Analysis of an Acute Diarrhea Piecewise Modified ABC Fractional Model: Optimal Control, Stability and Simulation

Abstract

Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional behaviors. We analyze the local and global stability of the disease-free equilibrium and calculate the basic reproduction number $R_0$ using the next-generation matrix method. Furthermore, we formulate an optimal control model that incorporates both strategies to reduce contact between susceptible and infected individuals, and to treat infected patients. Numerical simulations demonstrate the model’s behavior, illustrating that enhanced hygiene compliance reduces $R_0$ by decreasing contact rates, while increased effective contact rates elevate $R_0$. Additionally, the simulations reveal a positive correlation between higher concentrations of acute diarrhea bacteria and increased rates of subsequent infections.

1. Introduction

Mathematical modeling, particularly through systems of fractional differential equations (FDEs), has emerged as a powerful and versatile tool for dissecting the complex dynamics of infectious diseases and evaluating the effectiveness of various control strategies [1,2,3,4,5,6]. These models offer a lens through which to capture the intricate, non-linear processes inherent in disease transmission, providing valuable insights that can inform evidence-based interventions. The ability to represent disease spread mathematically allows for a deeper understanding of the underlying mechanisms, moving beyond mere observation to enable targeted strategies. At the heart of many recent advancements in this area lies the ABC operator, a powerful mathematical operator introduced by Atangana and Baleanu [7], which has proven adept at modeling real-world systems with complex dynamics. This operator facilitates the accurate representation and analysis of diverse phenomena across numerous fields, ranging from the realms of physics and engineering to the intricacies of biological processes and the fluctuations of financial markets [8,9,10,11]. These diverse applications underscore the broad utility of fractional calculus in modeling complex systems.

In the specific context of diarrheal diseases, recent research has demonstrated the potential of mathematical models to capture the nuances of these infections. For instance, Berhe et al. [12] developed a non-linear, autonomous model for dysentery diarrhea that not only illuminated the influence of environmental factors on disease transmission rates but also validated its findings with real-world statistical data. Simultaneously, Iqbal et al. [13] utilized the Mittag-Leffler kernel in their novel model of diarrhea, examining key qualitative characteristics like solution existence, uniqueness, equilibrium points, and asymptotic stability, thus offering deeper theoretical insights into the behavior of the disease. These works demonstrate a growing appreciation for the value of sophisticated modeling techniques in understanding and controlling diarrheal outbreaks. Furthermore, the influence of sanitation on these complex systems was investigated by Lasisi et al. [14], where the modeling of acute diarrhea was taken into account, and possible control measures were explored. Qureshi et al. [15] studied the diarrhea transmission dynamics using real data with fractal-fractional derivative.

Recently, Atangana and Araz [16] pushed the boundaries of this field by extending the modified ABC (mABC) operator, previously introduced by Al-Refai and Baleanu [17], into a piecewise mABC fractional operator (pmABC). This innovation combines classical and modified fractional derivatives. This approach has garnered significant interest in the scientific community due to its ability to examine crossover behavior—the intriguing transition or switch in a model’s dynamics between time intervals governed by either classical or mABC fractional derivatives. The ability to capture these changes in behavior enhances the realism and applicability of models in dynamic and non-linear environments. The pmABC fractional operator is a powerful tool, and for more insights into this area, readers can consult these key references [18,19,20,21]. Researchers have already demonstrated the potential of the pmABC fractional operator for modeling a diverse range of real-world issues, while also exploring the crossover behavior in these models [22,23,24,25]. These investigations have convincingly shown that the use of pmABC operator allows for a more profound understanding of a disease’s dynamics and underlying characteristics.

Building on these advances, and recognizing the significance of the pmABC operator and its ability to investigate the crossover behavior of disease dynamics, we introduce a novel mathematical model for diarrhea dynamics that incorporates this pmABC fractional operator. Our study aims to extend existing knowledge by focusing on the specific advantages of this operator within the complex context of diarrhea. We aim to go beyond the capabilities of prior models, utilizing pmABC’s power to address gaps and to provide new perspectives on the disease mechanisms, using a powerful and versatile tool. Our contributions to this work are as follows:

• Novel Application of Piecewise mABC for Multi-Scale Dynamics: The study introduces the piecewise mABC fractional operator to model acute diarrhea dynamics, enabling the analysis of multi-scale dynamics and the influence of varying time scales on disease behavior within a unified framework. This approach provides a novel perspective on capturing complex disease dynamics.
• Enhanced Modeling Accuracy and Disease Insight: Simulations using the piecewise mABC operator demonstrate improved precision over existing methods, offering a more detailed and accurate representation of acute diarrhea dynamics. This advancement contributes to a deeper understanding of the disease’s complex behavior.
• Exploration of Crossover Effects and Theoretical Analysis: This study investigates crossover effects within the acute diarrhea model, revealing critical insights into dynamic transitions during disease progression. Additionally, a comprehensive theoretical analysis is conducted, including the establishment of an invariant region, solution positivity, equilibrium points, and the basic reproduction number, all within a social hierarchy context.
• Fractional-Order System Analysis and Numerical Visualization: A detailed examination of the fractional-order model is presented, focusing on the existence, uniqueness, and stability of solutions. This study also provides numerical solutions and graphical interpretations, enhancing the understanding of model behavior and facilitating effective result interpretation. Comparisons of dynamics with and without control around the crossover point are also included.

To provide a clear roadmap for the remainder of this paper, the subsequent sections are structured as follows. Section 2 will lay the foundation by introducing the pmABC fractional model, detailing its formulation and its capacity to capture the complex dynamics of acute diarrhea transmission. Shifting to a practical perspective, Section 3, Crossover Behavior, will delve into the qualitative aspects of the model, while Section 4 will focus on critical mathematical properties including the existence and uniqueness of the model’s solutions, their boundedness and positivity, the calculation of the basic reproduction number ($R_0$), and an assessment of the local and global stability of the disease-free equilibrium point. Then, in Section 5, we will transition to considering the application of control strategies, formulating a fractional optimal control model that incorporates two distinct control measures, denoted as ${\alpha }_1$ and ${\alpha }_2$. Control ${\alpha }_1$ focuses on minimizing the interaction between susceptible individuals and bacteria, thereby directly reducing transmission, while ${\alpha }_2$ emphasizes the effective treatment of infected individuals who have already developed symptoms of the disease. Then, in Section 6, we will address sensitivity analysis to provide insights into the key factors influencing disease transmission and the effectiveness of controls. The numerical scheme of the fractional optimal control model utilizing the pmABC fractional derivative is described in detail in Section 7, providing the methodological foundation for our results. The simulation and comparative results are presented in Section 8. We will then discuss the numerical simulation results and findings in Section 9, focusing on the practical insights gained from our modeling approach. In Section 10, we present valuable insights for developing effective public health strategies. Finally, Section 11 will provide concluding remarks that summarize the significance of our work and provide a direction for future research.

2. Mathematical Model

Here, we extend the integer-order model presented in [14] by formulating a pmABC fractional model, noting that the fractional order $\varsigma$ influences all model parameters. The total population is divided into five groups as follows:

• Susceptible ($\mathbb{S}$): This group represents individuals at risk of acquiring acute diarrhea infection.
• Infected ($\mathbb{I}$): This group represents individuals who have been infected and are exhibiting symptoms of the infection.
• Vaccination ($\mathbb{V}$): This group represents individuals who have received vaccination against the infection.
• Recovery ($\mathbb{R}$): This group represents individuals who have recovered from the infection and are no longer exhibiting symptoms. However, they may still be susceptible to contracting the disease again.
• Bacteria ($\mathbb{B}$): This group represents the concentration of bacteria associated with the infection. It interacts with the susceptible population $\mathbb{S}$, leading to infection $\mathbb{I}$ through a nonlinear incidence rate known as the force of infection ${\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}.$

The pmABC fractional model of diarrhea disease in the interval $\left[0,T\right]$ is presented as follows:

$$\left\{\begin{array}{c}{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)=\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }+w_1^{\varsigma }\mathbb{V}+{\psi }^{\varsigma }\mathbb{R}-\left({\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}+w_2^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{S},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)={\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\lambda }^{\varsigma }{\Lambda }^{\varsigma }+w_2^{\varsigma }\mathbb{S}-\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{V},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)={\gamma }^{\varsigma }\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B},\hfill \end{array}\right.$$

(1)

under initial conditions $\mathbb{S}\left(0\right)>0,\mathbb{I}\left(0\right)>0,\mathbb{V}\left(0\right)>0$, and $\mathbb{R}\left(0\right)>0,$ where ${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\eta \left(\iota \right)$ is the pmABC fractional derivative defined as [17]

$${}_0{}^{pmABC}\mathbf{D}_t^{\varsigma }\eta \left(\iota \right)=\left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\eta \left(\iota \right),\iota \in \left[0,{\iota }_1\right],\\ \\ {}^{mABC}\mathbf{D}_0^{\varsigma }\eta \left(\iota \right),\iota \in \left[{\iota }_1,T\right].\end{array}\right.$$

(2)

The model incorporates additional factors, such as the human population growth rate $\Lambda$, the rates at which individuals lose immunity from vaccination $w_1$ and then recover to become susceptible again $\psi$, the vaccination rate $w_2$, an effective contact rate or exposure to contaminated sources $\beta$, the compliance rate of water and food hygiene $\phi$, the recovery rate of infected individuals $\gamma$, the production rate of bacteria infection from infected individuals $\theta$, the natural human mortality rate $\mu$, the disease-induced death rate $\delta$, the proportion of unvaccinated individuals $\lambda$, the concentration of bacteria in contaminated water K, the mortality rate for bacteria ${\mu }_p$, and the rate of water sanitation leading to a reduction in bacteria $\varphi$. These variables and parameters are important for our understanding of diarrhea transmission dynamics by considering population dynamics, vaccination, exposure, recovery, mortality, and environmental factors related to bacterial contamination and sanitation. They help in modeling and analyzing the dynamics of diarrhea within the population, aiding in the development of effective control and prevention strategies.

The population size is described in

Table 1. The population size for the model under consideration.

Variable	Definition
$\mathbb{S}$	Number of susceptible individuals at time $\iota$.
$\mathbb{I}$	Number of individuals infected and showing symptoms of the infection.
$\mathbb{V}$	Number of individuals who have been vaccinated against the infection.
$\mathbb{R}$	Number of individuals who have recovered from the infection

, and the definitions and values of the parameters are described in

Table 2. The parameters for the model under consideration.

Parameter	Definition of Parameter	Value	Units
$\Lambda$	Rate of human populations	100	day−1
$w_1$	Rate of immune decline in vaccinated individuals	0.55	day−1
$\psi$	Rate of recovered humans becoming susceptible	0.003	day−1
$w_2$	Vaccination rate among individuals	0.45	day−1
$\beta$	Contaminated food and water exposure rate	0.9	day−1
$\varphi$	Rate of compliance with water and food hygiene practices	0.6	day−1
$\gamma$	Recovery rate of infected humans	0.002	day−1
$\theta$	Bacterial infection transmission rate from infected humans	0.8	cell/mL/day
$\mu$	Natural human mortality	0.0247	day−1
$\delta$	Mortality rate caused by the disease	0.052	day−1
$\lambda$	Proportion of unvaccinated individuals	0.075	day−1
K	Bacterial concentration in contaminated water	50,000
${\mu }_p$	Mortality rate associated with bacterial infections	0.001	day−1
$\phi$	Water sanitation leading to dearth of bacteria	0.05	day−1

.

The schematic diagram of the acute diarrhea model is shown in Figure 1.

3. Crossover Behavior

The pmABC fractional operator facilitates the exploration of various scenarios and conditions within the model. By partitioning the model into distinct segments or intervals, each defined by specific parameters or rules, this approach accommodates the influence of various factors or mechanisms on disease transmission dynamics at different stages or under specific conditions. Specifically, this operator is used within the context of our diarrhea model to investigate the crossover behavior. The pmABC methodology combines the classical derivative with the mABC operator. The entire interval $\left[0,T\right]$ is therefore divided into two parts: $\left[0,t_1\right]$, where the classical derivative is used in this early stage of the disease to describe the dynamics of the disease, and $\left[t_1,T\right]$, where the mABC operator is applied in the second stage of the disease, when long-range and non-local effects become more prominent. This partitioning enhances the description of the system’s behavior, effectively capturing long-range memory effects and non-local dependencies during the latter portion of the interval. The use of the pmABC operator in our diarrhea model specifically aims to capture the crossover behavior of the system, which manifests as transitions or switches in the model’s dynamics between time intervals characterized by classical and mABC derivatives. By incorporating both classical and fractional calculus, the model enables a more comprehensive understanding of the system’s dynamics and the emergence of crossover behavior. The crossover point, denoted as ${\iota }_1$, is a key point that requires us to consider different formulations for our pmABC fractional model of diarrhea disease, as described in (1), which we will now rewrite into two distinct cases as follows:

$$\begin{array}{ccc}\hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)& =& \left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{S}\left(\iota \right)=\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }+w_1^{\varsigma }\mathbb{V}+{\psi }^{\varsigma }\mathbb{R}-\left({\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}+w_2^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{S}, \iota \in \left[0,{\iota }_1\right],\\ \\ {}_0{}^{mAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)=\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }+w_1^{\varsigma }\mathbb{V}+{\psi }^{\varsigma }\mathbb{R}-\left({\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}+w_2^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{S}, \iota \in \left[{\iota }_1,T\right],\end{array}\right.\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)& =& \left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{I}\left(\iota \right)={\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\iota \in \left[0,{\iota }_1\right],\\ \\ {}_0{}^{mAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)={\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\iota \in \left[{\iota }_1,T\right],\end{array}\right.\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)& =& \left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{V}\left(\iota \right)={\lambda }^{\varsigma }{\Lambda }^{\varsigma }+w_2^{\varsigma }\mathbb{S}-\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{V},\iota \in \left[0,{\iota }_1\right],\\ \\ {}_0{}^{mAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\lambda }^{\varsigma }{\Lambda }^{\varsigma }+w_2^{\varsigma }\mathbb{S}-\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{V},\iota \in \left[{\iota }_1,T\right],\end{array}\right.\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)& =& \left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{R}\left(\iota \right)={\gamma }^{\varsigma }\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\iota \in \left[0,{\iota }_1\right],\\ \\ {}_0{}^{mAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)={\gamma }^{\varsigma }\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\iota \in \left[{\iota }_1,T\right],\end{array}\right.\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)& =& \left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B},\iota \in \left[0,{\iota }_1\right],\\ \\ {}_0{}^{mAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B},\iota \in \left[{\iota }_1,T\right],\end{array}\right.\hfill \end{array}$$

where ${\displaystyle \frac{d}{dt}}$ is the classical derivative and ${}^{mABC}\mathbf{D}_0^{\varsigma }\eta \left(\iota \right)$ is the modified ABC fractional derivative defined as

$${}^{mAB}\mathbf{D}_0^{\varsigma }\eta \left(\iota \right)={\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}\left[\eta \left(t\right)-\eta \left(0\right)\right]+{\displaystyle \frac{\varsigma }{1-\varsigma }}\left[{}^{RL}I_0^{\varsigma }(\eta \left(t\right)-\eta \left(0\right))\right].$$

4. Mathematical Properties of the Model (1)

In this part, we seek to study the qualitative behavior of the diarrhea model (1).

4.1. Existence and Uniqueness of Solutions

Theorem 1.

The solution of the pmABC fractional diarrhea model (1) exists and is unique.

Proof. 

By the definition of the pmABC fractional (2), the solutions of the pmABC fractional diarrhea model (1) are given as

$$\mathbb{X}\left(\iota \right)=\left\{\begin{array}{c}\mathbb{X}\left(0\right)+{\int }_0^{{\iota }_1}\mathbb{G}\left(s,\mathbb{X}\left(s\right)\right)ds,0<\iota \le {\iota }_1,\\ \\ \mathbb{X}\left({\iota }_1\right)+{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}\mathbb{G}\left(\iota ,\mathbb{X}\left(\iota \right)\right)+{\displaystyle \frac{\varsigma }{▿\left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_{{\iota }_1}^{\iota }{\left(\iota -s\right)}^{\varsigma -1}\mathbb{G}\left(s,\mathbb{X}\left(s\right)\right)ds\\ -{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}\mathbb{G}\left(0,\mathbb{X}\left(0\right)\right)\left(1+{\displaystyle \frac{\varsigma }{1-\varsigma }}{\displaystyle \frac{{\iota }^{\varsigma }}{\Gamma \left(\varsigma +1\right)}}\right),{\iota }_1<\iota \le T,\end{array}\right.$$

where

$$\mathbb{X}\left(\iota \right)={\left(\mathbb{S}\left(\iota \right),\mathbb{I}\left(\iota \right),\mathbb{V}\left(\iota \right),\mathbb{R}\left(\iota \right),\mathbb{B}\left(\iota \right)\right)}^T,$$

and $\mathbb{G}\left(\iota ,\mathbb{X}\left(\iota \right)\right)$ represents the right-hand side of the equations in model (1). The proof can be demonstrated using fixed-point theory, but for brevity, it is not included here and can be found in existing references such as [26,27]. □

4.2. Boundedness and Positivity of the Solutions

This section shows that the system solution of the model (1) is positive for all $\iota >0$, demonstrating that the model is well posed and biologically feasible. We define the norm

$$∥G∥=\underset{\iota \in \left[0,T\right]}{sup}\left|G\left(\iota \right)\right|.$$

We assume that

$${\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}>0,\forall \iota \ge 0.$$

Now, from the second equation in model (1), we have

$$\begin{array}{ccc}\hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)& =& {\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}\hfill \\ & \ge & -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}.\hfill \end{array}$$

(3)

By (2), inequality (3) becomes

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)=\left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{I}\left(\iota \right)\ge -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\iota \in \left[0,{\iota }_1\right],\\ \\ {}^{mABC}\mathbf{D}_0^{\varsigma }\mathbb{I}\left(\iota \right)\ge -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\iota \in \left[{\iota }_1,T\right].\end{array}\right.$$

For $\iota \in \left[0,{\iota }_1\right],$ we have

$$\mathbb{I}\left(\iota \right)\ge \mathbb{I}\left(0\right)exp\left(-{\int }_0^{\iota }\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}dx\right)>0.$$

For $\iota \in \left[{\iota }_1,T\right],$ we have

$${}^{mABC}\mathbf{D}_0^{\varsigma }\mathbb{I}\left(\iota \right)\ge -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}.$$

Taking the Laplace transform of both sides of the above inequality, we have

$${\displaystyle \frac{s\mathcal{L}\left[\mathbb{I}\left(\iota \right)\right]\left(s\right)-\mathbb{I}\left(0\right)}{s+\varsigma \left(1-s\right)}}\ge -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathcal{L}\left[\mathbb{I}\left(\iota \right)\right]\left(s\right).$$

This implies that

$$\mathcal{L}\left[\mathbb{I}\left(\iota \right)\right]\left(s\right)\ge {\displaystyle \frac{\mathbb{I}\left(0\right)}{\left(1-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma \right)\left(s+\frac{\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma }{1-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma }\right)}}.$$

Taking the inverse Laplace transformation, we have

$$\mathbb{I}\left(\iota \right)\ge {\displaystyle \frac{\mathbb{I}\left(0\right)}{\left(1-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma \right)}}{\mathbb{E}}_{1,1}\left(-{\displaystyle \frac{\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma }{1-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\varsigma }}\iota \right).$$

Due to $\mathbb{I}\left(0\right)>0$ and $0\le {\mathbb{E}}_{1,1}\le 1,$ we deduce that $\mathbb{I}\left(\iota \right)>0,$ $\forall \iota \in \left[{\iota }_1,T\right].$ Thus, in view of the above cases, we deduce that $\mathbb{I}\left(\iota \right)>0$, $\forall \iota \in \left[0,T\right].$ Since $\mathbb{I}\left(\iota \right)>0,$ then we have

$$\begin{array}{ccc}\hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)& =& {\gamma }^{\varsigma }\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R}\hfill \\ & \ge & -\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R}.\hfill \end{array}$$

(4)

By (2), inequality (4) becomes

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)=\left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathbb{R}\left(\iota \right)\ge -\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\iota \in \left[0,{\iota }_1\right],\hfill \\ \\ {}^{mABC}\mathbf{D}_0^{\varsigma }\mathbb{R}\left(\iota \right)\ge -\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\iota \in \left[{\iota }_1,T\right].\hfill \end{array}\right.$$

Using the same technique, we can conclude that $\mathbb{R}\left(\iota \right)>0$, $\forall \iota \in \left[0,T\right]$. Similarly, we can demonstrate that $\mathbb{S}\left(\iota \right)>0,\mathbb{V}\left(\iota \right)>0$, $\forall \iota \in \left[0,T\right]$. Thus, the system solution of model (1) is positive for all $\iota >0.$ On the other hand, the total population $\mathcal{N}\left(\iota \right)=\mathbb{S}\left(\iota \right)+\mathbb{I}\left(\iota \right)+\mathbb{V}\left(\iota \right)+\mathbb{R}\left(\iota \right).$ Applying ${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }$ on both sides, we obtain

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{N}\left(\iota \right){=}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right){+}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right){+}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right){+}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right).$$

Under the model (1), we have

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{N}\left(\iota \right)\le {\Lambda }^{\varsigma }-{\mu }^{\varsigma }\mathcal{N}\left(\iota \right).$$

(5)

By (2), inequality (5) becomes

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{N}\left(\iota \right)=\left\{\begin{array}{c}{\displaystyle \frac{d}{d\iota }}\mathcal{N}\left(\iota \right)\le {\Lambda }^{\varsigma }-{\mu }^{\varsigma }\mathcal{N}\left(\iota \right),\iota \in \left[0,{\iota }_1\right],\hfill \\ \\ {}^{mABC}\mathbf{D}_0^{\varsigma }\mathcal{N}\left(\iota \right)\le {\Lambda }^{\varsigma }-{\mu }^{\varsigma }\mathcal{N}\left(\iota \right),\iota \in \left[{\iota }_1,T\right].\hfill \end{array}\right.$$

In case $\iota \in \left[0,{\iota }_1\right],$ we have

$$\mathcal{N}\left(\iota \right)\le \mathcal{N}\left(0\right)e^{-{\mu }^{\varsigma }\iota }+{\displaystyle \frac{{\Lambda }^{\varsigma }}{{\mu }^{\varsigma }}}\left(1-e^{-{\mu }^{\varsigma }\iota }\right).$$

Thus, $\mathcal{N}\left(\iota \right)$ is bounded by ${\displaystyle \frac{{\Lambda }^{\varsigma }}{{\mu }^{\varsigma }}}.$ In case $\iota \in \left[{\iota }_1,T\right],$ we have

$${}^{mABC}\mathbf{D}_0^{\varsigma }\mathcal{N}\left(\iota \right)\le {\Lambda }^{\varsigma }-{\mu }^{\varsigma }\mathcal{N}\left(\iota \right).$$

Taking the Laplace transform, we obtain

$$\mathcal{L}\left[{}^{mABC}\mathbf{D}_0^{\varsigma }\mathcal{N}\left(\iota \right)\right]\left(s\right)\le {\displaystyle \frac{{\Lambda }^{\varsigma }}{s}}-{\mu }^{\varsigma }\mathcal{L}\left[\mathcal{N}\left(\iota \right)\right]\left(s\right).$$

By definition of the Laplace transform of the mABC fractional operator, and utilizing the asymptotic behavior of the Mittag-Leffler function, we obtain

$$\mathcal{N}\left(\iota \right)\le {\displaystyle \frac{{\Lambda }^{\varsigma }}{{\mu }^{\varsigma }}}.$$

Thus, $\mathcal{N}\left(\iota \right)$ is bounded by ${\displaystyle \frac{{\Lambda }^{\varsigma }}{{\mu }^{\varsigma }}},\forall \iota \in \left[0,T\right].$ Therefore, all solutions of model (1) remain non-negative and bounded for all time periods $t>0$ within a biologically and mathematically feasible region $\Omega ,$ where

$$\Omega =\left\{\left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B}\right);\mathcal{N}\left(\iota \right)=\mathbb{S}+\mathbb{I}+\mathbb{V}+\mathbb{R}+\mathbb{B}\le {\displaystyle \frac{{\Lambda }^{\varsigma }}{{\mu }^{\varsigma }}}\right\}.$$

The DFE point of model (1) is given as

$${\ell }_0=\left({\displaystyle \frac{\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)+{\lambda }^{\varsigma }{\Lambda }^{\varsigma }{w}_1^{\varsigma }}{\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\left(w_2^{\varsigma }+{\mu }^{\varsigma }\right)-w_1^{\varsigma }{w}_2^{\varsigma }}},0,{\displaystyle \frac{{\lambda }^{\varsigma }{\Lambda }^{\varsigma }{\mu }^{\varsigma }+{\Lambda }^{\varsigma }{w}_2^{\varsigma }}{\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\left(w_2^{\varsigma }+{\mu }^{\varsigma }\right)-w_1^{\varsigma }{w}_2^{\varsigma }}},0,0\right).$$

4.3. Basic Reproduction Number $R_0$

To obtain $R_0,$ we use the next-generation operator technique [28]. We consider the non-negative matrix $\mathcal{F}$ expressing the inflow of the infected class, and the non-singular M-matrix $\mathcal{V}$ expressing the outflow of the infected. Our diarrhea model (1) is defined as follows:

$$\left\{\begin{array}{c}{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)={\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B}.\hfill \end{array}\right.$$

Thus, the above infected model can be expressed as ${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{Z}\left(\iota \right)=\mathcal{F}\left(\iota \right)-\mathcal{V}\left(\iota \right),$ where $\mathcal{Z}\left(\iota \right)={\left(\mathbb{I}\left(\iota \right),\mathbb{B}\left(\iota \right)\right)}^T,$ and

$$\mathcal{F}\left(\iota \right)=\left(\begin{array}{c}{\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}\\ {\theta }^{\varsigma }\mathbb{I}\end{array}\right),\mathcal{V}\left(\iota \right)=\left(\begin{array}{c}\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}\\ \left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B}\end{array}\right).$$

Now, the Jacobian matrix of $\mathcal{F}\left(\iota \right)$ and $\mathcal{V}\left(\iota \right)$ is given by

$$F=\left(\begin{array}{cc}0& {\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\\ {\theta }^{\varsigma }& 0\end{array}\right),V=\left(\begin{array}{cc}{\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }& 0\\ 0& {\mu }_p^{\varsigma }+{\phi }^{\varsigma }\end{array}\right).$$

Using the fact that the basic reproduction number $R_0$ [28] is the largest eigenvalue of the matrix $F{V}^{-1}$, we have

$$R_0=\sqrt{{\displaystyle \frac{{\theta }^{\varsigma }\left(1-{\varphi }^{\varsigma }\right){\beta }^{\varsigma }\left[\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)+{\lambda }^{\varsigma }{\Lambda }^{\varsigma }{w}_1^{\varsigma }\right]}{\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)K^{\varsigma }\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\left[\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\left(w_2^{\varsigma }+{\mu }^{\varsigma }\right)-w_1^{\varsigma }{w}_2^{\varsigma }\right]}}}.$$

4.4. Local and Global Stability of DFE Point

Understanding the behavior and progression of infectious diseases requires a thorough analysis of the stability of the equilibrium point, both locally and globally. Local stability analysis focuses on the short-term response of the system to disturbances. This type of analysis reveals whether small fluctuations or changes will grow larger (leading to instability) or will diminish over time, returning the system to its initial state (leading to stability). In contrast, global stability analysis examines the system’s long-term behavior. It determines whether the system will eventually settle at a specific equilibrium point, regardless of the initial conditions, and whether it will remain there. Both local and global stability analyzes are vital for several reasons. They are crucial for evaluating the effectiveness of control measures, forecasting the likely course of a disease outbreak, and developing effective public health strategies. Specifically, these analyses help us to understand if interventions will truly contain a disease in the short term and if the disease is likely to remain under control in the long term.

Theorem 2.

The DFE ${\ell }_0$ of the pmABC diarrhea model (1) is locally asymptotically stable within the region Ω when $R_0<1$, and unstable when $R_0>1$.

Proof. 

The Jacobian matrix for the diarrhea model (1) is given by

$$J\left({\ell }_0\right)=\left[\begin{array}{ccccc}-\left(w_2^{\varsigma }+{\mu }^{\varsigma }\right)& 0& w_1^{\varsigma }& {\psi }^{\varsigma }& 0\\ 0& -\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)& 0& 0& 0\\ w_2^{\varsigma }& 0& -\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)& 0& 0\\ 0& {\gamma }^{\varsigma }& 0& -\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)& 0\\ 0& {\theta }^{\varsigma }& 0& 0& -\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\end{array}\right].$$

The trace of the matrix $J\left({\ell }_0\right)$ is

$$trJ\left({\ell }_0\right)=-\left(2{\mu }^{\varsigma }+w_1^{\varsigma }+w_2^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }+{\psi }^{\varsigma }+{\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)<0.$$

Also, the determinant of $J\left({\ell }_0\right)$ is

$$detJ\left({\ell }_0\right)=\left(w_2^{\varsigma }+{\mu }^{\varsigma }\right)\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)>0.$$

It is known that the DFE ${\ell }_0$ is said to be asymptotically stable if the trace of the Jacobian matrix at ${\ell }_0$, $trJ\left({\ell }_0\right)<0$, and $detJ\left({\ell }_0\right)>0$. This implies that the DFE ${\ell }_0$ of the diarrhea model is locally asymptotically stable. □

Theorem 3.

The DFE ${\ell }_0$ of the diarrhea model (1) is globally asymptotically stable within the region Ω when the basic reproduction number $R_0\le 1$.

Proof. 

We define the Lyapunov function $\mathcal{P}$ by

$$\mathcal{P}={\theta }^{\varsigma }\mathbb{I}+\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{B}.$$

Applying the pmABC fractional derivative on both sides, we obtain

$$\begin{array}{ccc}\hfill {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{P}& =& {\theta }^{\varsigma }{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}+{\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{B}\hfill \\ & =& {\theta }^{\varsigma }\left({\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\mathbb{I}\right)+\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\left({\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B}\right)\hfill \end{array}$$

After some simplifications, we have

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{P}={\displaystyle \frac{\left({\mu }^{\varsigma }+{\gamma }^{\varsigma }+{\delta }^{\varsigma }\right)\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)K^{\varsigma }\mathbb{B}}{K^{\varsigma }+\mathbb{B}}}\left(R_0-{\displaystyle \frac{K^{\varsigma }+\mathbb{B}}{K^{\varsigma }}}\right).$$

(6)

If $R_0<1$, then it follows that ${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{P}<1$. Also, if $\mathbb{B}=0$, then ${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathcal{P}=0$. Therefore, by Lasalle’s invariance principle Equation (6), when $R_0\le 1$, all solutions of model (1) with initial conditions in the region $\Omega$ approach the DFE as time tends to infinity. This implies that the DFE is globally asymptotically stable within the feasible region $\Omega$. □

5. The Fractional Optimal Control Model

Here, we incorporate two control strategies, ${\alpha }_1$ and ${\alpha }_2$, where ${\alpha }_1$ focuses on minimizing the contact between susceptible individuals and bacteria, aiming to reduce transmission. ${\alpha }_2$ involves treating individuals who have developed symptoms of the infection, aiming to improve health outcomes and limit further transmission. The model of the pmABC fractional order can be described as follows

$$\left\{\begin{array}{c}{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)=\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }+w_1^{\varsigma }\mathbb{V}+{\psi }^{\varsigma }\mathbb{R}-\left(\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}+w_2^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{S},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)=\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+\left({\alpha }_2+{\gamma }^{\varsigma }\right)+{\delta }^{\varsigma }\right)\mathbb{I},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\lambda }^{\varsigma }{\Lambda }^{\varsigma }+w_2^{\varsigma }\mathbb{S}-\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{V},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)=\left({\alpha }_2+{\gamma }^{\varsigma }\right)\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B}.\hfill \end{array}\right.$$

(7)

We consider the state model (7) in R⁵ and define the admissible control set $\Omega$ by

$$\Omega =\left\{\left({\alpha }_1,{\alpha }_2\right) are Lebesgue-measurable,0\le {\alpha }_i\le 1,for all \iota \in \left[0,T_f\right]\right\}.$$

Now, the aim is to minimize the following objective functional:

$$J\left({\alpha }_1,{\alpha }_2\right)={\int }_0^{T_f}\left(b_1\mathbb{I}+{\displaystyle \frac{w_1^{*}{\alpha }_1^2+w_2^{*}{\alpha }_2^2}{2}}\right)d\iota ,$$

subject to the constraints

$${}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)={\Phi }_1{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)={\Phi }_2{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\Phi }_3{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)={\Phi }_4{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\Phi }_5,$$

where ${\Phi }_i=\Phi \left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B},{\alpha }_1,{\alpha }_1,\iota \right),i=1,2,...,5,$ where $w_1^{*}$ and $w_2^{*}$ are positive weights that measure the relative costs of implementing the respective intervention strategies over the period $\left[0,T_f\right]$, while the terms ${\displaystyle \frac{w_1^{*}{\alpha }_1^2+w_2^{*}{\alpha }_2^2}{2}}$ measure the cost of the intervention strategies. Thus, we seek an optimal control quadruple $\left({\alpha }_1^{*},{\alpha }_2^{*}\right)$ such that

$$J\left({\alpha }_1^{*},{\alpha }_2^{*}\right)=min\left\{J\left({\alpha }_1,{\alpha }_2\right):\left({\alpha }_1,{\alpha }_2\right)\in \Omega \right\}.$$

The modified objective functional is defined as follows ([28]):

$$\widehat{J}\left({\alpha }_1,{\alpha }_2\right)={\int }_0^{T_f}\left[H\left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B},{\alpha }_1,{\alpha }_2,\iota \right)-\sum _{i=1}^5{\sigma }_i{\Phi }_i\left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B},{\alpha }_1,{\alpha }_2,\iota \right)\right]d\iota ,$$

(8)

where the Hamiltonian is given as follows:

$$H\left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B},{\alpha }_1,{\alpha }_2,{\sigma }_i,\iota \right)=\left(b_1\mathbb{I}+{\displaystyle \frac{w_1^{*}{\alpha }_1^2+w_2^{*}{\alpha }_2^2}{2}}\right)+\sum _{i=1}^5{\sigma }_i{\Phi }_i\left(\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R},\mathbb{B},{\alpha }_1,{\alpha }_2,\iota \right).$$

(9)

From (8) and (9), the necessary conditions for the fractional optimal control model (7) are

$$\left\{\begin{array}{c}{}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_1={\displaystyle \frac{\partial H}{\partial \mathbb{S}}}{,}_0^{pmAB}{\mathbf{D}}_{T_f}^{\varsigma }{\sigma }_2={\displaystyle \frac{\partial H}{\partial \mathbb{I}}}{,}_0^{pmAB}{\mathbf{D}}_{T_f}^{\varsigma }{\sigma }_3={\displaystyle \frac{\partial H}{\partial \mathbb{V}}},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_4={\displaystyle \frac{\partial H}{\partial \mathbb{R}}}{,}_0^{pmAB}{\mathbf{D}}_{T_f}^{\varsigma }{\sigma }_5={\displaystyle \frac{\partial H}{\partial \mathbb{B}}},{\displaystyle \frac{\partial H}{\partial {\alpha }_k}}=0,\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)={\displaystyle \frac{\partial H}{\partial {\sigma }_1}}{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)={\displaystyle \frac{\partial H}{\partial {\sigma }_2}}{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\displaystyle \frac{\partial H}{\partial {\sigma }_3}},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)={\displaystyle \frac{\partial H}{\partial {\sigma }_4}}{,}_0^{pmAB}{\mathbf{D}}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\displaystyle \frac{\partial H}{\partial {\sigma }_5}},{\sigma }_i\left(T_f\right)=0,\hfill \end{array}\right.$$

(10)

where ${\sigma }_i,i=1,2,...,5$ are the Lagrange multipliers.

Theorem 4.

If ${\alpha }_1^{*},{\alpha }_2^{*}$ are the optimal controls with corresponding states ${\mathbb{S}}^{*},{\mathbb{I}}^{*},{\mathbb{V}}^{*},{\mathbb{R}}^{*}$ and $\mathbb{B},$ then there exist adjoint variables ${\sigma }_i^{*},i=1,2,...,5$ satisfying the following: (i) Adjoint equations:

$$\begin{array}{ccc}\hfill {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_1^{*}& =& -{\sigma }_1^{*}\left(-\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}-{\mu }^{\varsigma }+w_2^{\varsigma }\right)-{\sigma }_2^{*}\left(\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\right)-{\sigma }_3^{*}{w}_2^{\varsigma },\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_2^{*}& =& -b_1-{\sigma }_2^{*}\left(-{\alpha }_2-{\gamma }^{\varsigma }-{\mu }^{\varsigma }-{\delta }^{\varsigma }\right)-{\sigma }_4^{*}\left({\alpha }_2+{\gamma }^{\varsigma }\right)-{\sigma }_5^{*}{\theta }^{\varsigma },\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_3^{*}& =& -{\sigma }_1^{*}{w}_1^{\varsigma }-{\sigma }_3^{*}\left(-w_1^{\varsigma }-{\mu }^{\varsigma }\right),\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_4^{*}& =& -{\sigma }_1^{*}{\psi }^{\varsigma }+{\sigma }_4^{*}\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right),\hfill \\ \hfill {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_5^{*}& =& -{\sigma }_1^{*}\left(-{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{K^{\varsigma }+\mathbb{B}}}+{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{{\left(K^{\varsigma }+\mathbb{B}\right)}^2}}\right)\hfill \\ & & -{\sigma }_2^{*}\left(-{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{K^{\varsigma }+\mathbb{B}}}+{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{{\left(K^{\varsigma }+\mathbb{B}\right)}^2}}\right)+{\sigma }_5^{*}\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right),\hfill \end{array}$$

(11)

(ii) Transversality conditions

$${\sigma }_i^{*}\left(T_f\right)=0,i=1,2,...,5.$$

(iii) Optimality conditions:

$$H\left({\mathbb{S}}^{*},{\mathbb{I}}^{*},{\mathbb{V}}^{*},{\mathbb{R}}^{*},\mathbb{B},{\alpha }_1^{*},{\alpha }_2^{*},{\sigma }_i^{*}\right)=\underset{0\le {\alpha }_1,{\alpha }_1\le 1}{min}H\left({\mathbb{S}}^{*},{\mathbb{I}}^{*},{\mathbb{V}}^{*},{\mathbb{R}}^{*},\mathbb{B},{\alpha }_1,{\alpha }_2,{\sigma }_i^{*}\right).$$

Furthermore, the control functions ${\alpha }_1^{*},{\alpha }_2^{*}$ are given by

$$\begin{array}{ccc}\hfill {\alpha }_1^{*}& =& max\left\{0,min\left\{1,{\displaystyle \frac{\mathbb{S}\left({\sigma }_1{\beta }^{\varsigma }\mathbb{B}{\varphi }^{\varsigma }-{\sigma }_2{\beta }^{\varsigma }\mathbb{B}{\varphi }^{\varsigma }\right)}{\left(K^{\varsigma }+\mathbb{B}\right)w_1^{*}}}\right\}\right\},\hfill \\ \hfill {\alpha }_2^{*}& =& max\left\{0,min\left\{1,{\displaystyle \frac{\mathbb{I}\left({\sigma }_2-{\sigma }_4\right)}{w_2^{*}}}\right\}\right\}.\hfill \end{array}$$

(12)

Proof. 

We obtain the Hamiltonian function by using (9)

$$\begin{array}{ccc}\hfill H\left(\iota \right)& =& \left(b_1\mathbb{I}+{\displaystyle \frac{w_1^{*}{\alpha }_1^2+w_2^{*}{\alpha }_2^2}{2}}\right)+{\sigma }_1^{*}\left[{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)\right]\hfill \\ & & +{\sigma }_2^{*}\left[{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)\right]+{\sigma }_3^{*}\left[{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)\right]\hfill \\ & & +{\sigma }_4^{*}\left[{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)\right]+{\sigma }_5^{*}\left[{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)\right].\hfill \end{array}$$

Using conditions (10) with the above Hamiltonian equation, we can obtain the adjoint Equations (11). Furthermore, the conditions ${\sigma }_i^{*}\left(T_f\right)=0,i=1,2\dots ,,5$ hold. Also, by condition ${\displaystyle \frac{\partial H}{\partial {\alpha }_k^{*}}}=0$, we can determine the optimal control characterization (12). Thus, the optimality model (7) and the accompanying adjoint equation (11) encompass the characteristics of optimal control, initial conditions, and transversality conditions. □

6. Optimality Model

The optimality model constructed from the adjoint model and the pmABC fractional optimal control model is

$$\left\{\begin{array}{c}{}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{S}\left(\iota \right)=\left(1-{\lambda }^{\varsigma }\right){\Lambda }^{\varsigma }+w_1^{\varsigma }\mathbb{V}+{\psi }^{\varsigma }\mathbb{R}-\left(\left(1-{\alpha }_1^{*}\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}+w_2^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{S},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{I}\left(\iota \right)=\left(1-{\alpha }_1^{*}\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\mathbb{S}-\left({\mu }^{\varsigma }+\left({\alpha }_2^{*}+{\gamma }^{\varsigma }\right)+{\delta }^{\varsigma }\right)\mathbb{I},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{V}\left(\iota \right)={\lambda }^{\varsigma }{\Lambda }^{\varsigma }+w_2^{\varsigma }\mathbb{S}-\left(w_1^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{V},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{R}\left(\iota \right)=\left({\alpha }_2^{*}+{\gamma }^{\varsigma }\right)\mathbb{I}-\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right)\mathbb{R},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{\iota }^{\varsigma }\mathbb{B}\left(\iota \right)={\theta }^{\varsigma }\mathbb{I}-\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right)\mathbb{B},\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_1^{*}=-{\sigma }_1^{*}\left(-\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}-{\mu }^{\varsigma }+w_2^{\varsigma }\right)-{\sigma }_2^{*}\left(\left(1-{\alpha }_1\right){\displaystyle \frac{{\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right)}{K^{\varsigma }+\mathbb{B}}}\right)-{\sigma }_3^{*}{w}_2^{\varsigma },\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_2^{*}=-b_1-{\sigma }_2^{*}\left(-{\alpha }_2-{\gamma }^{\varsigma }-{\mu }^{\varsigma }-{\delta }^{\varsigma }\right)-{\sigma }_4^{*}\left({\alpha }_2+{\gamma }^{\varsigma }\right)-{\sigma }_5^{*}{\theta }^{\varsigma },\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_3^{*}=-{\sigma }_1^{*}{w}_1^{\varsigma }-{\sigma }_3^{*}\left(-w_1^{\varsigma }-{\mu }^{\varsigma }\right),\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_4^{*}=-{\sigma }_1^{*}{\psi }^{\varsigma }+{\sigma }_4^{*}\left({\psi }^{\varsigma }+{\mu }^{\varsigma }\right),\hfill \\ {}_0{}^{pmAB}\mathbf{D}_{T_f}^{\varsigma }{\sigma }_5^{*}=-{\sigma }_1^{*}\left(-{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{K^{\varsigma }+\mathbb{B}}}+{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{{\left(K^{\varsigma }+\mathbb{B}\right)}^2}}\right)\hfill \\ -{\sigma }_2^{*}\left(-{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{K^{\varsigma }+\mathbb{B}}}+{\displaystyle \frac{\left(1-{\alpha }_1\right){\beta }^{\varsigma }\mathbb{B}\left(1-{\varphi }^{\varsigma }\right){\mathbb{S}}^{*}}{{\left(K^{\varsigma }+\mathbb{B}\right)}^2}}\right)+{\sigma }_5^{*}\left({\mu }_p^{\varsigma }+{\phi }^{\varsigma }\right),\hfill \end{array}\right.$$

subject to conditions ${\sigma }_i^{*}\left(T_f\right)=0,i=1,2,...,5,$ $\mathbb{S}\left(0\right)>0,\mathbb{I}\left(0\right)>0,\mathbb{V}\left(0\right)>0$, and $\mathbb{R}\left(0\right)>0.$

7. Sensitivity Analysis

Sensitivity analysis is crucial in mathematical modeling, particularly when dealing with complex biological or epidemiological systems. This technique helps us determine to what extent the output of a model changes (in this case, $R_0$) in response to variations in its input parameters. Essentially, it quantifies the `influence’ that each parameter has on the model’s outcome. To this end, we calculate sensitivity indices using the following formula:

$$Se{n}_{\ell }^{R_0}={\displaystyle \frac{\ell }{R_0}}\left[{\displaystyle \frac{\partial R_0}{\partial \ell }}\right].$$

The calculated sensitivity values for each parameter are presented in

Table 3. Sensitivity values for parameters.

Parameter	Sensitivity Value
$\Lambda$	0.0000
$w_1$	−0.0762
$\psi$	0.0000
$w_2$	0.0762
$\beta$	0.5000
$\varphi$	−0.5000
$\gamma$	−0.0023
$\theta$	0.5000
$\mu$	−0.0446
$\delta$	−0.0286
$\lambda$	−0.0375
K	−0.5000
${\mu }_p$	−0.0011
$\phi$	−0.0257

.

Sensitivity analysis reveals that certain parameters significantly influence the basic reproduction number ($R_0$), a key indicator of disease spread. The parameters with the strongest positive impact are the exposure rate to contaminated food and water ($\beta$) and the transmission rate of bacterial infection from infected individuals ($\theta$). A higher exposure rate ($\beta$) directly increases the chance of infection, and an elevated transmission rate ($\theta$) from infected individuals accelerates the spread. Similarly, a higher population growth rate directly creates more susceptible individuals. In contrast, the concentration of bacteria in contaminated water (K) and compliance with water and food hygiene practices ($\varphi$) exhibit the strongest negative impact. This means that any reduction in bacterial contamination of water or food, as well as greater community adherence to good hygiene, will correspondingly reduce $R_0$. Other parameters, such as the natural human mortality rate ($\mu$), the disease-induced mortality rate ($\delta$), the rate of immune decline in vaccinated individuals ($w_1$), the proportion of unvaccinated individuals ($\lambda$), the mortality rate associated with bacterial infections (${\mu }_p$), and the rate of sanitation leading to a dearth of bacteria (${\phi }_p$) exhibit less significant influences, but a negative sign indicates that an increase in these will result in a reduction in $R_0$. Finally, the rate at which recovered individuals become susceptible again ($\psi$) has no impact on $R_0$, implying that this only has long-term effects, and not in the very first stages of an outbreak. These sensitivity values highlight the direct and indirect pathways through which the disease spreads and is controlled, emphasizing the importance of factors affecting transmission, hygiene, and population levels.

The calculated sensitivities offer valuable insights for developing effective public health strategies that will be presented in the Strategies section.

8. Numerical Scheme of Fractional Optimal Control Model with pmABC Fractional Derivative

In this section, we use the numerical method introduced by [16]. For a general initial value problem

$$\left\{\begin{array}{c}{}_0{}^{pmABC}\mathbf{D}_t^{\varsigma }\eta \left(\iota \right)=G\left(\iota ,\eta \left(\iota \right)\right),\hfill \\ \eta \left(0\right)>0.\hfill \end{array}\right.$$

(13)

With the help of the fundamental definition of the pmABC fractional derivative [16] to problem (13), we have

$$\eta \left(\iota \right)=\left\{\begin{array}{c}\eta \left(0\right)+{\int }_0^{{\iota }_1}G\left(s,\eta \left(s\right)\right)ds,0<\iota \le {\iota }_1,\hfill \\ \\ \eta \left({\iota }_1\right)+{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left(\iota ,\eta \left(\iota \right)\right)+{\displaystyle \frac{\varsigma }{▿\left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_{{\iota }_1}^{\iota }{\left(\iota -s\right)}^{\varsigma -1}G\left(s,\eta \left(s\right)\right)ds\hfill \\ -{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left(0,\eta \left(0\right)\right)\left(1+{\displaystyle \frac{\varsigma }{1-\varsigma }}{\displaystyle \frac{{\iota }^{\varsigma }}{\Gamma \left(\varsigma +1\right)}}\right),{\iota }_1<\iota \le T.\hfill \end{array}\right.$$

(14)

By discretizing Equation (14) at $ı={ı}_{k+1}=(k+1)h$, where h represents the time step size, we obtain the following discrete equations

$$\eta \left({ı}_{k+1}\right)=\left\{\begin{array}{c}\eta \left(0\right)+{\int }_0^{{\iota }_1}G\left(s,\eta \left(s\right)\right)ds,0<{ı}_{k+1}\le {\iota }_1,\hfill \\ \\ \eta \left({\iota }_1\right)+{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left({\iota }_k,\eta \left({\iota }_k\right)\right)+{\displaystyle \frac{\varsigma }{▿\left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_{{\iota }_1}^{{ı}_{k+1}}{\left({ı}_{k+1}-s\right)}^{\varsigma -1}G\left(s,\eta \left(s\right)\right)ds\hfill \\ -{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left(0,\eta \left(0\right)\right)\left(1+{\displaystyle \frac{\varsigma }{1-\varsigma }}{\displaystyle \frac{{\iota }_k^{\varsigma }}{\Gamma \left(\varsigma +1\right)}}\right),{\iota }_1<{ı}_{k+1}\le T.\hfill \end{array}\right.$$

(15)

By employing Lagrange’s interpolation polynomial by two-step [29] in terms of the modified ABC segment of the piecewise operator, we can represent Equation (14) as follows:

$$\eta \left({ı}_{k+1}\right)=\left\{\begin{array}{c}\eta \left(0\right)+\sum _{m=1}^k\left[{\displaystyle \frac{5}{12}}G\left({\iota }_{m-2},\eta \left({\iota }_m\right)\right)-{\displaystyle \frac{4}{3}}G\left({\iota }_{m-1},\eta \left({\iota }_m\right)\right)+G\left({\iota }_m,\eta \left({\iota }_m\right)\right)\right],\\ \\ \eta \left({\iota }_1\right)+\left\{\begin{array}{c}+{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left({\iota }_k,\eta \left({\iota }_k\right)\right)+{\displaystyle \frac{\varsigma h^{\varsigma }}{▿\left(\varsigma \right)\Gamma \left(\varsigma +2\right)}}\sum _{m=1}^{k}G\left({\iota }_k,\eta \left({\iota }_k\right)\right)\hfill \\ \times \left[{(k+1-m)}^{\varsigma }(2+k+\varsigma -m)-{(k-m)}^{\varsigma }(k+2+2\varsigma -m)\right]\hfill \\ -{\displaystyle \frac{\varsigma h^{\varsigma }}{▿\left(\varsigma \right)\Gamma (\varsigma +2)}}\sum _{m=1}^{k}G\left({\iota }_{m-1},\eta \left({\iota }_{m-1}\right)\right)\hfill \\ \times \left[{(k+1-m)}^{\varsigma +1}-(k+1+\varsigma -m){(k-m)}^{\varsigma }\right]\hfill \\ -{\displaystyle \frac{1-\varsigma }{▿\left(\varsigma \right)}}G\left(0,\eta \left(0\right)\right)\left(1+{\displaystyle \frac{\varsigma }{1-\varsigma }}{\displaystyle \frac{{\left(kh\right)}^{\varsigma }}{\Gamma (\varsigma +1)}}\right).\hfill \end{array}\right.\end{array}\right.$$

9. Simulation and Comparative Results

Here, we simulate the model (1) for the following three cases using the numerical values of Table 2 and initial values as given by

$$\mathbb{S}\left(0\right)=600000,\mathbb{I}\left(0\right)=109000,\mathbb{V}\left(0\right)=400000,\mathbb{R}\left(0\right)=108810,\mathbb{B}\left(0\right)=100000.$$

In the following cases, ${\alpha }_1$ refers to minimizing contact between individuals infected with acute diarrhea and susceptible individuals, and ${\alpha }_2$ refers to individual treatment.

9.1. Case 1: ${\alpha }_1={\alpha }_2=0$ (Without Control)

In this case, we simulate the results without control for all classes in the period $\left[0,300\right]$, as shown in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, respectively. Due to the pmABC derivatives used in this work, we observe that the crossover behaviors of all classes $\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R}$, and $\mathbb{B}$ occur at ${\iota }_1=10$. The decreases and increases over time in all classes can be easily observed. The populations of the susceptible, infected, and vaccinated individuals increase and reach their peak values around $\iota =50$, but in $\iota >50$, they remain steady.

9.2. Case 2: ${\alpha }_1=0.5$ and ${\alpha }_2=0$ (Without Control ${\alpha }_2$)

In this case, we simulate the results for given values of control variables for all classes, as shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, respectively, using different fractional orders. We observed that the crossover behaviors of all classes $\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R}$, and $\mathbb{B}$ occur at ${\iota }_1=10$. In this case, we noted that the populations of the susceptible, infected, and vaccinated individuals increase and reach their peak values around $\iota =50$, but in $\iota >50$, they remain steady.

9.3. Case 3: ${\alpha }_1=0$ and ${\alpha }_2=0.005$ (Without Control ${\alpha }_1$)

In this case, we simulate the results for the given values of control variables for all classes, as shown in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, respectively, using different fractional orders. It can be inferred that this control is very effective in controlling the spread of the diarrhea disease.

9.4. Comparison Between Classes Without and with Control

Here, we compare the dynamical curves for all classes in the period $\left[0.100\right]$ using fractional order values $\varsigma =0.95$, as shown in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21. We observe that the crossover behaviors of all classes $\mathbb{S},\mathbb{I},\mathbb{V},\mathbb{R}$, and $\mathbb{B}$ occur at ${\iota }_1=10$. From these figures, we observed that the population of the susceptible, infected, and vaccinated individuals model classes without and with control remained steady in the first interval $\left[0.10\right]$ before the crossover point (classical derivative). However, in the second interval $\left[10,100\right]$ after the crossover point (mABC fractional derivative), the population of the susceptible, infected, and vaccinated individuals decreased with control compared to the population without control. This means that controls ${\alpha }_1$ and ${\alpha }_2$ are very effective in controlling the spread of the diarrhea disease. Additionally, the pmABC fractional derivative is a powerful tool to illustrate sudden or abrupt changes in diarrhea disease in a more brilliant way.

10. Discussion

From the above figures, we conclude the following:

• Case 1: Without control, with ${\alpha }_1={\alpha }_2=0$ (Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6). In the absence of any control measures, the acute diarrhea model demonstrates an initial rise in susceptible individuals before their numbers peak and decline, stabilizing as the infection spreads. Concurrently, the infected, vaccinated, and recovered populations rapidly increase and reach steady-state levels, alongside a sharp increase in bacterial load. Notably, lower values of the fractional order parameter ($\varsigma$) are associated with higher levels of infection, vaccination, recovery, and bacterial concentration, indicating its influence on the overall scale of the outbreak.
  Biological Implications:
  Unmitigated Outbreak: This case demonstrates what an outbreak looks like without interventions. The model shows the rapid rise of infection, followed by a stabilization of population as the number of individuals in each class becomes steady.
  Impact of Fractional Order: The fractional order seems to have an impact on the overall size of each class. For example, lower fractional order values result in a higher bacterial load, which can lead to more infections. This suggests that the "memory effect" captured by the fractional order plays a role in how the infection spreads.
• Case 2: Control with ${\alpha }_1=0.5$ and ${\alpha }_2=0$ (without control ${\alpha }_2$) (Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11).
  Impact of Reduced Contact: The intervention of the control ${\alpha }_1=0.5$, which represents a measure to reduce contact between susceptible and infected individuals (e.g., implementing public health measures to encourage hand washing and sanitation), clearly shows an impact. It leads to a lower number of infections and susceptible individuals at equilibrium compared to the no-control scenario. This is expected since reducing interactions between susceptible and infected individuals reduces the possibility of transmission.
  Disease Reduction, Not Elimination: Despite the intervention, the disease is not eliminated. The disease will reach a steady state but at a reduced level compared to the scenario with no interventions. This means that public health measures such as sanitation are essential to reduce the disease spread but cannot eliminate it entirely.
• Case 3: Control with ${\alpha }_1=0$ and ${\alpha }_2=0.005$ (without control ${\alpha }_1$) (Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16). The intervention of the control (${\alpha }_2=0.005$), which targets the treatment of infected individuals, has a strong impact on the number of infected individuals. The model shows that treatment has the biggest impact on disease dynamics, as it reduces the overall number of infections compared to the intervention that reduces contact.
  Higher Recovery: This scenario clearly shows that treatment reduces infection numbers and increases the number of recovered individuals. This demonstrates the importance of treating patients to reduce both infections and transmission.
  Bacterial Load Reduction: The model shows that treatment reduces the bacterial load in the environment. This is consistent with reality since reducing the duration of infection would reduce bacterial production and environmental load.

From Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 (comparison between classes without and with control), we conclude the following: The susceptible class (Figure 17) shows that the number of susceptible individuals is lower at any given time during the infection. The control intervention reduces the number of susceptible individuals by preventing infections.

The infected class (Figure 18) shows that the combined controls are very effective at lowering the number of infected individuals. The infection reaches a much lower peak with the control interventions. This emphasizes the importance of both treatment and prevention measures in controlling the spread of the disease.

The vaccinated class (Figure 19) shows that the integrated controls reduce the need for vaccination. This does not mean that vaccinations are not necessary, but that in the scenario where integrated control measures are in place, infections and the need for vaccination will not be as high as in the absence of controls.

The recovered class (Figure 20) shows that the decrease in the recovered population with the integrated control indicates that the combined controls are very effective in reducing disease and the overall impact on populations, reducing the overall number of infections and resulting in fewer people requiring recovery.

Bacterial concentration (Figure 21) shows that combined controls also have an impact on the overall bacterial load in the environment. This also demonstrates the impact of combined prevention and treatment strategies on overall disease reduction.

11. Strategies

The above analysis and figures offer valuable insights for developing effective public health strategies. We summarize it as follows:

• Preventative Contact Control: Implement public health measures to minimize contact between susceptible individuals and the bacterial sources, utilizing sanitation, hygiene campaigns, and source reduction methods.
• Targeted Treatment: Ensure timely access to treatment for all individuals exhibiting symptoms, bolstering health infrastructures to achieve maximum effect. Integrated Intervention: Employ combined prevention and treatment approaches, using both sanitation and individual treatment strategies, for the optimized control of disease levels.
• Resource Allocation: Allocate sufficient resources to support preventative and treatment interventions, including strengthening the healthcare infrastructure and increasing public health programs.
• Community Engagement: Implement campaigns and programs with community involvement to ensure proper engagement with all health policies.

The calculated sensitivities in the Sensitivity Analysis section offer valuable insights for developing effective public health strategies. The parameters with the strongest positive impact on $R_0$ ($\beta$ and $\theta$) should be primary targets for intervention, and conversely those with high negative impact should be strengthened. Specifically, measures to reduce the exposure rate ($\beta$), such as improved food and water safety, are critical. Strategies for decreasing the transmission rate ($\theta$) could include interventions such as isolating and treating infected individuals. However, efforts should focus on minimizing the impact of overpopulation on sanitation and healthcare capacities. Moreover, the strong negative sensitivity associated with the concentration of bacteria in contaminated water (K) and hygiene compliance ($\phi$) indicates that public health resources should prioritize water sanitation and community hygiene campaigns. The model suggests that to maximize resource utilization, public health should prioritize those factors that have the most influence on $R_0$, but in a real-world scenario, those should be combined with other factors, such as cost-effectiveness and ease of implementation, while not neglecting the secondary effects of the other parameters. Vaccination rate ($w_2$) is also a key target, as an increased rate can create more protected individuals and thus decrease the basic reproduction rate of the disease. The numerical results from this study help focus public health efforts on those factors that matter the most for controlling the spread of diarrhea.

12. Conclusions

Through the development of a modified ABC order (pmABC) model, we examined the crossover effect and identified effective control strategies tailored to the challenges faced in some community, where access to resources may be limited. Our findings highlight the importance of minimizing contact between infected and susceptible individuals and implementing individual treatment as key strategies for disease control. Furthermore, by calculating the basic reproduction number and establishing the local and global stability of the DFE, we found that the DFE of the diarrhea model is globally asymptotically stable within the feasible region $\Omega$ when $R_0\le 1$. Our results indicate that an increase in the effective contact rate corresponds to a higher reproduction number, underscoring the importance of preventive measures in reducing disease transmission. Moreover, our analysis reveals that improving compliance with good hygiene practices can lead to a decrease in the number of reproductions by reducing the contact rate. The implementation of strategies related to the controls ${\alpha }_1$ and ${\alpha }_2$ is shown to be very effective in controlling the spread of the diarrhea disease. From the simulations, we concluded that the fractional pmABC derivative is a powerful tool to illustrate sudden or abrupt changes in the incidence of diarrhea disease, allowing for a closer match to observed disease patterns. These findings suggest that a combination of preventive measures, such as improving hygiene and sanitation, and the targeted treatment of infected individuals provides the most effective approach for controlling acute diarrhea outbreaks in resource-limited settings. Future research could expand this work by incorporating additional factors such as spatial heterogeneity, the impact of climate change, and the influence of specific strains of bacteria to more effectively improve local public health interventions.