EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5881 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mathematical Insights into Zoonotic Disease Spread: Application of the Milstein Method Sayed Saber1,2, Abdullah A. Alahmari3,∗ 1 Mathematics Department, Faculty of Science, Al-Baha University, Saudi Arabia 2 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt 3 Mathematics Department, Faculty of Science, Umm Al-Qura University, Saudi Arabia Abstract. This study develops a mathematical model to analyze the transmission dynamics of zoonotic diseases between baboons and humans in the Al-Baha region, incorporating key interven- tion strategies such as sterilization, restricted food access, and reduced human-baboon interactions. The model is formulated using both deterministic and stochastic frameworks to capture the effects of environmental and social variability. Theoretical analysis establishes the existence, uniqueness, and stability of the solutions, while the basic reproduction number R0 determines conditions for disease eradication or persistence. A stochastic extension, implemented via the Milstein method, accounts for random fluctuations in disease dynamics. Sensitivity analysis identifies critical trans- mission parameters influencing disease spread, and numerical simulations evaluate the effectiveness of different intervention strategies. The findings provide actionable insights for public health and wildlife management, offering a robust framework for controlling zoonotic disease risks in human- wildlife interactions. 2020 Mathematics Subject Classifications: 92D30, 60H10 Key Words and Phrases: Zoonotic disease transmission, Milstein method, stochastic modeling, basic reproduction number, intervention strategies 1. Introduction Zoonotic diseases, which are transmitted between animals and humans, pose a signif- icant threat to global public health. Human populations’ increasing encroachment into wildlife habitats has escalated zoonotic transmission risk. This is driven by factors such as habitat loss, competition for resources, and direct human-animal interactions [1, 2]. Among wildlife reservoirs, baboons (Papio spp.) are particularly noteworthy for their fre- quent interactions with humans in regions like Al-Baha, Saudi Arabia, where shared use ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5881 Email addresses: Sayed011258@science.bsu.edu.eg (S. Saber), aaahmari@uqu.edu.sa (A. A. Alahmari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 2 of 38 of resources exacerbates the potential for disease spillover [3, 4]. Human-baboon interac- tions, often facilitated by food provisioning from residents and tourists, create conditions for zoonotic disease transmission. Diseases such as tuberculosis and viral infections have been identified as significant risks, spreading through direct contact or contaminated re- sources [5, 6]. The complexity of these transmission pathways underscores the urgent need for effective intervention strategies to mitigate zoonotic disease risks at human-wildlife in- terfaces. Mathematical modeling is a powerful tool for understanding infectious disease dynam- ics. Deterministic models, which assume a fixed transmission pattern, have traditionally been used to analyze disease propagation [7, 8]. Mathematical modeling has gained signifi- cant attention for its applications in engineering [9], plant epidemiology [10], mathematical biology [11], medicine [12], as well as psychological and life sciences [13], viscoelasticity [14], electromagnetic wave propagation [15], quantum dynamics [16], Langevin systems [17], physics [18–26], Diabetes [27–37], Computer Virus [38], tobacco smoking models [39], medicine [40, 41], influenza [42], infectious diseases [43], epidemics [44], cancer [45], coron- avirus [46], monkeypox [47], zoonotic viral infections [48], and alcohol-related models [49]. In many disciplines, including medicine, engineering, chemistry, physics, economics, and many more, epidemiology plays an important role, seee [50, 51]. Stochastic models provide a more realistic representation by incorporating randomness into disease transmission processes in biological systems [52–59]. The Michaelis-Menten function, frequently employed in enzyme kinetics, is particularly suitable for modeling the nonlinear degradation rate of insulin, providing a more nuanced understanding of diabetes pathogenesis [60]. By using numerical techniques such as the Milstein method to solve SDEs, these models achieve a high level of accuracy in simulating zoonotic disease dynamics [61, 62]. In this study, we develop a mathematical model to describe the transmission dynamics of zoonotic diseases between baboons and humans, incorporating key control measures such as sterilization, restricted food access, and reduced human-baboon interactions. Extend- ing beyond a deterministic framework, we introduce a stochastic model using the Milstein method to capture the effects of environmental and social variability on disease dynam- ics. Through theoretical analysis, we establish the fundamental properties of the model, including the existence, uniqueness, and boundedness of solutions. Stability analysis deter- mines the conditions for disease eradication or persistence based on the basic reproduction number, R0. Sensitivity analysis identifies the most influential transmission parameters, guiding the development of effective intervention strategies. Numerical simulations as- sess the impact of different control measures, evaluating the effectiveness of sterilization, food access restrictions, and interaction reductions in mitigating disease spread. By in- tegrating deterministic and stochastic approaches, this study provides a comprehensive understanding of zoonotic disease transmission. The incorporation of stochasticity allows for a more realistic representation of disease spread by accounting for environmental and social uncertainties. This research offers actionable insights for public health officials and wildlife managers to design robust, evidence-based strategies for minimizing disease risks in human-baboon interactions. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 3 of 38 The primary objective of this study is to develop a compartmental mathematical model that captures the dynamics of zoonotic disease transmission between baboons and humans. The model’s fundamental properties, including the existence, uniqueness, and boundedness of solutions, are analyzed to ensure its well-posedness. The basic reproduction number (R0) is determined, and the local and global stability of both the disease-free and endemic equilibria are examined. To account for environmental and social variability, the model is extended into a stochastic framework using the Milstein method. A sensitivity analysis is conducted to identify key parameters influencing disease spread and control effective- ness. Finally, different intervention strategies—such as sterilization, restricted food access, and reduced human-baboon interactions—are simulated to assess their impact on disease mitigation. The findings of this study provide insights into optimal public health and wildlife management strategies to mitigate zoonotic disease risks. By integrating deterministic and stochastic approaches, this study offers a robust framework for understanding and controlling zoonotic disease risks at the human-wildlife interface. The findings provide actionable insights for public health officials and wildlife managers, aiding in the design of effective intervention strategies to mitigate zoonotic disease transmission in the Al-Baha region and similar ecological settings. 2. Deterministic model It aims to simulate baboon and human populations dynamics in the context of zoonotic disease transmission. It incorporates control strategies such as sterilization, restricted food access, and reduced interactions between humans and baboons. We model baboon population dynamics using these equations, which account for natural growth, disease transmission, and control actions. To model baboon and human populations with zoonotic disease transmission, the following equations are used: dx1 dt = rx1 ( 1− x1 + x2 + x3 K ) − ηbx1x2 −Hs(t)x1 −Hf (t)x1, dx2 dt = ηbx1x2 − κbx2 −Hs(t)x2 −Hf (t)x2, dx3 dt = κbx2 −Hf (t)x3, dx4 dt = −ηhx4x2 −Hi(t)x4, dx5 dt = ηhx4x2 − κhx5 −Hi(t)x5, dx6 dt = κhx5. (1) The parameters of the model, existed in Table 1, were determined through empirical data, assumptions, and numerical calibration. Epidemiological data from the Al-Baha region informed estimates for key parameters such as the transmission rate from baboons S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 4 of 38 Parameter Description Units x1 Susceptible baboons Individuals x2 Infected baboons Individuals x3 Recovered baboons Individuals x4 Susceptible humans Individuals x5 Infected humans Individuals x6 Recovered humans Individuals r Intrinsic growth rate of baboons 1/time K Carrying capacity of baboons Individuals ηb Transmission rate among baboons 1/time ηh Transmission rate from baboons to humans 1/time κb Recovery rate of infected baboons 1/time κh Recovery rate of infected humans 1/time Hs Sterilization control measure 1/time Hf Food restriction control measure 1/time Hi Interaction reduction measure 1/time Table 1: Definitions of variables and parameters used in the model. to humans (βh) and the human recovery rate (γh), while control parameters (Hs, Hf , Hi) were based on plausible intervention strategies and validated through sensitivity analysis. Numerical simulations ensured model outputs aligned with observed disease dynamics. Human and baboon populations share similar susceptibility and interaction rates, simpli- fying analysis while retaining essential disease dynamics. Food access control parameters like (Hf ) represent direct contact and indirect pathways, such as contamination of shared food resources. Sterilization (Hs), food restrictions (Hf ), and reduced human-baboon interactions (Hi) are considered constant over time. Furthermore, the model assumes a closed system, excluding human and wildlife migration to focus on internal disease dy- namics without external influences. 3. Stochastic model The parameters of the system (1) are all deterministic and do not take into account ran- domness or environmental fluctuations. It is more appropriate and reasonable to consider how environmental noise affects Zoonotic Disease in a biological environment. Environ- mental fluctuations affect the propagation of epidemics on the population (see, [63–69]). Therefore, there are various approaches to consider random fluctuations in deterministic systems. One of them assumes that the epidemic is subject to some small and stan- dard random fluctuations. This paper examines the effect of environmental noise on the Zoonotic Disease model (1) by assuming that the stochastic perturbations are white noise in nature and proportional to the variable. Next, we introduce a stochastic Zoonotic S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 5 of 38 Disease model defined by the following system of differential equations dx1 = ( rx1 ( 1− x1 + x2 + x3 K ) − ηbx1x2 −Hs(t)x1 −Hf (t)x1 ) dt+ σ1x1(t)dB1(t), dx2 = ( ηbx1x2 − κbx2 −Hs(t)x2 −Hf (t)x2 ) dt+ σ2x2(t)dB2(t), dx3 = ( κbx2 −Hf (t)x3 ) dt+ σ1S(t)dB1(t)dt+ σ3x3(t)dB3(t), dx4 = ( − ηhx4x2 −Hi(t)x4 ) dt+ σ4x4(t)dB4(t), dx5 = ( ηhx4x2 − κhx5 −Hi(t)x5 ) dt+ σ5x5(t)dB5(t), dx6 = ( κhx5 ) dt+ σ6x6(t)dB6(t), (2) where Bi(t), (i = 1, 2, 3, 4, 5, 6) are a standard Brownian motion (see Definition 1 ) defined on a complete probability space ( Ω,F , {Ft}t≥0 ,P ) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F0 contains all P-null sets), σi, (i = 1, 2, 3, 4, 5, 6) represent the intensity of noise. Definition 1 ([70]). The positive real-time numbers process (B(t))t≥0 is called the Wiener process if (i) B(0) = 0,P− a.s. (ii) ∀0 ≤ s < t,B(t)−B(s) are independent of Fs. (iii) ∀0 ≤ s < t,B(t)−B(s) ∽ N (0, t− s). The n-dimensional stochastic differential equation (briefly, SDE) is defined by: dy(t) = F (t, y(t))dt+G(t, y(t))dB(t), with y0 = y(0), where B(t) is a n-dimensional standard Wiener process, F is the drift coefficient, and G is the diffusion coefficient. 4. Existence, uniqueness, non-negativity of the solution This section provides evidence that model (1) is epidemiologically relevant by demon- strating its non-negativity and boundedness. According to Derrick and Groosman theorem [71], if Lipchitz’s condition as in Definition 1 is satisfied, then the solution of model (1) exists and is unique. Definition 2 ([72]). In system (1), f⃗ satisfies Lipchitz’s condition if there exists a positive constant k, such as∥∥∥f⃗ (X⃗1 ) − f⃗ ( X⃗2 )∥∥∥ < k ∥∥∥X⃗1 − X⃗2 ∥∥∥ , ∀X⃗1, X⃗2 ∈ Ω. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 6 of 38 The following theorem guarantee the existence and uniqueness of solution of model (1). Theorem 1. Under non-negative initial conditions, the model (1) is unique in Ω for all t ≥ 0. Proof. The right side of model (1) can be written as follows. f1 = rx1 ( 1− x1 + x2 + x3 K ) − ηbx1x2 −Hs(t)x1 −Hf (t)x1, f2 = ηbx1x2 − κbx2 −Hs(t)x2 −Hf (t)x2, f3 = κbx2 −Hf (t)x3, f4 = −ηhx4x2 −Hi(t)x4, f5 = ηhx4x2 − κhx5 −Hi(t)x5, f6 = κhx5. Then, it can be shown that ∂fi ∂xj is continuous and ∣∣∣ ∂fi∂xj ∣∣∣ < ∞ for all i, j = 1, 2, .., 6. As a result of Derrick and Groosman theorem in [71], the system (1) is Lipchitz’s unique solution. Theorem 2. The solutions of model (1) are all non-negative and ultimately bound under the assumption that the initial values are non-negative. Proof. Let us consider the model equations. Since f1, f2, f3, f4, f5, f6 are constructed from non-negative parameters and functions, we conclude that if the initial conditions are non-negative, then the solution remains non-negative for all t ≥ 0. In other words, we analyze the equations for the fractional-order baboon-human population model at the boundary conditions where the population variables are zero: ẋ1(t) ∣∣ x1=0 = 0, ẋ2(t) ∣∣ x2=0 = 0, ẋ3(t) ∣∣ x3=0 = γbx2 ≥ 0, ẋ4(t) ∣∣ x4=0 = −βhx4x2 ≥ 0, ẋ5(t) ∣∣ x5=0 = βhx4x2 ≥ 0, ẋ6(t) ∣∣ x6=0 = γhx5 ≥ 0. From these equations, we can deduce that for any non-negative initial conditions, the deterministic model ensures that the solution remains non-negative for all t ≥ 0. By employing similar techniques as demonstrated in Lemmas 5 and 6 in [73], it follows that the baboon-human population model solutions are non-negative. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 7 of 38 Theorem 3. The model (1) possesses uniformly bounded solutions within the region de- fined as: B = { (x1, x2, x3, x4, x5, x6) ∈ R6 + ∪ −→ 0 : N = x1 + x2 + x3 + x4 + x5 + x6 ≤ rK λ } . Proof. Let the total population at time t be defined as: N(t) = x1(t) + x2(t) + x3(t) + x4(t) + x5(t) + x6(t). By summing the equations of the baboon-human population model, we arrive at: dN(t) dt = rx1 ( 1− x1 + x2 + x3 K ) −Hs(t)x1 −Hf (t)x1 −Hi(t)x4 − γbx2 − γhx5. This leads to: dN(t) dt ≤ rK− µN(t), (3) where µ represents the minimum of the death rates and control actions. The solution of the equation (3) satisfies N(t) ≤ rK µ + ( N(0)− rK µ ) exp(−µt), where N(0) is the initial value. It is clear that lim t→∞ N(t) ≤ 0, and thus N(t) is bounded with N(t) ≤ rK µ . Hence, we can see that the feasible region of model (1) is B which is positively invariant region. Remark 1. The region B is positively invariant concerning the baboon-human population model. Theorem 4. Consider the stochastic differential equation (SDE) system: dXt = f(Xt)dt+ g(Xt)dWt, where Xt ∈ Rn is the state vector, Wt is an m-dimensional Wiener process, and the functions f : Rn → Rn and g : Rn → Rn×m satisfy the following conditions: (i) (Lipschitz Continuity) There exists a constant L > 0 such that for all x, y ∈ R∗n, ∥f(x)− f(y)∥+ ∥g(x)− g(y)∥ ≤ L∥x− y∥. (ii) (Linear Growth Condition) There exists a constant C > 0 such that for all x ∈ Rn, ∥f(x)∥2 + ∥g(x)∥2 ≤ C(1 + ∥x∥2). S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 8 of 38 Under these conditions, there exists a unique global solution Xt for all t ≥ 0, and if the initial condition X0 ≥ 0, then Xt ≥ 0 almost surely for all t ≥ 0. Proof. The existence and uniqueness of a global solution follow from the standard results of stochastic differential equations based on the Lipschitz continuity and linear growth conditions (see, e.g., [74] ). To prove the nonnegativity of the solution, we apply Itô’s formula to the function ϕ(Xt) = max(0, Xt) component-wise. Since the drift term f(Xt) and the diffusion term g(Xt) are assumed to preserve nonnegativity (i.e., they do not push the solution into neg- ative values if started from a nonnegative initial condition), we conclude that Xt remains nonnegative almost surely. A more rigorous approach involves considering a stopping time τϵ = inf{t ≥ 0 : Xt ̸≥ 0} and using a contradiction argument, showing that P(τϵ < ∞) = 0. Theorem 5. For any initial value (Sb(0), Ib(0), Rb(0), Sh(0), Ih(0), Rh(0)) ∈ R6 +, there is a unique solution (Sb(t), Ib(t), Rb(t), Sh(t), Ih(t), Rh(t)) to the model (2). The solution will remain in R6 + with probability one, namely (Sb(t), Ib(t), Rb(t), Sh(t), Ih(t), Rh(t)) ∈ R6 + for all t ≥ 0 almost surely (briefly a.s.). By using the same steps of proof of Theorem 2.1 in [75] we can prove the above theorem. 5. Equilibrium points and their stability analysis At equilibrium, we solve for x1 and x4, leading to the infection-free equilibrium of model (1): E0 = (x01, 0, 0, x 0 4, 0, 0), where: x01 = Kr Hs +Hf , x04 = K Hi . The basic reproduction number R0 is given by: R0 = ηbx 0 1 κb +Hs + ηhx 0 4 κh +Hi . Now, we prove the local and global stability of the infection-free equilibrium point. Lemma 1. E0 = (x01, 0, 0, x 0 4, 0, 0) is locally asymptotically stable in Ω if R0 < 1, and unstable if R0 > 1. Proof. For the baboon-human population model at E0 = (x01, 0, 0, x 0 4, 0, 0) , the Jaco- bian matrix J(E0) is given by: J(E0) =  −r 0 −ηbx 0 1 0 0 0 0 −Hi 0 −ηhx 0 4 0 0 0 0 ηbx 0 1 − (κb +Hs) 0 0 0 0 0 0 ηhx 0 4 − (κh +Hi) 0 0 0 0 0 0 −Hf 0 0 0 0 0 0 −Hf  . S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 9 of 38 The characteristic equation is obtained by solving det(J(E0) − κI) = 0, where κ is the eigenvalue: det (J(E0)− κI) = 0. The characteristic equation becomes: (κ+ r)(κ+Hi)(κ− ηbx 0 1 + κb +Hs)(κ− ηhx 0 4 + κh +Hi) = 0. Thus, the eigenvalues are: κ1 = −r, κ2 = −Hi, κ3 = ηbx 0 1 − (κb +Hs), κ4 = ηhx 0 4 − (κh +Hi). For stability, all eigenvalues must have negative real parts. κ1 and κ2 are always negative. The eigenvalues κ3 and κ4 depend on the basic reproduction number R0: R0 = ηbx 0 1 κb +Hs + ηhx 0 4 κh +Hi . If R0 < 1, then κ3 < 0 and κ4 < 0, and E0 is locally asymptotically stable. If R0 > 1, then κ3 > 0 or κ4 > 0, and E0 is unstable. Lemma 2. E0 in Ω is globally asymptotically stable if R0 < 1, and unstable if R0 > 1. Proof. To establish the global asymptotic stability, we construct an appropriate Lya- punov function: L = x2 + x5, where x2 and x5 represent the infected populations of baboons and humans, respectively. The time derivative of L can be expressed as: dL dt = dx2 dt + dx5 dt . Using the model equations for the dynamics of x2 and x5: dx2 dt = ηbx1x2 − (κb +Hs)x2, dx5 dt = ηhx4x2 − (κh +Hi)x5, we obtain: dL dt = (ηbx1 − (κb +Hs))x2 + (ηhx4 − (κh +Hi))x5. At the infection-free equilibrium, we have x1 = x01 and x4 = x04. Thus, the derivative simplifies to: dL dt = ( ηbx 0 1 − (κb +Hs) ) x2 + ( ηhx 0 4 − (κh +Hi) ) x5. If R0 < 1, then it follows that: dL dt ≤ 0. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 10 of 38 The largest invariant set where dL dt = 0 corresponds to the infection-free equilibrium E0. Therefore, by LaSalle’s Invariance Principle, E0 is globally asymptotically stable if R0 < 1. Now, we prove the local and global stability of the endemic equilibrium point. For I>0, E∗ = (x∗1, x ∗ 2, x ∗ 3, x ∗ 4, x ∗ 5, x ∗ 6) is the endemic equilibrium point and is defined as the steady-state solution for the model (1). dxi dt = 0, i = 1, 2, 3, 4, 5, 6. From the system equations, we can use the second and third equations to express x∗2 and x∗3: x∗3 = κb Hf x∗2. Substituting this expression into the fourth equation yields: x∗5 = ηhx ∗ 4 κh +Hi x∗2. Next, substituting into the first equation allows us to solve for x∗1 as: x∗1 = Kr Hs + ηb a2 Hs x∗2. Thus, E∗ = (x∗1, x ∗ 2, x ∗ 3, x ∗ 4, x ∗ 5, x ∗ 6). Lemma 3. If R0 > 1, E∗ is locally asymptotically stable in Ω. Proof. For the first-order integer model (1), the Jacobian matrix J(E∗) at the endemic equilibrium point E∗ is given by: J(E∗) =  −r 0 −ηbx ∗ 1 0 0 0 0 −Hi 0 −ηhx ∗ 4 0 0 0 0 κb −Hf 0 0 0 0 0 0 κh −Hi 0 0 0 0 0 0 −Hf 0 0 0 0 0 0 −Hf  . Evaluating the Jacobian matrix at E∗ gives the characteristic equation: P (ξ) = det (ξI − J(E∗)) = 0, where I is the identity matrix. The characteristic polynomial is: P (ξ) = ξ6 + a1 ξ 5 + a2 ξ 4 + a3 ξ 3 + a4 ξ 2 + a5 ξ + a6, S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 11 of 38 where the coefficients a1, a2, . . . , a6 are defined in terms of the parameters of the model (1). If the Routh-Hurwitz criterion is satisfied, the endemic equilibrium is locally stable: a1 > 0, a2 > 0, a3 > 0, a4 > 0, a5 > 0, a6 > 0. Thus, E∗∗ is locally asymptotically stable when R0 > 1, as all conditions of the Routh- Hurwitz criterion are satisfied. Lemma 4. E∗ is globally asymptotically stable if R0 > 1. Proof. We construct a suitable Lyapunov function L to prove the global asymptotic stability of E∗∗: L = (x1 − x∗1) + (x2 − x∗2) + (x3 − x∗3) + (x4 − x∗4) + (x5 − x∗5) + (x6 − x∗6). As a result dL dt = dx1 dt + dx2 dt + dx3 dt + dx4 dt + dx5 dt + dx6 dt . Thus dx1 dt = rx1 ( 1− x1 + x2 + x3 K ) − ηbx1x2, and similarly for the other variables. At equilibrium, the terms involving x2, x5, and x3 vanish, leading to: dL dt ≤ 0. Thus, dL dt = 0 if and only if x1 = x∗1, x2 = x∗2, x3 = x∗3, x4 = x∗4, x5 = x∗5, and x6 = x∗6. E ∗ is asymptotically stable under the LaSalle’s Invariance Principle when R0 > 1. 6. An optimal control of the proposed model In this section, we extend the baboon-human population model (1) into an optimal control framework to determine the most effective intervention strategies for disease eradi- cation over a finite time period. The extended model (1) introduces three control variables, each representing a different intervention method: • u1(t): Preventive measures to reduce contact between susceptible and infected indi- viduals. • u2(t): Treatment efforts aimed at reducing the number of infectious individuals. • u3(t): Screening efforts to identify and isolate infected individuals early. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 12 of 38 The inclusion of these controls modifies the original baboon-human disease transmis- sion model (1) into the following system of differential equations: dx1 dt = rx1 ( 1− x1 + x2 + x3 K ) − (1− u1)ηbx1x2 N −Hsx1, dx2 dt = (1− u1)ηbx1x2 N − (u2+κb)x2, dx3 dt = u2 x2 −Hfx3, dx4 dt = −(1− u1)ηhx4x2 N −Hix4, dx5 dt = (1− u1)ηhx4x2 N − (u2+κh)x5, dx6 dt = u2 x5 −Hfx6, (4) where N = x1 + x2 + x3 + x4 + x5 + x6 represents the total population of baboons and humans. The control variables u1,u2, and u3 are bounded between 0 and 1, representing the proportion of effort applied in each intervention strategy. To determine the optimal control strategy, we aim to minimize the number of infected individuals while balancing the costs associated with implementing the control measures. The objective functional J(u1,u2, u3) is defined as: J = min u1,u2,u3 ∈ t∗0 tf ( b1x2 + b2x5 + 1 2 ∗∑ i=1 3wiu 2 i ) dt, where: • b1 and b2 are weights that reflect the relative importance of reducing the infected baboon and human populations, respectively. • w1, w2, w3 represent the costs associated with each control measure (prevention, treatment, and screening). • The quadratic terms 1 2wiu 2 i capture the increasing cost of higher control efforts. Our objective is to determine which control u∗1,u ∗ 2, u ∗ 3 that minimize J . Given these conditions, an optimal control (u,1 u , 2 u ) 3 exists, ensuring that the objective functional J is minimized. We examine the following control scenarios to evaluate the impact of different inter- ventions on the disease dynamics: • Control with Prevention Only: Preventing with u1 without treatment (u2 = 0) and screening (u3 = 0). The results show a significant reduction in the infected pop- ulation, as prevention reduces contact between susceptible and infected individuals. Implementing only prevention measures reduces the contact rate between susceptible and infected individuals, leading to a significant reduction in infections. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 13 of 38 • Control with Treatment Only: Using only the treatment effort u2 without pre- vention (u1 = 0) and screening (u3 = 0). The treatment reduces the number of infectious individuals initially, but without prevention, new infections still occur. Treatment efforts (u2) decrease the number of infectious individuals, but without prevention, the infection still spreads. • Control with Screening Only: Using only the screening effort u3, without pre- vention or treatment. Screening helps identify infected individuals and reduce the spread of the disease, but without prevention, new infections occur. Screening efforts (u3) help to isolate infected individuals but are less effective without simultaneous prevention measures. • Control with Prevention and Treatment: Using both prevention (u1) and treatment (u2) without screening. This strategy effectively reduces both infectious individuals and intervention costs. The combination of prevention and treatment significantly reduces infection and is more cost-effective than either measure alone. • Control with Prevention, Treatment, and Screening: Using all three control measures (u1, u2,u3) together. The results show that this strategy provides the most effective disease control, with the number of infected individuals rapidly declining. The combined use of prevention, treatment, and screening provides the best outcome, rapidly decreasing the number of infections. The extension of the baboon-human population model (4) into an optimal control frame- work allows us to identify the most effective strategies for disease prevention and treat- ment. By simulating different combinations of control measures, we can minimize both the number of infected individuals and the associated costs. Implementing all three control measures simultaneously provides the best outcomes for disease eradication. The optimization output provides the best values for control parameters u1,u2, u3, which are used to define the time-dependent control terms Hs(t), Hf (t), and Hi(t) in the differential equations. These controls, applied to susceptible, infected, and recovered pop- ulations, directly influence the disease dynamics between baboons and humans. As the optimization progresses, the objective function J decreases, indicating effective control measures that reduce infection rates while adhering to model (1) constraints. The con- vergence in the optimality column shows that the algorithm finds near-optimal control values, ensuring minimal disease transmission between the populations, thereby achieving the goal of the model (4). Figure 1: Population Dynamics with Different Control Scenarios. Subfigures (a-f): These subfigures depict the dynamics of compartments for baboons (x1, x2, x3) and hu- mans (x4, x5, x6) under varying intensities of control measures Hs, Hf , and Hi. • (a)-(d): Moderate control intensities (Hs = 0.1, Hi = 0.01, varying Hf ). These scenarios show gradual reductions in infected populations, demonstrating the effec- tiveness of sterilization and food restriction. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 14 of 38 • (e)-(f): Higher control intensities (Hs = 0.5, Hf = 0.5) lead to significant declines in infections, highlighting the impact of increased intervention efforts. Increased control measures reduce infection rates in both populations. Sterilization (Hs) and food restriction (Hf ) significantly affect infected populations, while interaction reduc- tion (Hi) supports prevention. Figure 1: Baseline Dynamics of Populations with Different 0 10 20 30 40 50 60 70 80 90 100 Time 0 100 200 300 400 500 600 700 800 P o p u la ti o n Dynamics with Hs = 0.10, Hi = 0.01, Hf = 0.10 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) 0 10 20 30 40 50 60 70 80 90 100 Time 0 100 200 300 400 500 600 700 800 P o p u la ti o n Dynamics with Hs = 0.10, Hi = 0.01, Hf = 0.50 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) 0 10 20 30 40 50 60 70 80 90 100 Time 0 100 200 300 400 500 600 700 800 P o p u la ti o n Dynamics with Hs = 0.10, Hi = 0.05, Hf = 0.10 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) 0 10 20 30 40 50 60 70 80 90 100 Time 0 100 200 300 400 500 600 700 800 P o p u la ti o n Dynamics with Hs = 0.10, Hi = 0.05, Hf = 0.50 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) 0 10 20 30 40 50 60 70 80 90 100 Time 0 100 200 300 400 500 600 700 800 P o p u la ti o n Dynamics with Hs = 0.10, Hi = 0.05, Hf = 0.50 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) 0 10 20 30 40 50 60 70 80 90 100 Time -100 0 100 200 300 400 500 P o p u la ti o n Dynamics with Hs = 0.50, Hi = 0.01, Hf = 0.50 S b (Susceptible Baboons) I b (Infected Baboons) R b (Recovered Baboons) S h (Susceptible Humans) I h (Infected Humans) R h (Recovered Humans) Figure 1: Comparision between the deterministic and stochastic model (2) for x1(t) and x4(t). Control Scenarios. Figures 1(a) to 1(f) depict the population dynamics for compartments of baboons and humans over time with various values of Hs, Hf , and Hi: Subfigures (a)-(d) display scenarios with moderate control intensities (e.g., Hs = 0.1, Hi = 0.01, S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 15 of 38 and varying values of Hf ). Subfigures (e)-(f) represent higher-intensity controls, such as Hs = 0.5 and increased Hf , showing a pronounced reduction in infected populations over time. Increased control values generally reduce infection rates in both populations. Par- ticularly, high values of Hs and Hf significantly lower the infected baboon and human populations, demonstrating the effectiveness of sterilization and food restriction controls. 7. Sensitivity Analysis The values of parameters and initial populations used in this study are obtained from various sources, including WHO situation reports and previous literature. The initial population values are set as follows: x1(0) = 5000, x2(0) = 150, x3(0) = 50, x4(0) = 8000, x5(0) = 100, and x6(0) = 100. The following Table 1, provides a summary of the parameter values and their respective sources: A sensitivity statistical analysis is essential Parameter ηb ηh κb κh Hs Hi Hf Value 0.3 0.1 0.07 0.15 0.1 0.05 0.05 Source Estimated Estimated Estimated Estimated Assumed Assumed Assumed Table 2: Parameter values for the baboon-human population model (1). to evaluating whether various factors affect a model (1) stability if data is unknown. Analyzing these parameters helps identify critical parameters. With the baboon-human model (1), we compute the sensitivity indices of the basic reproduction number, R0. Local sensitivity analysis uses the normalized forward sensitivity index R0. According to our model (1), R0 has the following sensitivity index: ΓR0 υ = ∂R0 ∂υ × υ R0 , where υ represents a parameter from Table 3, and R0 is derived from the model (1) dynamics. The basic reproduction number R0 is calculated as follows: R0 = ηbx1x2 κb +Hs + ηhx4x5 κh +Hi . Substituting the parameter values from Table 1 gives: R0 = R0b + R0h = 0.3 · 5000 · 150 0.07 + 0.1 + 0.1 · 8000 · 100 0.15 + 0.05 = 1723529.41. Also ∂R0 ∂ηb = x01 κb +Hs > 0, ∂R0 ∂ηh = x04 κh +Hi > 0, ∂R0 ∂κb = − ηbx 0 1 (κb +Hs)2 < 0, ∂R0 ∂κh = − ηhx 0 4 (κh +Hi)2 < 0, ∂R0 ∂Hs = − ηbx 0 1 (κb +Hs)2 < 0, ∂R0 ∂Hi = − ηhx 0 4 (κh +Hi)2 < 0, ∂R0 ∂µ = −ηbx1(0) + ηhx4(0) µ2 < 0. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 16 of 38 These equations illustrate that increases in the transmission parameters ηb and ηh result in higher values of R0, indicating a greater potential for disease spread. In contrast, increases in recovery rates κb and κh, or in the handling rates Hs and Hi, lead to a decrease in R0, implying a reduction in disease transmission potential. The computed sensitivity indices of R0 for the parameters in the baboon-human model (1) are summarized in Table 1. The Parameter x1(0) x4(0) ηb ηh κb Sensitivity Index Positive Positive Positive Positive Negative Parameter κh Hs Hi µ Sensitivity Index Negative Negative Negative Negative Table 3: Sensitivity indices of R0 for the baboon-human model (1). sensitivity indices indicate that R0 increases with the values of x1(0), x4(0), ηb, and ηh, suggesting an increased risk of disease transmission as these parameters rise. Conversely, increases in κb, κh, Hs, Hi, and µ lead to a decrease in R0, reflecting reduced transmission potential and suggesting effective control measures. By focusing on these key parameters, we can significantly influence baboon-human disease dynamics. The results indicate that the values of R0 increase when the parameters x1, x2, x4, x5, ηb, and ηh increase, while the values of κb and κh have a decreasing effect. As a result of the sensitivity analysis, potential strategies for disease control and prevention are identified as critical parameters that affect the dynamics of the baboon-human model (1). 8. Numerical investigation 8.1. Application of the Runge-Kutta method on the deterministic model The Runge-Kutta method of order 4 can be expressed as follows: yn+1 = yn+ h 6 (k1+2k2+2k3+k4) , where the intermediate stages k1, k2, k3, and k4 are defined as follows: k1 = f(tn, yn), k2 = f ( tn+ h 2 , yn+ h 2 k1 ) , k3 = f ( tn+ h 2 , yn+ h 2 k2 ) , k4 = f(tn+h, yn+h k3), where: h is the time step, yn is the approximation of the solution at time step tn, f(t, y) is the function representing the system of ODEs. Now, we apply the Runge-Kutta method on the determinstic model: By using the val- ues of the parameters existed in Table 1 with Initial Conditions: x1(0) = 5000, x2(0) = 150, S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 17 of 38 Figure 2: Sensitivity indices and behaviour of R0 x3(0) = 50, x4(0) = 8000, x5(0) = 100, x6(0) = 100. The time step: ∆t = 0.01 and the noise intensities: σ1, σ2, σ3, σ4, σ5, σ6 set to desired levels (e.g., 0.1, 0.3, 0.5, 0.8). Simu- late the model (1) for 100 time units. Compare deterministic and stochastic trajectories. Plot x1, x2, x3, x4, x5, x6 over time for different noise levels. The numerical simulations in this study explore the dynamics of zoonotic disease transmission between baboons and humans under various control scenarios. The simulations focus on how changes in con- trol measures, represented by parameters Hs, Hf , and Hi, impact susceptible, infected, and recovered populations for both baboons and humans. These parameters correspond to sterilization, food access restriction, and interaction reduction controls, respectively. Below is an interpretation of each figure in the manuscript, followed by explanations of the trends and outcomes observed in each simulation. Finally, the simulations reveal that control measures targeting baboon populations—such as sterilization and limiting food access—significantly reduce infection rates in both populations. In addition, strategies that reduce human-baboon interactions are highly effective in minimizing zoonotic trans- mission risks. In light of this, holistic control strategies that consider the environmental and behavioral factors influencing disease dynamics between humans and wildlife are es- sential. Figure 3 examines how different levels of the control parameters Hs, Hf , and Hi affect the infected baboon population: Subfigure (a) shows how increasing Hs gradually reduces x2, indicating that sterilization effectively limits disease spread. Subfigure (b) shows similar effects for varying Hf values, suggesting that limiting food access substan- S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 18 of 38 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 In fe c te d B a b o o n s ( I b ) Infected Baboons (I b ) with different H s values H s = 0.05 H s = 0.10 H s = 0.15 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 In fe c te d B a b o o n s ( I b ) Infected Baboons (I b ) with different H f values H f = 0.02 H f = 0.05 H f = 0.08 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 In fe c te d B a b o o n s ( I b ) Infected Baboons (I b ) with different H i values H i = 0.03 H i = 0.05 H i = 0.07 Figure 3: The effect of varying Hs(t), Hf (t), Hi(t) on x2. tially decreases infection rates. Subfigure (c) illustrates that increased human interaction control (Hi) leads to lower infection levels in baboons, as reduced contact with humans helps prevent disease spillover. Figure 3 (a-c): Effect of Varying Hs(t), Hf (t), Hi(t) on x2 (Infected Baboons). • (a): Increasing Hs reduces x2, showing that sterilization effectively limits disease spread. • (b): Increasing Hf (food restriction) significantly lowers x2, emphasizing the role of reducing food-based human-baboon interactions. • (c): Increasing Hi (interaction reduction) decreases x2, as fewer interactions reduce direct transmission risks. All control measures reduce infected baboons, with Hf having the greatest impact. Figure 4 shows the impact of different control intensities Hs, Hf , and Hi on the infected human population (x5): Subfigure (a) illustrates that increasing Hs has a limited direct effect on infected humans. Subfigure (b) indicates that food restriction (Hf ) significantly reduces x5, as this measure limits baboon-human encounters around food sources. Subfigure (c) highlights that higher Hi values effectively lower x5, demonstrating that reducing baboon- human interactions is crucial for controlling zoonotic transmission. Figure 4 (a-c) illustrate the effect of varying Hs(t), Hf (t), Hi(t) on x5 (Infected Humans). S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 19 of 38 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 In fe c te d H u m a n s ( I h ) Infected Humans (I h ) with different H s values H s = 0.05 H s = 0.10 H s = 0.15 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 In fe c te d H u m a n s ( I h ) Infected Humans (I h ) with different H f values H f = 0.02 H f = 0.05 H f = 0.08 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 In fe c te d H u m a n s ( I h ) Infected Humans (I h ) with different H i values H i = 0.03 H i = 0.05 H i = 0.07 Figure 4: The effect of varying Hs(t), Hf (t), Hi(t) on x5. • (a): Increasing Hs has a limited effect on x5 since sterilization primarily targets baboons. • (b): Increasing Hf significantly reduces x5, as controlling food interactions limits exposure. • (c): Higher Hi values effectively lower x5, demonstrating that reduced human- baboon interactions are critical for controlling zoonotic transmission. Interaction reduction (Hi) and food restriction (Hf ) are particularly effective in reducing human infections. Figure 5 explores how Hs, Hf , and Hi influence the recovered baboon population: Subfigure (a) shows that increasing Hs leads to a steady rise in x3 as more baboons recover. Subfigure (b) and (c) reveal similar trends forHf andHi, confirming that higher control measures indirectly promote recovery by reducing infection rates. Figure 5(a-c): Effect of Varying Hs(t), Hf (t), Hi(t) on x3 (Recovered Baboons). • (a): Increasing Hs steadily increases x3, showing more baboons recover when ster- ilization limits new infections. • (b): Increasing Hf leads to a rise in x3, as food restriction reduces x2. • (c): Higher Hi values also promote recovery (x3) by preventing disease spread through reduced interactions. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 20 of 38 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 200 400 600 800 1000 1200 1400 R e c o v e re d B a b o o n s ( R b ) Recovered Baboons (R b ) with different H s values H s = 0.05 H s = 0.10 H s = 0.15 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 500 1000 1500 R e c o v e re d B a b o o n s ( R b ) Recovered Baboons (R b ) with different H f values H f = 0.02 H f = 0.05 H f = 0.08 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 200 400 600 800 1000 1200 R e c o v e re d B a b o o n s ( R b ) Recovered Baboons (R b ) with different H i values H i = 0.03 H i = 0.05 H i = 0.07 Figure 5: The effect of varying Hs(t), Hf (t), Hi(t) on x3. Control measures indirectly promote recovery in baboons by reducing infection rates. Figure 6 illustrates the impact of control measures Hs, Hf , and Hi on the recovered human population (x6): Subfigure (a) indicates minimal effect from Hs, as sterilization directly impacts baboons. Subfigures (b) and (c) show that higher Hf and Hi values promote recovery in humans, likely due to reduced exposure to infected baboons. Figure 6 (a-c) illustrate the effect of varying Hs(t), Hf (t), Hi(t) on x6 (Recovered Humans). • (a): Minimal effect from Hs, as sterilization does not directly target humans. • (b): Higher Hf values significantly increase x6, as reduced exposure to infected baboons promotes recovery. • (c): Higher Hi values also increase x6, reflecting the importance of reducing human- baboon interactions. Food restriction (Hf ) and interaction reduction (Hi) have the most substantial impact on human recovery. 8.2. Application of the Milstein method To study the dynamics of the model (2), we present numerical simulations by using the Milstein numerical approximation method for the SDE presented in [76]. Then, we can S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 21 of 38 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 R e c o v e re d H u m a n s ( R h ) Recovered Humans (R h ) with different H s values H s = 0.05 H s = 0.10 H s = 0.15 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 R e c o v e re d H u m a n s ( R h ) Recovered Humans (R h ) with different H f values H f = 0.02 H f = 0.05 H f = 0.08 0 10 20 30 40 50 60 70 80 90 100 Time (days) 0 1000 2000 3000 4000 5000 6000 7000 R e c o v e re d H u m a n s ( R h ) Recovered Humans (R h ) with different H i values H i = 0.03 H i = 0.05 H i = 0.07 Figure 6: The effect of varying Hs(t), Hf (t), Hi(t) on x6. write the system (2) by the following discretized form: The Milstein scheme for numerically approximating these equations is: Yi+1 = Yi + f(Yi)∆t+ g(Yi) √ ∆tξi + 1 2 g(Yi)g ′(Yi) ( ξ2i − 1 ) ∆t, where: • Yi is the state at time step i, • f(Yi) is the deterministic term (right-hand side of the ODE), • g(Yi) is the diffusion coefficient representing noise intensity, • ξi ∼ N (0, 1) is a Gaussian random variable, • ∆t is the time step. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 22 of 38 The Milstein updates for each compartment of model (1) are: x1,i+1 = x1,i + [ rx1,i ( 1− x1,i + x2,i + x3,i K ) − ηbx1,ix2,i −Hsx1,i −Hfx1,i ] ∆t+ σ1x1,i √ ∆tξ1,i + σ1 2 x1,i ( ξ∗1,i2− 1 ) ∆t, x2,i+1 = x2,i + [ηbx1,ix2,i − κbx2,i −Hsx2,i −Hfx2,i] ∆t+ σ2x2,i √ ∆tξ2,i + σ2 2 x2,i ( ξ∗2,i2− 1 ) ∆t, x3,i+1 = x3,i + [κbx2,i −Hfx3,i] ∆t+ σ3x3,i √ ∆tξ3,i + σ3 2 x3,i ( ξ∗3,i2− 1 ) ∆t, x4,i+1 = x4,i + [−ηhx4,ix2,i −Hix4,i] ∆t+ σ4x4,i √ ∆tξ4,i + σ4 2 x4,i ( ξ∗4,i2− 1 ) ∆t, x5,i+1 = x5,i + [ηhx4,ix2,i − κhx5,i −Hix5,i] ∆t+ σ5x5,i √ ∆tξ5,i + σ5 2 x5,i ( ξ∗5,i2− 1 ) ∆t, x6,i+1 = x6,i + [κhx5,i] ∆t+ σ6x6,i √ ∆tξ6,i + σ6 2 x6,i ( ξ∗6,i2− 1 ) ∆t, (5) where {ξu,i, i ≥ 0} are sequences of random numbers uniformly distributed in [0, 1] and ∆t is a time step. Now, we apply the Milstein method to the stochastic model (2): Figure 7 illustrates the comparison of the deterministic and stochastic model (2) for all compartments. Subfigures (a-f): These subfigures compare deterministic and stochastic dynamics for all compart- ments (x1, x2, x3, x4, x5, x6) with varying levels of noise. The deterministic model (1) shows smooth trajectories, while the stochastic model (2) exhibits fluctuations due to random transmission dynamics. Larger noise levels amplify variability, highlighting the importance of the stochastic model (2) in capturing real-world uncertainty. Stochastic model (2) provides a more realistic representation of disease dynamics, especially under conditions of environmental and social variability. 8.3. Comparison of stochastic and deterministic model Comparison of Deterministic and stochastic model (2) for x1(t) and x4(t). This figure highlights the dynamics of susceptible baboons (x1) and humans (x4) over time, comparing deterministic and stochastic model (2). Variability in the stochastic model (2) reflects the randomness in disease transmission dynamics due to environmental or social factors. The stochastic model (2) captures fluctuations in susceptible populations, particularly in human-baboon interactions. Compared to Runge-Kutta and Milstein methods, FDM is Method Advantages Disadvantages Runge-Kutta (RK4) High accuracy, stable for stiff problems Computationally expensive Milstein Method Captures stochastic effects well Requires stochastic differen- tial equation formulation Table 4: Comparison of Numerical Methods S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 23 of 38 0 50 100 150 200 250 300 Time 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 S b P o p u la ti o n S b Population Dynamics: Deterministic vs Stochastic Deterministic 0 50 100 150 200 250 300 Time 0 2000 4000 6000 8000 10000 12000 S h P o p u la ti o n S h Population Dynamics: Deterministic vs Stochastic Deterministic 0 50 100 150 200 250 300 Time 0 1000 2000 3000 4000 5000 6000 7000 I b P o p u la ti o n I b Population Dynamics: Deterministic vs Stochastic Deterministic 0 50 100 150 200 250 300 Time 0 1000 2000 3000 4000 5000 6000 7000 I h P o p u la ti o n I h Population Dynamics: Deterministic vs Stochastic Deterministic 0 50 100 150 200 250 300 Time 0 2000 4000 6000 8000 10000 12000 14000 R b P o p u la ti o n R b Population Dynamics: Deterministic vs Stochastic Deterministic 0 50 100 150 200 250 300 Time 0 1 2 3 4 5 6 R h P o p u la ti o n 10 4 R h Population Dynamics: Deterministic vs Stochastic Deterministic Figure 7: A comparison of deterministic and stochastic diabetes models with varying levels of noise=0.1, 0.2, 0.3, 0.4. easier to implement but may require smaller time steps to maintain stability. In future studies, implicit FDM techniques could be explored for better stability in stiff systems. 9. Discussion This study presents a mathematical model for zoonotic disease transmission between baboons and humans, incorporating both deterministic and stochastic frameworks. By integrating key control strategies such as sterilization, restricted food access, and reduced S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 24 of 38 human-baboon interactions, we provide a comprehensive analysis of disease dynamics at the human-wildlife interface. The deterministic model establishes fundamental properties such as solution existence, uniqueness, and boundedness, while the stochastic extension accounts for environmental and social variability. Theoretical and numerical analyses provide insights into disease persistence, eradication conditions, and optimal interven- tion strategies. The model (1) incorporates key intervention strategies such as steriliza- tion, restricted food access, and reduced human-baboon interactions to mitigate disease spread. Analyzing both deterministic and stochastic versions of the model (1), we provide insights into how environmental and social variability influence disease dynamics. The deterministic model (1) establishes the theoretical foundations of the system, including solutions’ existence, uniqueness, non-negativity, and boundlessness. Stability analysis of the infection-free and endemic equilibrium points reveals that the basic reproductive num- ber, R0, is critical for determining disease persistence or eradication. If R0 < 1, the disease will eventually die out, while R0 > 1 indicates sustained transmission within the popula- tion. Sensitivity analysis identifies key parameters, such as transmission rates (ηb, ηh) and recovery rates (κb, κh), as significant drivers of disease spread. In a stochastic version of the model, random fluctuations are incorporated into disease dynamics using Milstein’s method. In this manner, real-world variables, such as population interactions and en- vironmental conditions, are captured, which are not incorporated into the deterministic model (1). In stochastic simulations, higher noise levels increase the variability of pop- ulation dynamics. Therefore, when designing robust interventions, randomness must be taken into account. Numerical simulations demonstrate the effectiveness of the proposed control measures. Sterilization (Hs) reduces the infected baboon population by limiting reproduction, while restricted food access (Hf ) significantly decreases transmission rates by decreasing human-baboon interactions over shared food resources. Interaction reduc- tion (Hi) is particularly effective at minimizing human infections. In both populations, a combination of these strategies results in a more comprehensive control strategy that results in fewer infections and higher recovery rates. A combination of environmental and behavioral factors affecting the transmission of disease should be addressed in integrated control strategies. Public health efforts should focus on reducing human-baboon interac- tions and improving recovery rates through medical interventions for humans. Wildlife management strategies, including sterilization and habitat control, are essential for miti- gating risks at the source. To account for geographical factors influencing disease spread, future research should incorporate spatial dynamics. Additionally, collecting detailed em- pirical data on baboon-human interactions and disease prevalence in the Al-Baha region will enhance model (1) accuracy and provide better insights for intervention planning. By including migrations and seasonal variations in the stochastic framework, the model (1) could be further improved. Overall, this study highlights the value of combining deter- ministic and stochastic model (2) approaches to better understand and control zoonotic diseases. By identifying critical parameters and evaluating control strategies, this work contributes to the development of effective public health and wildlife management policies aimed at reducing zoonotic disease risks. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 25 of 38 9.1. Key Findings and Contributions One of the primary findings of this study is the role of the basic reproduction num- ber, R0, in determining disease dynamics. Stability analysis reveals that when R0 < 1, the disease-free equilibrium is stable, implying that zoonotic disease transmission can be controlled or eradicated under appropriate intervention measures. Conversely, if R0 > 1, the disease persists within the population, highlighting the need for sustained control ef- forts. Sensitivity analysis identifies transmission rates between baboons and humans as the most influential parameters affecting R0. This underscores the importance of interventions targeting direct contact and environmental contamination pathways. By incorporating stochasticity using the Milstein method, we demonstrate that envi- ronmental and behavioral fluctuations significantly impact disease transmission dynamics. The stochastic model shows higher variability in infection trends, particularly under con- ditions where control measures are inconsistently applied. This finding emphasizes the necessity of accounting for randomness in zoonotic disease modeling, as real-world trans- mission patterns rarely follow deterministic trajectories. 9.2. Novel Contributions and Medical Relevance of the Study This study presents a novel mathematical model that integrates both deterministic and stochastic frameworks to analyze zoonotic disease transmission between baboons and hu- mans. While classical compartmental models, such as the Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR) frameworks, have been widely used to describe disease transmission, they often assume homogeneous mixing and overlook interspecies interactions, which are fundamental in zoonotic dynamics. Similarly, ecolog- ical models like the Lotka-Volterra framework, commonly applied to predator-prey in- teractions, have limited applicability in modeling human-animal disease transmission. By explicitly incorporating human-baboon interaction terms and control strategies—including sterilization, food restrictions, and reduced contact—this study extends classical epidemi- ological frameworks to better capture the complexities of zoonotic disease transmission. Unlike prior deterministic approaches to wildlife-related disease modeling, such as those applied to rabies or avian influenza, this study employs a stochastic extension using the Milstein method. This accounts for environmental and social variability, making the re- sults more reflective of real-world conditions. The region-specific application to Al-Baha, Saudi Arabia, where human-baboon interactions are prevalent due to food provisioning and shared resources, further highlights the necessity of tailoring intervention strategies to localized ecological and social conditions. The medical relevance of this study lies in its ability to inform public health officials and wildlife conservationists on designing data-driven intervention strategies to mitigate disease transmission risks from wildlife to humans. Through sensitivity analysis, this work systematically identifies the most influ- ential transmission parameters, offering valuable insights for optimizing control measures and minimizing infection risks. A key distinction of this study is its comprehensive parametric sensitivity analysis, which identifies the transmission rates among baboons (ηb) and from baboons to humans S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 26 of 38 (ηh) as the most critical factors influencing disease spread. Traditional models primarily establish threshold conditions for disease persistence but often neglect to quantify the relative importance of individual parameters. By highlighting the sensitivity of R0 to ηb and ηh, this study emphasizes the need for targeted interventions focused on limiting direct human-baboon interactions and environmental contamination pathways. Comparisons with established numerical methods, such as the Runge-Kutta method and Generalized Differential Transform Methods (GDTM), further demonstrate that the stochastic approach provides a more accurate representation of disease fluctuations un- der uncertain environmental conditions. Stochastic modeling better accounts for random perturbations in transmission dynamics, which are particularly relevant in ecological and wildlife disease settings where deterministic models may oversimplify real-world variability. Furthermore, this study advances classical epidemiological modeling by integrating an optimal control framework, allowing for a quantitative assessment of intervention strate- gies in terms of both effectiveness and cost. While many existing studies analyze inter- ventions qualitatively, they often lack a structured optimization approach to determine the most efficient long-term implementation of control measures. Our results demonstrate that combining sterilization and reduced human-baboon interaction strategies provides the most effective long-term control of zoonotic disease transmission. The numerical sim- ulations confirm that a multi-pronged approach, incorporating sterilization, food access restrictions, and minimized human-baboon interactions, is essential for achieving sustained reductions in disease prevalence. Unlike generalized epidemiological models, our framework is specifically designed to evaluate the impact of region-specific interventions, enhancing its real-world applicability. The findings of this study can directly inform public health policies and wildlife man- agement strategies in regions with significant human-wildlife interactions. Moreover, the model can be extended to similar zoonotic transmission scenarios involving other wildlife species that frequently interact with human populations. Overall, this study represents an original research contribution that advances the mathematical modeling of zoonotic diseases. The novelty of this work lies in the hybrid deterministic-stochastic approach, the detailed parametric sensitivity analysis, the region- specific application to Al-Baha, and the integration of an optimal control framework for intervention strategies. These elements distinguish this study from previous research and provide a comprehensive and practical framework for managing zoonotic disease transmis- sion. Future research directions could enhance this framework by incorporating vaccination strategies, spatial dynamics, or climate-driven seasonal variations to further refine disease control policies. Expanding the model to account for host-pathogen co-evolution and behavioral adaptations in both humans and wildlife could also provide deeper insights into long-term disease dynamics. As human-wildlife interactions continue to evolve due to environmental changes and urban expansion, mathematical models such as the one presented in this study will be increasingly crucial for guiding disease prevention and control efforts. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 27 of 38 9.3. New Achievements in This Study This study introduces several significant advancements in zoonotic disease modeling. First, explicit inter-species transmission is modeled by fully incorporating zoonotic disease dynamics between baboons and humans, accounting for both direct and indirect pathways of infection. Second, control strategies are integrated by introducing three intervention measures—sterilization, food control, and interaction reduction—each evaluated through numerical simulations to assess their effectiveness. Third, the Milstein method is employed to extend the deterministic model into a stochastic framework, allowing for the capture of random fluctuations in disease spread, thereby enhancing the model’s realism. Fourth, a comprehensive sensitivity analysis identifies the most influential parameters affecting the transmission dynamics, providing insight for targeted interventions. Fifth, the model is applied to a real-world case study in the Al-Baha region, making the findings directly relevant to regional disease management efforts. Finally, the study expands to an optimal control framework, evaluating strategies to minimize infections while balancing the associ- ated costs through various control scenarios. Overall, this study addresses previous limita- tions by developing a comprehensive, intervention-based, hybrid deterministic-stochastic model for zoonotic disease transmission, with real-world applicability, advanced numeri- cal techniques, and an in-depth evaluation of control strategies, representing a significant advancement in epidemic modeling. 9.4. Modeling Zoonotic Disease Transmission Between Baboons and Hu- mans in Al-Baha This study presents a mathematical model aimed at analyzing and mitigating the transmission of zoonotic diseases between baboons and humans in the Al-Baha region. The model integrates both deterministic and stochastic approaches to evaluate disease dynamics under various environmental and social conditions. Key findings include: The model incorporates three primary intervention measures—sterilization (Hs), food access restriction (Hf ), and human-baboon interaction reduction (Hi)—which significantly reduce infection levels. Among these, food access restriction and interaction reduction have the strongest impact. The basic reproduction number, R0, derived as R0 = ηbx 0 1 κb +Hs + ηhx 0 4 κh +Hi , determines disease persistence or eradication. If R0 < 1, the disease dies out, whereas if R0 > 1, long-term control measures are necessary. Increasing recovery rates (κb, κh) and enhancing control strategies (Hs, Hf , Hi) effectively reduce R0. Sensitivity analysis reveals that transmission rates (ηb, ηh) are the most influential parameters in increasing R0, while increasing recovery rates and intervention measures reduce it. Among the interventions, food access restriction (Hf ) is identified as the most effective in controlling disease spread. A comparison of deterministic and stochastic mod- els highlights important differences. Deterministic models assume uniform transmission dynamics, while stochastic models, implemented using the Milstein method, account for S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 28 of 38 real-world fluctuations in disease spread. Stochastic fluctuations introduce variability in infection levels, with higher noise levels amplifying uncertainty and providing a more accu- rate representation of unpredictable transmission patterns. These findings emphasize the importance of using stochastic models to predict real-world outbreaks and evaluate control strategies. The implications for public health and wildlife management include designing more effective strategies to prevent zoonotic outbreaks, implementing targeted control measures such as sterilization campaigns and restricted food access, and using mathe- matical modeling as a predictive tool to assess intervention strategies before real-world implementation. Overall, this study provides actionable insights into controlling zoonotic disease transmission, emphasizing the need for data-driven public health measures and adaptive wildlife management strategies. 9.5. Previous Shortcomings in Existing Models Several limitations exist in previous zoonotic disease models. First, limited consider- ation of inter-species transmission has been a key issue, as many past models primarily focused on human-to-human transmission or generalized animal-to-human interactions, without explicitly modeling baboon-human interactions. Some models also assumed sim- plified transmission pathways, failing to capture the complexity of zoonotic spillovers. Second, there has been a lack of integrated control strategies, as previous studies often neglected explicit intervention measures such as sterilization, food restriction, and inter- action reduction. When control measures were included, they were not tested through optimal control theory to determine their effectiveness over time. Third, many models lacked a stochastic framework, relying solely on deterministic approaches that do not ac- count for random environmental and social variations affecting disease spread. Stochastic- ity plays a critical role in zoonotic diseases, where unpredictable factors such as seasonal changes, human behavior, and ecological shifts can significantly influence transmission dynamics. Fourth, sensitivity analysis in existing models has been limited, with most studies conducting only basic threshold analyses without fully investigating the most in- fluential parameters for disease control. Identifying key parameters is crucial for effective intervention, yet many models lacked detailed sensitivity analysis. Finally, a generalized application without regional context has reduced the practical impact of past studies, as many models were developed without applying them to specific real-world cases, making their conclusions harder to implement for policy-making. The absence of localized data has further limited their applicability in designing effective disease control strategies. 9.6. Study Limitations, Implications, and Future Research Recommen- dations This study has certain limitations, including the assumption of homogeneous mixing of populations, which may not accurately represent real-world interactions, the lack of spatial considerations in disease spread, uncertainty in parameter estimations due to lim- ited empirical data, and the exclusion of vaccination strategies as a potential intervention. Additionally, the model assumes a closed population with no migration effects, which may S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 29 of 38 not fully capture the complexity of disease dynamics in open ecosystems. While the model considers key control measures, other factors such as climate change, habitat fragmenta- tion, and pathogen evolution may also influence zoonotic disease transmission. Expanding the model to include these elements could provide a more holistic understanding of dis- ease spread. Moreover, the assumption of constant control parameters over time does not account for the potential variability in intervention strategies due to resource availability, policy changes, or behavioral adaptations among humans and baboons. Future research could develop adaptive control models that account for dynamic intervention strategies and improve model accuracy by incorporating migration patterns and seasonal variations. Empirical validation of the model through field data collection is also essential for refining parameter estimates and enhancing model predictions. Despite these limitations, the find- ings have important implications for public health and wildlife management, emphasizing the effectiveness of targeted control strategies such as restricting food access and reducing human-baboon interactions, the integration of stochastic modeling into policy decision- making, and the necessity for cross-disciplinary collaboration among epidemiologists, con- servation biologists, and policymakers. Future research should focus on incorporating spatial and seasonal disease dynamics, collecting more empirical data on baboon-human interactions and disease prevalence in the Al-Baha region, evaluating the impact of vacci- nation strategies, and enhancing stochastic modeling by integrating additional sources of randomness to improve real-world applicability. 9.7. Comparison with Existing Studies and Justification for Variations This study builds upon previous research on zoonotic disease transmission but intro- duces novel aspects that enhance its real-world applicability. Unlike traditional models that focus primarily on human-to-human disease spread or generalized animal-human in- teractions, this study explicitly models interspecies transmission between baboons and humans, incorporating region-specific control measures such as sterilization, restricted food access, and reduced human-baboon interactions. Studies conducted in other regions, such as Africa and Asia, have focused on zoonotic transmission from other wildlife species, often overlooking the unique dynamics of baboon-human interactions. Additionally, many previous models rely solely on deterministic approaches, whereas this work integrates a stochastic framework using the Milstein method, making it more suitable for capturing real-world variability in disease spread. The observed variations between this study and others may arise from differences in ecological settings, social behaviors, and intervention strategies. For instance, while some studies emphasize vaccination as a primary control measure, this research focuses on behavioral interventions due to the lack of existing vac- cines for certain zoonotic diseases transmitted by baboons. The findings of this study are directly applicable to zoonotic disease control in the Al-Baha region but can also be adapted to similar wildlife-related transmission scenarios worldwide. Previous studies on zoonotic disease transmission have primarily focused on deterministic models [7, 8]. While these models provide valuable insights into disease spread, they often fail to cap- ture the randomness inherent in real-world epidemiology. Our study builds upon this S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 30 of 38 work by introducing a stochastic framework, which better reflects the uncertainties in human-wildlife interactions and environmental factors. The use of the Milstein method enhances the model’s accuracy in representing stochastic fluctuations, making it a more realistic tool for policy recommendations. Additionally, while traditional models have ex- amined zoonotic transmission in broad ecological contexts, few studies have specifically analyzed human-baboon disease interactions. Our work fills this gap by focusing on the unique socio-environmental conditions of the Al-Baha region, where human-baboon en- counters are frequent. This region-specific application provides a foundation for localized intervention strategies that can be adapted to similar ecological settings worldwide. 9.8. Comparative Analysis of Epidemic Models and Numerical Approaches for Zoonotic Disease Transmission This study extends traditional epidemic models by incorporating zoonotic disease transmission between baboons and humans using a hybrid deterministic-stochastic frame- work. While classical compartmental models such as the Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR) frameworks have been widely employed to describe disease transmission in human and animal populations, they often assume homogeneous mixing and overlook interspecies interactions, which are fundamen- tal to zoonotic dynamics. Similarly, models like the Lotka-Volterra framework, commonly used in ecological systems to examine predator-prey dynamics, have limited applicability to human-animal disease interactions. By incorporating human-baboon interaction terms and control measures such as sterilization, food restrictions, and reduced contact, this study enhances classical frameworks to better capture the complexities of zoonotic disease transmission. Unlike previous deterministic approaches for wildlife-related disease transmission, such as those applied to rabies or avian influenza, this work employs a stochastic extension using the Milstein method, which accounts for environmental and social variability, making the results more applicable to real-world scenarios. A region-specific application to the Al- Baha region of Saudi Arabia, where human-baboon interactions are particularly common due to food provisioning and shared resources, further underscores the necessity of tailoring intervention strategies to local conditions. Numerical analysis, particularly sensitivity analysis, reveals that the transmission rates among baboons (ηb) and from baboons to humans (ηh) are the most influential parame- ters, highlighting the need for targeted interventions. Comparisons with numerical meth- ods such as the Runge-Kutta method and Generalized Differential Transform Methods (GDTM) demonstrate that the stochastic approach provides a better representation of disease prevalence fluctuations under uncertain environmental conditions. Additionally, the study leverages optimal control theory to assess the cost-effectiveness of various inter- ventions, drawing parallels with models used for malaria and COVID-19, where targeted control strategies such as sterilization and food restrictions significantly reduce infections. The integration of deterministic and stochastic modeling, along with numerical meth- ods and optimal control, offers a robust framework for understanding zoonotic disease S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 31 of 38 transmission. These insights provide valuable guidance for public health officials and wildlife managers seeking to mitigate disease risks at the human-wildlife interface. Future research could enhance this framework further by incorporating vaccination strategies or spatial dynamics to refine disease control policies. 9.9. Previous Shortcomings and New Achievements Many previous studies on zoonotic disease modeling exhibit several limitations, in- cluding limited consideration of interspecies transmission, as most past models primarily focused on human-to-human transmission or generalized animal-to-human interactions without explicitly modeling baboon-human interactions. Additionally, few models have integrated targeted interventions such as sterilization, food restriction, and interaction reduction into a comprehensive framework. Many studies rely solely on deterministic models, which fail to capture environmental and social variability affecting disease trans- mission. Prior models often conduct only simple threshold analyses without investigating the most influential parameters governing disease dynamics, and most are developed with- out region-specific applications, reducing their practical value for policy implementation. This study addresses these limitations by introducing several key advancements, in- cluding explicit interspecies transmission modeling, capturing both direct and indirect pathways of infection, and integrating three intervention strategies—sterilization, food access restriction, and reduced human-baboon interactions—evaluated through numeri- cal simulations. A stochastic extension using the Milstein method is applied to account for random fluctuations in transmission, enhancing real-world applicability. Additionally, a detailed parametric sensitivity analysis identifies the most influential factors affecting disease spread, providing insights for optimal intervention strategies. Unlike previous generalized models, this study is tailored to baboon-human interac- tions in the Al-Baha region, ensuring direct applicability to regional disease management. Furthermore, an optimal control framework is incorporated, evaluating strategies to min- imize infection rates while balancing intervention costs. By addressing past limitations and introducing novel modeling techniques, this study significantly advances the field of zoonotic disease epidemiology. The integration of deterministic and stochastic approaches, coupled with real-world applications, contributes valuable insights to public health and wildlife management efforts. 9.10. Advantages of the Proposed Method Compared to Existing Ap- proaches The manuscript focuses on theoretical predictions for the epidemic using a mathe- matical modeling framework that integrates deterministic and stochastic elements. While various numerical solution algorithms exist for epidemiological models, the proposed ap- proach offers several key advantages compared to traditional methods such as Runge-Kutta methods, which provide accurate numerical solutions but struggle with stochastic effects crucial for real-world disease modeling; finite difference methods, which are often used S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 32 of 38 for solving partial differential equations in spatially distributed models but may not ef- fectively capture memory effects in fractional-order systems; and variational iteration and Adomian decomposition methods, which are suitable for solving nonlinear equations but can be computationally expensive and less efficient for large-scale epidemic simulations. The proposed method improves upon these approaches by incorporating a stochastic extension through the Milstein numerical scheme, which accounts for random fluctua- tions in disease transmission, making predictions more realistic; fractional-order mod- eling, which captures memory effects and long-term epidemic behavior more accurately than Markovian dynamics; and comprehensive sensitivity analysis, which identifies key parameters affecting disease spread to optimize intervention strategies. Additionally, this hybrid deterministic-stochastic approach bridges the gap between theoretical predictions and real-world uncertainties, while the inclusion of an optimal control framework allows policymakers to evaluate and implement the most effective disease mitigation strategies. Compared to traditional numerical methods, the proposed approach provides improved accuracy in modeling zoonotic disease dynamics by integrating stochasticity, fractional cal- culus, and sensitivity analysis, making it more applicable to real-world epidemic scenarios and offering valuable insights for public health interventions. 9.11. Implications for Public Health and Wildlife Management The insights from this study have direct applications for public health policies and wildlife management strategies. Implementing a combination of sterilization, restricted food access, and reduced human-baboon interactions can significantly lower disease trans- mission risks. Among these measures, food restriction appears to have the greatest im- pact, as it reduces direct contact between baboons and humans while also limiting indi- rect transmission via contaminated food sources. Additionally, sterilization programs can help manage baboon population sizes, further reducing the likelihood of sustained disease transmission. From a public health perspective, these findings support policies aimed at regulating human behavior in areas with high wildlife interaction. Community education on the risks of feeding baboons and improved waste management strategies can mini- mize exposure to zoonotic pathogens. Furthermore, early disease surveillance programs that monitor baboon populations for potential outbreaks can enhance proactive response strategies. 10. Conclusion This study develops a comprehensive mathematical model to analyze zoonotic disease transmission between baboons and humans in the Al-Baha region, integrating both deter- ministic and stochastic frameworks to assess the impact of key control measures—sterilization, food access restriction, and human-baboon interaction reduction—on mitigating disease spread. Through theoretical analysis, we established the existence, uniqueness, and bound- edness of solutions, and stability analysis revealed that the basic reproduction number (R0) is a critical threshold for disease persistence. The stochastic model, using the Mil- S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5881 33 of 38 stein method, provides a realistic representation of disease dynamics, capturing the effects of random environmental and social factors. Sensitivity analysis highlighted that trans- mission rates and control measures significantly affect disease spread, with food access restriction and interaction reduction being the most effective interventions. Key findings indicate that a strong intervention is needed when R0 > 1 and that stochastic models offer a more accurate reflection of real-world disease dynamics compared to determinis- tic models. This study provides valuable insights for public health officials and wildlife managers, suggesting that future zoonotic disease control strategies should prioritize food access restrictions, human-baboon interaction reduction, and adopt stochastic models to account for real-world uncertainties. The research also lays the groundwork for future studies exploring seasonality, climate effects, vaccination strategies, and spatial dynamics in zoonotic disease transmission, offering practical recommendations for protecting both human and wildlife populations from emerging infectious diseases. This study highlights the importance of integrating deterministic and stochastic modeling approaches to under- stand and control zoonotic disease transmission. By identifying key transmission path- ways, assessing control strategies, and incorporating stochastic fluctuations, our findings provide actionable insights for public health officials and wildlife managers. The results emphasize that a multi-faceted approach—combining sterilization, food restriction, and reduced human-baboon interactions—is essential for mitigating disease risks. Future re- search should focus on refining the model to include migration effects, dynamic control strategies, and empirical validation through real-world data. The integration of advanced computational techniques and field-based studies will further enhance our understanding of zoonotic disease dynamics, ultimately contributing to more effective disease prevention and wildlife management strategies. Acknowledgements The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4220004GSSR03. Data Availability All data is included in the manuscript. https://doi.org/10.32604/cmc.2022.020732 Funding This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4220004GSSR03. References [1] K. E. Jones, N. G. Patel, M. A. Levy, A. Storeygard, D. Balk, J. L. Gittleman, and P. Daszak. Global trends in emerging infectious diseases. Nature, 451(7181):990–993, 2008. S. Saber, A. A. 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