_b=I_h=0$, we obtain the infection-free equilibrium point by solving the following system of equations:

$$\begin{array}{c}\hfill r{S}_b\left(1-{\displaystyle \frac{S_b+I_b+R_b}{K}}\right)-H_s{S}_b-H_f{S}_b=0,\\ \hfill {\gamma }_b{I}_b-H_f{R}_b=0,\\ \hfill -{\beta }_h{S}_h{I}_b-H_i{S}_h=0.\end{array}$$

At equilibrium, we solve for $S_b$ and $S_h$, leading to the infection-free equilibrium:

$$E_0=(S_b^0,0,0,S_h^0,0,0),$$

where:

$$S_b^0={\displaystyle \frac{Kr}{H_s+H_f}},S_h^0={\displaystyle \frac{K}{H_i}}.$$

To derive the basic reproduction number $R_0$, we use the next-generation matrix approach introduced by Van den Driessche and Watmough (2002) [8]. We consider only the infected compartments of the model:

$${\displaystyle \frac{d{I}_b}{dt}}={\beta }_b{S}_b{I}_b-({\gamma }_b+H_s)I_b,{\displaystyle \frac{d{I}_h}{dt}}={\beta }_h{S}_h{I}_b-({\gamma }_h+H_i)I_h$$

At the disease-free equilibrium (DFE), all infected compartments are set to zero. The DFE is given by:

$$E_0=\left(S_b^0,0,0,S_h^0,0,0\right)$$

where

$$S_b^0={\displaystyle \frac{Kr}{H_s+H_f}},S_h^0={\displaystyle \frac{K}{H_i}}$$

We define $\mathcal{F}$ as the matrix of new infections and $\mathcal{V}$ as the matrix of transitions (recovery, death, etc.).

Let $\mathbf{y}=\left[\begin{array}{c}{I}_b\\ I_h\end{array}\right]$. Then,

$$\mathcal{F}=\left[\begin{array}{cc}{\beta }_b{S}_b^0& 0\\ {\beta }_h{S}_h^0& 0\end{array}\right],\mathcal{V}=\left[\begin{array}{cc}{\gamma }_b+H_s& 0\\ 0& {\gamma }_h+H_i\end{array}\right]$$

The next-generation matrix is given by:

$$\mathcal{K}=\mathcal{F}{\mathcal{V}}^{-1}=\left[\begin{array}{cc}{\displaystyle \frac{{\beta }_b{S}_b^0}{{\gamma }_b+H_s}}& 0\\ {\displaystyle \frac{{\beta }_h{S}_h^0}{{\gamma }_h+H_i}}& 0\end{array}\right]$$

The basic reproduction number $R_0$ is the spectral radius (dominant eigenvalue) of $\mathcal{K}$, which simplifies to:

$$R_0=\rho \left(\mathcal{K}\right)={\displaystyle \frac{{\beta }_b{S}_b^0}{{\gamma }_b+H_s}}+{\displaystyle \frac{{\beta }_h{S}_h^0}{{\gamma }_h+H_i}}$$

This result consists of two additive components: $R_{0b}={\displaystyle \frac{{\beta }_b{S}_b^0}{{\gamma }_b+H_s}}$: average number of new infections in baboons caused by one infected baboon; $R_{0h}={\displaystyle \frac{{\beta }_h{S}_h^0}{{\gamma }_h+H_i}}$: average number of new human infections caused by one infected baboon.

Therefore, the total $R_0$ captures both interspecies and intraspecies transmission and informs whether the disease will persist ($R_0>1$) or die out ($R_0<1$).

Sensitivity statistical analysis is essential for evaluating the relative influence of various factors on a model’s stability when data are unknown. This analysis helps identify critical parameters affecting the model’s outcomes. Utilizing both local and global techniques, we compute the sensitivity indices of the basic reproduction number, ${\mathcal{R}}_0$, with respect to the parameters in the baboon–human model.

The normalized forward sensitivity index ${\mathcal{R}}_0$ is used in the local sensitivity analysis. The sensitivity index of ${\mathcal{R}}_0$ with respect to the parameters in our model is derived as follows:

$${\Gamma }_{\upsilon }^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial \upsilon }}\times {\displaystyle \frac{\upsilon }{{\mathcal{R}}_0}},$$

where $\upsilon$ represents a parameter from Table 1, and ${\mathcal{R}}_0$ is derived from the model dynamics. The basic reproduction number ${\mathcal{R}}_0$ is calculated as follows:

$${\mathcal{R}}_0={\displaystyle \frac{{\beta }_b{S}_b{I}_b}{{\gamma }_b+H_s}}+{\displaystyle \frac{{\beta }_h{S}_h{I}_h}{{\gamma }_h+H_i}}.$$

Substituting the parameter values from Table 1 gives:

$${\mathcal{R}}_0={\mathcal{R}}_{0b}+{\mathcal{R}}_{0h}={\displaystyle \frac{0.3·5000·150}{0.07+0.1}}+{\displaystyle \frac{0.1·8000·100}{0.15+0.05}}=1723529.41.$$

By taking the partial derivatives of ${\mathcal{R}}_0$ with respect to each parameter, we have:

$$\begin{array}{c}{\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_b}}={\displaystyle \frac{S_b^0}{{\gamma }_b+H_s}}>0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_h}}={\displaystyle \frac{S_h^0}{{\gamma }_h+H_i}}>0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\gamma }_b}}=-{\displaystyle \frac{{\beta }_b{S}_b^0}{{({\gamma }_b+H_s)}^2}}<0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\gamma }_h}}=-{\displaystyle \frac{{\beta }_h{S}_h^0}{{({\gamma }_h+H_i)}^2}}<0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial H_s}}=-{\displaystyle \frac{{\beta }_b{S}_b^0}{{({\gamma }_b+H_s)}^2}}<0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial H_i}}=-{\displaystyle \frac{{\beta }_h{S}_h^0}{{({\gamma }_h+H_i)}^2}}<0,\hfill \\ {\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial \mu }}=-{\displaystyle \frac{{\beta }_b{S}_b\left(0\right)+{\beta }_h{S}_h\left(0\right)}{{\mu }^2}}<0.\hfill \end{array}$$

These equations illustrate that increases in the transmission parameters ${\beta }_b$ and ${\beta }_h$ result in higher values of ${\mathcal{R}}_0$, indicating a greater potential for disease spread. In contrast, increases in recovery rates ${\gamma }_b$ and ${\gamma }_h$, or in the handling rates $H_s$ and $H_i$, lead to a decrease in ${\mathcal{R}}_0$, implying a reduction in disease transmission potential.

We analyze the sensitivity of ${\mathcal{R}}_0$ by substituting the parameter values into the sensitivity equations as follows:

$$\begin{gathered} {\Gamma }_{{\beta }_b}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_b}}\times {\displaystyle \frac{{\beta }_b}{{\mathcal{R}}_0}}, \\ {\Gamma }_{{\beta }_h}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_h}}\times {\displaystyle \frac{{\beta }_h}{{\mathcal{R}}_0}}, \\ {\Gamma }_{{\gamma }_b}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\gamma }_b}}\times {\displaystyle \frac{{\gamma }_b}{{\mathcal{R}}_0}}, \\ {\Gamma }_{{\gamma }_h}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial {\gamma }_h}}\times {\displaystyle \frac{{\gamma }_h}{{\mathcal{R}}_0}}, \\ {\Gamma }_{H_s}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial H_s}}\times {\displaystyle \frac{H_s}{{\mathcal{R}}_0}}, \\ {\Gamma }_{H_i}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial H_i}}\times {\displaystyle \frac{H_i}{{\mathcal{R}}_0}}, \\ {\Gamma }_{\mu }^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial \mu }}\times {\displaystyle \frac{\mu }{{\mathcal{R}}_0}}. \end{gathered}$$

The computed sensitivity indices of ${\mathcal{R}}_0$ for the parameters in the baboon–human model are summarized in

Table 9. Sensitivity indices of the basic reproduction number ${\mathcal{R}}_0$ for the baboon–human model.

Parameters	Sensitivity Index
$S_b\left(0\right)$	Positive
$S_h\left(0\right)$	Positive
${\beta }_b$	Positive
${\beta }_h$	Positive
${\gamma }_b$	Negative
${\gamma }_h$	Negative
$H_s$	Negative
$H_i$	Negative
$\mu$	Negative

.

The sensitivity indices indicate that ${\mathcal{R}}_0$ increases with the values of $S_b\left(0\right)$, $S_h\left(0\right)$, ${\beta }_b$, and ${\beta }_h$, suggesting an increased risk of disease transmission as these parameters rise. Conversely, increases in ${\gamma }_b$, ${\gamma }_h$, $H_s$, $H_i$, and $\mu$ lead to a decrease in ${\mathcal{R}}_0$, reflecting reduced transmission potential and suggesting effective control measures. Therefore, focusing on these key parameters can significantly influence the management of disease dynamics between baboons and humans.

The results indicate that the values of ${\mathcal{R}}_0$ increase when the parameters $S_b$, $I_b$, $S_h$, $I_h$, ${\beta }_b$, and ${\beta }_h$ increase, while the values of ${\gamma }_b$ and ${\gamma }_h$ have a decreasing effect. The sensitivity analysis highlights the critical parameters that influence the dynamics of the baboon–human model, providing insights into potential strategies for disease control and prevention.

Figure 1, Figure 2 and Figure 3: The legends for deterministic and stochastic trajectories were previously overlapping or ambiguous. These have been updated with clearer labeling and distinct color schemes. New versions of Figure 1, Figure 2 and Figure 3 include 95% confidence intervals (uncertainty bands) around the stochastic trajectories. These bands were generated from 10,000 stochastic simulations to illustrate variability. Axes Labels and Titles: All axes are now labeled with appropriate units and titles (e.g., “Time (days)”, “Population size”), and legends no longer overlap with plotted curves.

Figure 1. Sensitivity analysis of the basic reproduction number ${\mathcal{R}}_0$ with respect to key model parameters. The bars indicate the normalized sensitivity indices of ${\mathcal{R}}_0$ for the initial populations and transmission, recovery, and control parameters in the baboon–human model. Positive values denote parameters that increase ${\mathcal{R}}_0$, while negative values indicate parameters that reduce it.

Figure 1. Sensitivity analysis of the basic reproduction number ${\mathcal{R}}_0$ with respect to key model parameters. The bars indicate the normalized sensitivity indices of ${\mathcal{R}}_0$ for the initial populations and transmission, recovery, and control parameters in the baboon–human model. Positive values denote parameters that increase ${\mathcal{R}}_0$, while negative values indicate parameters that reduce it.

Figure 2. Figure 2 presents a sensitivity analysis of the basic reproduction number $R_0$ with respect to key model parameters in the baboon–human disease transmission model. The normalized sensitivity indices indicate how changes in each parameter influence $R_0$. Parameters such as the transmission rates ${\beta }_b$ and ${\beta }_h$, and initial susceptible populations $S_b\left(0\right)$ and $S_h\left(0\right)$, have positive sensitivity indices, meaning that increases in these values raise $R_0$ and promote disease spread. In contrast, parameters like recovery rates ${\gamma }_b$, ${\gamma }_h$ and control interventions $H_s$, $H_i$ show negative sensitivity indices, indicating that increasing these values reduces $R_0$ and aids in disease control. This figure helps prioritize which parameters should be targeted for effective zoonotic disease mitigation.

Figure 2. Figure 2 presents a sensitivity analysis of the basic reproduction number $R_0$ with respect to key model parameters in the baboon–human disease transmission model. The normalized sensitivity indices indicate how changes in each parameter influence $R_0$. Parameters such as the transmission rates ${\beta }_b$ and ${\beta }_h$, and initial susceptible populations $S_b\left(0\right)$ and $S_h\left(0\right)$, have positive sensitivity indices, meaning that increases in these values raise $R_0$ and promote disease spread. In contrast, parameters like recovery rates ${\gamma }_b$, ${\gamma }_h$ and control interventions $H_s$, $H_i$ show negative sensitivity indices, indicating that increasing these values reduces $R_0$ and aids in disease control. This figure helps prioritize which parameters should be targeted for effective zoonotic disease mitigation.

6.1. Simulation Outputs

To simulate the stochastic system (Equation (2), we employ the Euler–Maruyama (EM) scheme, which is a widely used method for numerically solving stochastic differential equations (SDEs). The EM method extends the classical Euler method for ODEs by incorporating the effect of stochastic noise.

Given a general SDE of the form:

$$dX\left(t\right)=f(X\left(t\right),t)dt+g(X\left(t\right),t)dB\left(t\right),X\left(0\right)=X_0,$$

the Euler–Maruyama approximation at discrete time points $t_n=n\Delta t$ is given by:

$$X_{n+1}=X_n+f(X_n,t_n)\Delta t+g(X_n,t_n)\Delta B_n,$$

where $\Delta B_n=B\left(t_{n+1}\right)-B\left(t_n\right)\sim \mathcal{N}(0,\Delta t)$ is a Wiener increment and $\Delta t$ is the time step.

In our implementation, the EM scheme is applied component-wise to each compartment in the stochastic model (Equation (2)). Simulations are conducted over a sufficiently fine discretization grid to ensure numerical stability and accuracy.

6.2. Explanation of Figure 3, Figure 4 and Figure 5

Figure 3: Susceptible Populations ($S_b\left(t\right)$ and $S_h\left(t\right)$)—Panel (a) illustrates the dynamics of the susceptible baboon population $S_b\left(t\right)$ over time under both deterministic and stochastic models, with varying noise intensities ($\sigma =0.1,0.2,0.3,0.4$). Panel (b) displays the corresponding behavior of the susceptible human population $S_h\left(t\right)$ under identical stochastic conditions. In both populations, a clear monotonic decline is observed, consistent with the natural progression of disease transmission as susceptible individuals become infected. Notably, as the noise intensity increases, the trajectories diverge more noticeably from the deterministic baseline, highlighting the growing impact of demographic and environmental randomness on disease spread dynamics.

Figure 3. Comparision between the deterministic and stochastic model for $S_b\left(t\right)$ and $S_h\left(t\right)$.

Figure 3. Comparision between the deterministic and stochastic model for $S_b\left(t\right)$ and $S_h\left(t\right)$.

Figure 4: Infected Populations ($I_b\left(t\right)$ and $I_h\left(t\right)$)—Panel (a) presents the trajectory of the infected baboon population $I_b\left(t\right)$, while Panel (b) shows the corresponding infected human population $I_h\left(t\right)$, as simulated by both deterministic and stochastic models. As the intensity of stochastic noise increases, the infected populations exhibit sharper rises and attain higher peaks than those predicted by the deterministic framework. This amplification of infection levels under stochastic dynamics highlights the potential for more severe outbreaks in real-world settings, reinforcing the importance of accounting for demographic and environmental randomness in predictive disease modeling.

Figure 4. Comparision between the deterministic and stochastic model for $I_b\left(t\right)$ and $I_h\left(t\right)$.

Figure 4. Comparision between the deterministic and stochastic model for $I_b\left(t\right)$ and $I_h\left(t\right)$.

Figure 5: Recovered Populations ($R_b\left(t\right)$ and $R_h\left(t\right)$)—Panel (a) and Panel (b) illustrate the recovery trajectories for the baboon $R_b\left(t\right)$ and human $R_h\left(t\right)$ populations, respectively. Both compartments show a steady increase over time, reflecting the accumulation of individuals recovering from infection. As observed in the infected populations, stochastic effects become more pronounced at higher noise intensities, resulting in broader spreads around the mean trajectory. These patterns emphasize the critical role of stochastic modeling in systems influenced by environmental unpredictability and demographic fluctuations. Overall, Figure 3, Figure 4 and Figure 5 highlight the divergence between deterministic and stochastic modeling frameworks: while the deterministic model provides a useful baseline, the stochastic simulations capture a fuller spectrum of potential epidemic trajectories, offering a more realistic and reliable foundation for public health planning and zoonotic disease management.

Figure 5. Comparision between the deterministic and stochastic model for $R_b\left(t\right)$ and $R_h\left(t\right)$.

Figure 5. Comparision between the deterministic and stochastic model for $R_b\left(t\right)$ and $R_h\left(t\right)$.

1.

Susceptible Populations

Figure 6 compares the deterministic and stochastic trajectories for susceptible baboons $S_b\left(t\right)$ and susceptible humans $S_h\left(t\right)$. We observe a monotonic decrease in susceptible individuals over time, which aligns with the natural course of exposure in a spreading disease. The increasing spread between deterministic and stochastic curves with larger $\sigma$ values reflects the escalating uncertainty due to demographic noise. Table 2 quantifies this with narrow confidence intervals at early stages ($t=0$) and gradually widening bounds as time progresses, highlighting the growing influence of randomness.

2.

Infected Populations

In Figure 7, infected populations for both baboons ($I_b$) and humans ($I_h$) demonstrate significant growth under higher noise levels, with the stochastic curves diverging notably from the deterministic baseline. This indicates that stochastic fluctuations amplify the rate and magnitude of infections. Table 3 further supports this observation: the mean values of $I_b$ and $I_h$ increase steadily, with confidence intervals expanding at each time step. These results suggest that stochastic factors, such as irregular human–baboon contact or environmental variability, may intensify outbreak severity beyond what deterministic models predict.

3.

Recovered Populations

Recovered populations show an upward trend over time in all scenarios (Figure 8). The deterministic curve again remains within the envelope of the stochastic simulations, but greater deviation occurs at higher $\sigma$. Table 4 reveals that both $R_b$ and $R_h$ display steady increases with tight confidence intervals, especially early on. However, as time progresses and randomness accumulates, the spread of potential outcomes increases. These findings emphasize the role of noise in shaping recovery dynamics, particularly in uncertain ecological settings.

Overall, the comparison across all compartments illustrates how stochastic effects introduce variability that deterministic models may underrepresent. The results confirm that even modest noise levels ($\sigma \ge 0.2$) significantly impact the trajectories of infection and recovery. This reinforces the importance of incorporating stochastic components when modeling zoonotic diseases in complex human–wildlife interfaces.

7. Discussion

This study proposes and analyzes a novel deterministic-stochastic framework for modeling zoonotic disease transmission between baboon and human populations, explicitly incorporating ecological processes, epidemiological interactions, and practical control strategies. The integration of both deterministic and stochastic elements allows for a nuanced understanding of disease spread under varying degrees of environmental and demographic uncertainty.

7.1. Insights from Deterministic and Stochastic Modeling

The deterministic model provided foundational insights into the baseline transmission dynamics and facilitated analytical derivation of the basic reproduction number ${\mathcal{R}}_0$. Sensitivity analysis revealed that ${\beta }_b$ and ${\beta }_h$ (transmission rates) and the initial number of susceptibles significantly influence ${\mathcal{R}}_0$, while recovery (${\gamma }_b$, ${\gamma }_h$) and intervention parameters ($H_s$, $H_i$) exerted a mitigating effect.

The stochastic model extended these insights by incorporating random perturbations modeled through independent Brownian motions. The inclusion of noise led to outcomes that diverged from deterministic trajectories, especially in infected compartments. As noise intensity increased, some scenarios demonstrated noise-induced extinction even when ${\mathcal{R}}_0>1$. This behavior was captured using metrics like the stochastic reproduction number ${\mathcal{R}}_0^S$ and illustrated the crucial role of variability in epidemic persistence and extinction.

7.2. Quantitative Evaluation and Uncertainty Analysis

The use of Maximum Likelihood Estimation (MLE) enabled robust parameter estimation from synthetic time series data, yielding a minimized log-likelihood of $1.6710\times {10}^4$ with small standard errors. This confirmed the reliability of the fitted model under controlled conditions.

Stochastic simulations, repeated 10,000 times per scenario, allowed for comprehensive statistical evaluation. Mean Squared Error (MSE) analysis highlighted the divergence between stochastic and deterministic results under increasing noise. At $\sigma =0.4$, baboon recovered class MSE reached 2,577,800, underscoring the unpredictability induced by environmental fluctuations.

Power Spectral Density (PSD) analysis further enriched the findings by identifying dominant periodic components in the infection time series of both species. These frequency components are consistent with ecological observations of seasonal migration and human–primate interaction patterns in regions like Al-Baha.

7.3. Model Strengths and Implications

This model contributes several innovations to zoonotic disease modeling:

• It bridges deterministic predictions with stochastic realism, allowing simultaneous analysis of average trends and uncertainty bounds.
• It supports counterfactual testing of interventions (sterilization, food restriction, interaction reduction) under both predictable and noisy scenarios.
• It combines epidemiological modeling with advanced statistical validation, including confidence intervals, MLE, MSE, and PSD.
• It highlights key stochastic phenomena such as noise-induced extinction and endemic persistence in fluctuating environments.

Together, these features provide public health and wildlife officials with a flexible, data-informed decision-support tool capable of responding to real-time ecological complexity.

7.4. Limitations and Future Research Directions

Several assumptions and simplifications were necessary for model tractability, including homogeneous mixing, constant parameters, and Gaussian noise:

• Spatial heterogeneity was not included, which may be critical for understanding localized outbreaks.
• Latent or asymptomatic compartments were not modeled, potentially underestimating disease persistence.
• Parameter values were derived from the literature or synthetic data in the absence of field-confirmed transmission events.
• Environmental transmission and multi-pathogen dynamics were excluded, though they may significantly influence real-world outcomes.

Future work should integrate spatial structure, agent-based modeling, and real-time data assimilation via Bayesian or machine learning methods. Incorporating empirical field data and adapting the model for other zoonotic reservoirs (e.g., bats, rodents) would enhance its generalizability.

7.5. Potential Model Extensions: Environmental Reservoirs

An important direction for enhancing the realism of the current framework is the incorporation of environmental reservoirs as a secondary transmission pathway. In ecological settings, pathogens can persist on surfaces, in food waste, or in contaminated water, posing a risk of indirect transmission between baboons and humans. Shared environments such as garbage bins and park benches in ecotourism zones are common interfaces of such transmission.

Mathematically, this can be modeled by introducing an environmental compartment $E\left(t\right)$, with transmission rate ${\lambda }_e={\beta }_{e}E\left(t\right)$. The dynamics of the environmental reservoir may be described by:

$${\displaystyle \frac{dE}{dt}}=\eta I_b+\theta I_h-\delta E,$$

where $\eta$ and $\theta$ are the pathogen shedding rates from infected baboons and humans, respectively, and $\delta$ is the decay rate of the pathogen in the environment.

This modification would align the model more closely with the One Health framework, acknowledging the interconnectedness of animal, human, and environmental health.

7.6. Biological Interpretation of Stochastic Intensities

In the stochastic model, each state variable is influenced by a diffusion term ${\sigma }_i{X}_{i}d{B}_i\left(t\right)$, where ${\sigma }_i$ represents the stochastic intensity and $B_i\left(t\right)$ is a standard Brownian motion. These terms model real-world randomness arising from environmental variability, demographic heterogeneity, and behavioral shifts.

Biologically:

• ${\sigma }_1$: Captures baboon birth or movement variability due to environmental stressors.
• ${\sigma }_2$: Reflects unpredictable baboon infection rates from contact or immunity differences.
• ${\sigma }_5$: Represents human reporting variability or sudden exposure events, such as tourist surges.

These components enrich the model’s realism and ensure outcome distributions (e.g., confidence intervals) are consistent with field uncertainty.

Table 10. Biological interpretation of stochastic intensities ${\sigma }_i$.

Parameter	Biological Interpretation
${\sigma }_1$	Environmental fluctuations in baboon births or movements
${\sigma }_2$	Variability in baboon infection contact rates
${\sigma }_3$	Inconsistencies in baboon recovery (e.g., nutrition, stress)
${\sigma }_4$	Human exposure fluctuations (e.g., tourist behavior)
${\sigma }_5$	Reporting noise or healthcare access variability
${\sigma }_6$	Recovery outcome variance in humans

provides an interpretation of the stochastic intensities (${\sigma }_i$) used in the baboon-human zoonotic disease transmission model. The noise components (${\sigma }_1$ to ${\sigma }_6$) represent various sources of environmental and demographic variability. Specifically, ${\sigma }_1$ models fluctuations in baboon births or movements, ${\sigma }_2$ accounts for variability in baboon infection contact rates, and ${\sigma }_3$ captures inconsistencies in baboon recovery. Similarly, ${\sigma }_4$ represents fluctuations in human exposure to baboons, ${\sigma }_5$ models variability in healthcare access for humans, and ${\sigma }_6$ reflects inconsistencies in human recovery outcomes. By incorporating these noise intensities, the model effectively simulates the randomness inherent in real-world disease dynamics, providing a more accurate representation of disease transmission and control.

7.7. Synthetic Data Use for MLE Calibration

Due to the lack of detailed, real-time epidemiological data for baboon–human interactions in the Al-Baha region, we employed synthetic data for parameter calibration via MLE. These data were generated from numerically solved deterministic equations using biologically plausible parameters informed by the literature and ecological reports.

Random noise was added to simulate observational errors. While synthetic datasets cannot fully replicate complex ecological realities, they offer a valuable testing ground for methodological development. Future extensions will aim to incorporate empirical surveillance data where available.

7.8. Motivation Behind Noise Intensities

Stochastic noise intensities $\sigma \in \{0.1,0.2,0.3,0.4\}$ were chosen to reflect varying degrees of ecological uncertainty:

• Real-world representativeness: Captures demographic and environmental variability across compartments.
• Dynamic range testing: Assesses model behavior under escalating noise conditions.
• Validation through error metrics: Demonstrated with MSE trends and confidence interval widening.
• Robust decision-making support: Enables policy evaluation across realistic uncertainty scenarios.

These values were not arbitrarily chosen but represent plausible fluctuation scales in complex, ecologically sensitive zoonotic systems.

7.9. Visualization of Uncertainty in Comparative Outputs

To improve interpretability, we recommend plotting deterministic solutions alongside stochastic confidence bands or standard deviation envelopes. These can be computed using:

• Bootstrapped 95% confidence intervals (2.5th and 97.5th percentiles),
• Standard deviations from 10,000 Monte Carlo runs.

Shaded bands or vertical error bars can be added to time-series plots of $S_b\left(t\right), I_b\left(t\right),$$R_b\left(t\right), S_h\left(t\right),$$I_h\left(t\right), R_h\left(t\right)$, allowing for clear visualization of the system’s uncertainty and robustness. This visual insight supports clearer communication of risk to public health authorities and stakeholders.

7.10. Model Validation and Real-World Application

We acknowledge the absence of empirical data as a current limitation. While this study focuses on a theoretical and simulation-driven exploration of zoonotic transmission at the baboon–human interface, the incorporation of real-world data is essential for validating model predictions and enhancing their practical utility.

Future extensions will seek to integrate ecological surveillance data and public health records from regions with active baboon–human interaction—such as parts of East Africa and the Arabian Peninsula. Such integration will allow for empirical parameter calibration via techniques like Maximum Likelihood Estimation (MLE) or Bayesian inference using incidence and recovery data.

In the meantime, model robustness has been ensured through sensitivity analysis, confidence interval estimation from stochastic simulations, and validation via Mean Squared Error (MSE) and Power Spectral Density (PSD) analyses.

Table 11. Summary of mean, 95% CI, and MSE for each compartment at $t=1.0$.

Compartment	Mean	95% CI	MSE
$S_b$	87.5	[84.3, 90.7]	1.2
$I_b$	12.1	[10.8, 13.4]	0.9
$R_b$	7.8	[6.9, 8.6]	0.7
$S_h$	141.0	[138.2, 143.5]	1.5
$I_h$	9.2	[8.1, 10.4]	1.0
$R_h$	5.5	[4.9, 6.2]	0.6

Summary of mean, 95% CI, and MSE for each compartment at t = 1.0. These internal checks lend credibility to the simulation outcomes, even in the absence of empirical calibration.

7.11. Parameter Justification and Estimation Strategy

Parameter values for transmission (${\beta }_b$, ${\beta }_h$) and recovery rates (${\gamma }_b$, ${\gamma }_h$) were initially selected based on the literature from related zoonotic systems and plausible ecological assumptions. To reinforce their credibility, we implemented MLE on synthetic time-series outputs, minimizing the log-likelihood function to yield statistically optimal values. These estimates, reported in Section 4.3, are more data-informed and support the reliability of model-based insights.

As real transmission data become available, future work will refine these parameters using Bayesian updating or non-linear regression, thereby improving biological realism and predictive capability.

7.12. Justification for Power Spectral Density (PSD) Analysis

Power Spectral Density analysis was applied to time-series outputs to explore whether internal dynamics or stochastic feedbacks could induce periodic behavior. This decision is supported by ecological studies indicating that wildlife–human contact rates often follow seasonal rhythms influenced by resource availability and human activities.

Although our current model does not incorporate explicit seasonal forcing, the observed low-frequency peaks in PSD outputs—especially under moderate noise levels ($\sigma =0.2$, $0.3$)—suggest emergent periodicities. These quasi-periodic fluctuations mimic real-world epidemic waves and enhance the model’s value in anticipating outbreak timing.

Future models will explicitly include seasonally modulated contact or migration terms to verify how external periodicities interact with intrinsic system dynamics.

7.13. Time Scale and Interpretation of Parameters

The model operates on a daily time scale, appropriate for capturing the rapid progression of zoonotic outbreaks in environments with high contact frequency. All parameters, including ${\beta }_b$, ${\beta }_h$, ${\gamma }_b$, and ${\gamma }_h$, were either estimated directly in daily units or converted from the weekly/monthly literature values using standard scaling techniques.

This time resolution emphasizes short-term variability and enables fine-grained control strategy analysis. However, to avoid misinterpretation, we stress that simulated outbreaks represent relative trends rather than precise forecasts. Future studies will seek to cross-validate these temporal scales against epidemiological records.

7.14. Literature Gaps and Study Justification

Traditional zoonotic disease models rely heavily on deterministic structures that assume homogeneity, fixed parameter values, and average-case dynamics. While useful for baseline predictions, these models neglect critical randomness associated with demographic variability, environmental change, and irregular human–wildlife interaction patterns [4].

Stochastic models offer a more nuanced representation of such systems but often omit real-world interventions and rarely assess control strategies under uncertainty. Moreover, there is limited modeling attention paid to non-traditional wildlife hosts like baboons, particularly in arid ecosystems such as those in Saudi Arabia, where contact drivers differ from tropical zones.

This study fills these gaps by:

• Introducing a dual deterministic-stochastic model that integrates both average trends and variability.
• Explicitly incorporating intervention strategies relevant to human–baboon conflicts (e.g., sterilization, food restriction, and contact reduction).
• Using a rigorous suite of statistical tools—MLE, MSE, confidence intervals, PSD—to validate model reliability and performance.
• Providing counterfactual simulations to assess control effectiveness under different uncertainty levels.

Our approach aligns with modern One Health principles and equips decision-makers with an adaptable, empirically grounded framework for managing zoonotic risk in ecologically sensitive regions.

8. Conclusions

This study develops a comprehensive deterministic-stochastic modeling framework to analyze zoonotic disease transmission between baboons and humans, incorporating control strategies such as sterilization, restricted food access, and reduced interaction. By deriving and analyzing the basic reproduction number ${\mathcal{R}}_0$, calibrating parameters through Maximum Likelihood Estimation (MLE), and quantifying uncertainty using Mean Squared Error (MSE), confidence intervals, and Power Spectral Density (PSD) analysis, the model demonstrates the critical role of stochasticity in capturing real-world epidemic variability. While deterministic models offer baseline trends, the inclusion of stochastic components enables more realistic simulations of demographic and environmental fluctuations, especially in dynamic ecological settings. The results affirm the importance of early intervention targeting transmission pathways and validate the robustness of the proposed model as a decision-support tool for wildlife and public health agencies. By addressing the literature gaps—such as the lack of integrated intervention analysis, advanced statistical evaluation, and focus on non-traditional hosts—this work contributes a flexible and ecologically grounded platform for understanding and managing zoonotic risks. Future extensions will incorporate empirical data, seasonal forcing, and spatial heterogeneity to enhance predictive accuracy and operational relevance.