Dynamic Model for the Epidemiology of Diarrhea and Simulation Considering Multiple Disease Carriers

Diarrhea is responsible for killing around 525,000 children every year, even though it is preventable and treatable. This research focuses on both houseflies’ roles and humans’ roles in carrying pathogens causing diarrhea as multiple disease carriers. Both human and fly compartmental models are simulated with five diseases control strategies in order to identify the epidemic dynamics. The framework considers the life cycle of flies modeled into eggs, larvae, pupae, susceptible flies, and carrier flies, while the human system follows a compartment model as susceptible, infected, recovered, and back to susceptible again (SIRS). The relationships are modeled into an ordinary differential equation-based compartmental system. Then, the control parameters of the compartmental framework are analyzed. In order to propose effective control methods, five control strategies are considered: (1) elimination of flies’ breeding site, (2) sanitation, (3) installation of UV light trap, (4) good personal and food hygiene, and (5) water purification. Then, overall, ten control scenarios using the five control strategies are analyzed. Among them, effective control solutions considering various dynamic epidemiology are provided with the simulations and analyses. The proposed framework contributes to an effective control strategy in reducing the number of both flies and infected humans, since it minimizes the spread of the disease and considers cost-effectiveness.


Introduction
Infectious diarrhea, a disease causing fluid loss and dehydration, is the eighth leading cause of death, responsible for around 525,000 children deaths globally every year [1], mostly children in developing countries. Diarrhea is caused by infectious organisms, including viral and bacterial pathogens [2]. These are typically transmitted from the stool of one individual. This means the pathogen spreads through contaminated food and water or from an infectious person to a healthy person as a result of poor hygiene. Although diarrhea is both preventable and treatable, it is still fatal. This is due to several reasons, including a lack of awareness and shortages of existing lifesaving interventions.
Several existing studies showed that bacteria causing diarrhea are carried by houseflies [3,4] mainly. The house fly, Musca domestica L., is known to carry pathogens including bacteria, viruses, fungi, and parasites which cause life threatening diseases in humans and animals [5,6]. Houseflies breed in human feces [7] and the existing studies [8][9][10][11][12] have shown increased incidences of diarrhea during the periods of high fly density. Most importantly, several existing studies [8,[13][14][15] have shown that a reduction in housefly density may affect the reduction in diarrhea incidence as well.
Mathematical models [16] are widely used to convert the real case into mathematical epidemiologic representations and predict the dynamics of infectious disease transmission so that they have vital roles in developing public health strategies for disease control and prevention. The formulation process of 2 of 14 diarrhea considers multiple carriers (e.g., houseflies and infected humans). Then, the mathematical model is used for relevant disease control. Effective disease control has become an important part of computational epidemiology [17] that can provide useful guidelines for designing effective disease intervention strategies while balancing the costs of the control measures.
The objective of this paper is to understand and to formulate diarrhea's dynamic epidemiology through a set of differential equation-based mathematical models and to predict the possible future for the effectiveness of disease control strategies not only for reducing the infected population, but also decreasing or even eliminating fly population as principal carriers of the disease as well.
The following section presents the relevant epidemiologic network of diarrhea and its mathematical models along with multiple controls and their theoretical solutions. Then, the effectiveness of the proposed framework and simulation results are analyzed in Sections 3 and 4. In order to show the influence of each intervention towards the spreading of the disease, numerical simulations under several control scenarios are provided in Section 5. Figure 1 shows a flow diagram of diarrhea transmission through multiple carriers, including humans, flies and other environmental carriers. The system contains humans' and flies' epidemiologic systems, which are transformed into both (human and flies) compartmental models. As shown in Figure 1, the flies system shows the lifecycle of flies from egg stage (E f ), larva (L f ), pupa (U f ), until adult flies, which divides into susceptible (S f ), and carrier flies (C f ). A carrier means an agent that carries a pathogen causing the disease but it does not show any symptoms of illness.

Background and Epidemiologic Network Model
Mathematical models [16] are widely used to convert the real case into mathematical epidemiologic representations and predict the dynamics of infectious disease transmission so that they have vital roles in developing public health strategies for disease control and prevention. The formulation process of diarrhea considers multiple carriers (e.g., houseflies and infected humans). Then, the mathematical model is used for relevant disease control. Effective disease control has become an important part of computational epidemiology [17] that can provide useful guidelines for designing effective disease intervention strategies while balancing the costs of the control measures.
The objective of this paper is to understand and to formulate diarrhea's dynamic epidemiology through a set of differential equation-based mathematical models and to predict the possible future for the effectiveness of disease control strategies not only for reducing the infected population, but also decreasing or even eliminating fly population as principal carriers of the disease as well.
The following section presents the relevant epidemiologic network of diarrhea and its mathematical models along with multiple controls and their theoretical solutions. Then, the effectiveness of the proposed framework and simulation results are analyzed in Sections 3 and 4. In order to show the influence of each intervention towards the spreading of the disease, numerical simulations under several control scenarios are provided in Section 5. Figure 1 shows a flow diagram of diarrhea transmission through multiple carriers, including humans, flies and other environmental carriers. The system contains humans' and flies' epidemiologic systems, which are transformed into both (human and flies) compartmental models.

Background and Epidemiologic Network Model
As shown in Figure 1, the flies system shows the lifecycle of flies from egg stage (E ), larva (L ), pupa (U ), until adult flies, which divides into susceptible (S ), and carrier flies (C ). A carrier means an agent that carries a pathogen causing the disease but it does not show any symptoms of illness. The human system follows susceptible human (S ), infectious (I ), recovered (R ), and back to susceptible again, since no one has immunity against diarrhea. Figure 2 shows an epidemiologic The human system follows susceptible human (S h ), infectious (I h ), recovered (R h ), and back to susceptible again, since no one has immunity against diarrhea. Figure 2 shows an epidemiologic network where human can be infected by the disease by consuming contaminated food or water due to carrier flies laying pathogens on it.
network where human can be infected by the disease by consuming contaminated food or water due to carrier flies laying pathogens on it. As shown in Figure 2, a set of mathematical models are derived from the fly-human compartmental model. The model parameters in Figure 2 are provided in Table 1.

Symbol Description Initial Values Variables
Ef The number of eggs of flies 100 Lf The number of larvae of flies 100 Uf The number of pupae of flies 10 Sf The number of susceptible flies 10 Cf The number of carrier fly 10 Sh The number of susceptible humans 1000 Ih The number of infected humans 1 Rh The  As shown in Figure 2, a set of mathematical models are derived from the fly-human compartmental model. The model parameters in Figure 2 are provided in Table 1.  Table 2 shows the differential equation-based transition models with respect to the lifecycle of the fly and the epidemiologic processes of a human. Humans are classified with the Susceptible-Infected-Recovered-Susceptible (SIRS) reaction-diffusion compartmental model. The ordinary differential equations (ODE) are generated based on each of states in the network. The time unit is "day" in this research. Table 2. Ordinary differential equations (ODE) considering both the fly and the human system.

Compartment ODE
The solutions of the compartment models, as shown in Table 2, have to be non-negative (positiveness) and exist (boundedness). The below conditions show the positiveness and boundedness of the compartment model.
is positive and bounded interval for all time t > 0. Consider the Inequality (1) at time t.
Inequality (1) proves that the solution set is positive for all time t > 0. The same rule can be applied for Inequality (2) until Inequality (6) at time t.

Epidemic Model Dynamics
The dynamics of the models provided in Table 2 depend on the basic reproduction number, which is defined as the average number of secondary infections of an infectious human [18,19].
The basic reproduction number is denoted by R 0 as the number of secondary infections caused by an infected individual. If the value of R 0 < 1, then the disease dies out. While the value of R 0 > 1, then the number of infectious individuals increases and the disease invades the population. Let x = (x 1 , x 2 , . . . , x n ) T be the number of individuals in each compartment where the first m < n compartments contain infected individuals. The disease-free equilibrium (DFE) is given by (S,E,I,R) = (S 0 , 0, 0, 0). The terms S, E, I and R mean Susceptible, Exposed, Infected, and Recovered, respectively. In this model dynamics, it is assumed that the DFE exists and is stable in the absence of disease. Consider those equations written in the form is a non-singular matrix for 1 ≤ i, j ≤ m. F(X, Y) denotes a vector of new infection rates (flows from X to Y) and V(X, Y) is a vector of all other rates. For each compartment, an inflow in V is negative and an outflow in V is positive. It is assumed that F(0, Y) = 0 and V(0, Y) = 0. FV −1 is called the next generation matrix where the spectral radius of it is equal to R 0 , which is the largest eigenvalue of FV −1 . FV −1 is derived using Equations (7)- (10).
Then, R 0 is determined as the basic reproduction number using FV −1 .
After modeling the infection processes of diarrhea provided in Sections 2 and 3, the derived differential equations are solved with the initial conditions of E f = 10; L f = 10; U f = 10; S f = 10; C f = 10; S h = 1000; I h = 1; R h = 0. The initial parameters are supposed with real-world examples. Figure 3 shows the disease's infection simulations without any control method, in terms of humans' and flies' status. The simulation is performed using Matlab©. According to Figure 3, the disease starts with an outbreak at t = 0 and, the number of infected people is around 2700 at t = 20. The time unit is day in this research.
The provided ODE-based compartment model is used to check the epidemiologic processes of diarrhea and the effectiveness of control methods, which are provided in the following section.
Then, R is determined as the basic reproduction number using FV .
After modeling the infection processes of diarrhea provided in Sections 2 and 3, the derived differential equations are solved with the initial conditions of E = 10; L = 10; U = 10; S = 10; C = 10; S = 1000; I = 1; R = 0. The initial parameters are supposed with real-world examples. Figure 3 shows the disease's infection simulations without any control method, in terms of humans' and flies' status. The simulation is performed using Matlab©. According to Figure 3, the disease starts with an outbreak at t = 0 and, the number of infected people is around 2700 at t = 20. The time unit is day in this research.
The provided ODE-based compartment model is used to check the epidemiologic processes of diarrhea and the effectiveness of control methods, which are provided in the following section.

Effective Control Framework Considering Multiple Disease Carriers
In this section, the previous fundamental infection model is extended with several control strategies on the spread of the disease. The term "effectiveness" is evaluated with the blocks of additional infections and a cost-effective concept. This consideration may help to block the spread of SARS-2 or COIVID-19. The cost-effective criteria are determined using Pontryagin's [20] maximum principle on five variations of control methods. Pontryagin [20,21] introduced the idea of adjoint functions which has a similar purpose as Lagrange multipliers, to append the differential equation to the objective function.
This section focuses on effective control frameworks to prevent additional infections of humans. In order to control additional spreads of diarrhea, five control methods are considered: (1) elimination of fly's breeding site, (2) sanitation-related investment, (3) installation of UV light traps for killing flies, (4) good personal and food hygiene, and (5) water purification. In general, these methods intend to prevent additional infections among humans or to remove one of primary disease carriers-flies.
As the first control method, "Elimination of flies' breeding site" is considered. As flies feed on garbage, food waste, and animal feces, therefore one way to prevent additional infections is to keep the places clean, and to spray pesticide on prone areas. This intervention could reduce or even eliminate the maturation rate of fly eggs, larva, and pupa, which eventually impacts the number of adult flies. Let α denote the level of a breeding site elimination strategy (0 ≤ α(t) ≤ 1). The effect of breeding site elimination will decrease the maturation rate of eggs, larvae, and pupae, which is modelled as α·E f , α·L f and α·U f , respectively.
The second control option is sanitation efforts [22], which include the sterilization of cooking utensils, washing, and drinking water by boiling it properly. The parameter π(t) (0 ≤ π(t) ≤ 1) denotes the level of sanitation strategy. The effect of sanitation is to decrease the number of pathogens, which is modelled as a reduction in the rate of the carrier fly population by the term π·C f .
The third control method is to install light traps. When flies see ultra-violet (UV) light, they are naturally lured in the direction of the source of the UV rays. This strategy is more desirable than spray for indoor fly control as it keeps the surroundings clean. ϑ (0 ≤ ϑ(t) ≤ 1) denotes the effectiveness of the installation of UV light trap strategy. The effect of this effort is to decrease the number of susceptible and carrier flies by the terms ϑ·S f and ϑ·C f . The next consideration is to keep good personal and food hygiene. Good personal and food hygiene can be defined as handling, preparing, and storing food in a way that reduces the risk of becoming contaminated. Let ρ (0 ≤ ρ(t) ≤ 1) be the good personal and food hygiene effort. The effect of this effort is to suppress the number of infected people by the term ρ·I h .
The final option is to purify water. Chlorine is often a choice for water purification, since it effectively inactivates the bacteria causing diarrhea, leaves residual protection, has low cost and is easy to transport and use. There are two chlorine-based options [23] used in diarrhea outbreaks, such as tablets and liquid. τ (0 ≤ τ(t) ≤ 1) denotes the level of water purification strategy. The effect of water purification is to inactivate pathogens, which is modelled as a reduction in the disease transmission rate by the term τ·I h . Table 3 explains each control parameter for each control option. Based on the control parameters shown in Table 3, the initial transmission models are modified with additional terms to Equations (12)- (17). The additional term is underlined in each equation.
In this research, an effective control model is considered as a cost-effective control method within a controllable infection size. The objective function that aims to minimize cost in the control strategy J = {α, π, ϑ, ρ, τ} is obtained by Equation (18). In Equation (18), the parameter W 1 denotes the cost for elimination of breeding site, W 2 is the cost for sanitation-based works, W 3 is the cost for installing UV light trap, W 4 is the cost for isolation of infected person, and W 5 is the cost of water purification. Each C i,i∈ [1,5] denotes each control parameter-based cost.
Due to the fact that the cost function is nonlinear with the infection trends, time and other conditions, the mathematical programming belongs to the differential equation-based nonlinear mathematical programming. When denoting the prices associated with their respective classes by the nonlinear mathematical programming is solved using Pontryagin's maximum principle and the derivation of necessary conditions. As explained in [24,25], the existence of an optimal control is a sequence of the convexity of the integrand J with respect to α, π, ϑ, ρ, τ; a priori boundedness of the state variables, and the Lipschitz property of the state system with respect to the state variables. The differential equations are obtained by differentiating the Hamiltonian function, as shown in Appendix A. Finally, optimal control values are obtained as shown in Appendix B.
Therefore, it can be concluded by the standard control arguments involving the bounds on the controls in Table 4 as follows: Table 4. The optimal control parameter of each strategy.

Control Parameter
The Optimal Value of Control Parameter A *: optimal value of parameter of A.

Simulation and Analysis of Control Model Considering Multiple Disease Carriers
With the provided optimal control frameworks, numerical simulations are performed using the parameter values given in Tables 1 and 3. Several control scenarios are considered with combinations of intervention strategies for non-adult flies, adult flies, and infected humans, as summarized in Table 5. Table 5 shows the highest number of flies and humans using 50 days' of simulation.    Table 6 shows the assumed control costs and medical treatment costs. The costs are assigned with the consideration of existing relevant literatures [26][27][28][29][30]. After comparing all of those ten scenarios, the most effective strategy is considered Scenario X, which combines all control method under the assumptions. Table 7 shows the simulation result for each control scenario I to scenario X.    Table 6 shows the assumed control costs and medical treatment costs. The costs are assigned with the consideration of existing relevant literatures [26][27][28][29][30]. After comparing all of those ten scenarios, the most effective strategy is considered Scenario X, which combines all control method under the assumptions. Table 7 shows the simulation result for each control scenario I to scenario X.    -Ensure every healthy people to only consume hygiene water is like cutting the chain of the disease to be spreading in the environment. -It will affect the reduction rate of contaminated water to be consumed by human. Therefore, the number of infected people will decrease at t = 14 after experiencing its peak with 1641 people. In terms of cost and the number of infected persons, Scenario X shows the best performance. This indicates that the overall control methods are required for effective disease control and their portions influence its performance. It also denotes the significant reduction in number of infected people from 2795 to 877 as a result of the decreasing number of disease carriers. It means there is a 65.7% reduction in the number of infected people compared to the simulation without applying any control.
The provided framework can be used to check the effectiveness with different assumptions and numerical simulations considering multiple disease carriers.

Conclusions
This paper presents the ordinary differential equation-based epidemiologic models to understand diarrhea's infection dynamics and with the consideration of multiple carriers. Numerical simulation and mathematical analyses were performed to identify the relationship and the status of flies and humans when the disease outbreaks.
Then, each control strategy is applied, as shown in Scenarios I to V, and is combined in some combinations of these strategies to obtain the optimal condition, as shown in Scenarios VI to X. Under a certain simulation scenario, Scenario X, which combines all control methods, is proven to be the most effective strategy in reducing both of the number of flies and the infected population, as it minimizes the spread of the disease.
In the future, further study about broader conditions such as the effect of temperature on the transmission of a disease is required, as the population of housefly reaches its peak when the temperature is warmer.
The proposed framework is considered an effective disease control considering multiple carriers with epidemiologic dynamics and the relevant parameters.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Consider the objective function J(α, π, ϑ, ρ, τ) to investigate the optimal strategy needed to control the spreading of diarrhea provided in Equation, where A 1 , . . . , A 8 represent the weight constants of the eggs, larva, pupa, susceptible flies, carrier flies, susceptible human, infected human, and recovered human. J(α, π, ϑ, ρ, τ) = The Hamiltonian H is provided with J(α, π, ϑ, ρ, τ) and the ODEs, where φ E f , φ L f , φ U f , φ S f , φ C f , φ S h , φ I h , φ R h , φ C stf are the co-state variables given by the system: