Mathematical model of cholera spread based on SIR: Optimal control

Scan me: The bacterium Vibrio cholerae is the cause of cholera. Cholera is spread through the feces of an infected individual in a population. From a mathematical point of view, this problem can be brought into a mathematical model in the form of Susceptible-Infected-Recovered (SIR), which considers the birth rate. Because outbreaks that occur easily spread if not treated immediately, it is necessary to control the susceptible individual population by vaccination. The vaccine used is Oral Vibrio cholera. For this reason, the purposes of this study were to establish a model for the spread of cholera without vaccination, analyze the stability of the model around the equilibrium point, form a model for the spread of cholera with vaccination control, and describe the simulation results of numerical model completion. Based on the analysis of the stability of the equilibrium point of the model, it indicates that if the contact rate is smaller than the sum of the birth rate and the recovery rate, cholera will disappear over time. If the contact rate is greater than the sum of the birth rate and the recovery rate, then cholera is still present, or in other words, the disease can still spread. Because the spread is endemic, optimal control of the population of susceptible individuals is needed, in this case, control by vaccination, so that the population of susceptible individuals becomes minimum and the population of recovered individuals increases.


INTRODUCTION
Feces (human waste) that are contaminated with Vibrio cholerae bacteria are mostly found on water surfaces. The bacteria can transmit through food, such as uncleaned vegetables already fertilized with infected individual's feces through consumption. Therefore, the transmission of these bacteria can be through water, food, or poor environmental sanitation because they are contaminated with infected individual's feces (Cabral, 2010). Vibrio cholerae can attack gastroenteritis, then multiply. Vibrio cholerae also causes damage to the host characterizing their virulence (Finkelstein, 1996). The disease caused by the bacteria Vibrio cholera is called cholera. Individuals infected with cholera mostly show no symptoms, but their feces are contagious (World Health Organization (WHO), 2017). Cholera is an acute diarrheal disease caused by intestinal infection due to the bacterium Vibrio cholera. The disease is easily spread if not treated soon.
Africa was the continent with the most countries contracting this disease. According to the data obtained, there are about 3-5 million cases of cholera, and 100-120 thousand of them die every year in the world (World Bank, 1994). Moreover, in the 69 cholera-endemic countries studied, there were 2,9 million cases of cholera, even 95.000 annual deaths, with the highest cases in Sub-Saharan Africa (Ali et al., 2015). Even cholera cases from December 2020 to January 2021 in Nigeria, which means that during the COVID-19 pandemic, there were 266 cases with a case fatality rate (CFR) of 12.0% (World Health Organization Africa Office Regional, 2021). Due to the many cases of cholera, it is necessary to have prevention efforts. One of the recommendations is a vaccination (World Health Organization (WHO), 2012). In this section, we sketch the procedures for achieving the research aims. In the first part, a mathematical model of the spread of cholera without vaccination is developed. Based on the model, we determine the equili-brium point and analyze its stability. In the other part, we developed a new model with vaccination control. Analysis of the control problem uses the Pontryagin Minimum Principal (PMP). In the last, we describe numerically to confirm the analytical results. Figure 1 shows the flow of the studies in this article. There are two major groups discussed, namely models of cholera spread without control and models with control. Furthermore, both are illustrated numerically to strengthen the results analytically.

Assumptions of the Mathematical Model without Control
Cholera transmission occurs through contact with sufferers through food, water, and the environment contaminated with individual feces infected with Vibrio cholerae. The spread of this disease will be analyzed through the SIR model. In this case, the population is divided into three classes, namely susceptible class ( S ) indicating a healthy individual population and susceptible to cholera, the infected class ( I ) indicating the population of individuals infected with cholera and able to recover, and the recovered class ( R ) indicating the population of individuals recovering from cholera and immune to the disease. We define a population of susceptible individual class at timet as ( ) S t , a population of infected individual class at timet as ( ) states the rate of death in class R at timet . Fourth, the susceptible individual class population is infected with cholera disease if contact occurs through feces of the infected individual class population. Fifth, the recovered individual population from cholera is not susceptible. Sixth, population born in susceptible, infected, and recovered individual classes will be susceptible to choleric disease entering the susceptible class. Seventh, the population of individuals infected with cholera can recover from the disease with a recovery rateγ so that the rate of the population of the infected individuals who recover during timet is

Stability Analysis
In the second part, we determine its equilibrium points and analyze the stability. The equilibrium point will be determined by taking 0 dS dI dR dt dt dt = = = . Since our model is a nonlinear system of ordinary differential equations, it is not accessible to study the stability analysis. Then linearization using the Jacobian matrix can help our discussion as long as the equilibrium points are hyperbolic -usually, consideration of stability analysis through the value of basic reproduction number. One method to determine this value is by next generating matrix. Furthermore, we can adopt the Routh Hurwitz criterion to analyze the stability around the equilibrium point. Let  The iteration stops when the last row (2) is zero. If the first column in the arrangement (2) is all positive or all negative, then the real part of the polynomial (1) is negative (Niu & Wang, 2008;Olsder & van der Woude, 2004).

Assumptions of the Mathematical Model using Control
After the spread of the disease becomes endemic, it is necessary to control cholera by vaccination in case of cholera. Furthermore, it is assumed that the vaccine ( ) u t is given after an outbreak, meaning immediate control by vaccination. Vaccines ( ) u t are only given to the individual class S . We used the Pontryagin Minimum Principal (PMP) to analyze the optimal control following some articles (e.g., Mahmudah et al., 2013;Njagarah & Nyabadza, 2015;Sari et al., 2017).

RESULTS
This section discussed the SIR model formulation to cholera without control, analyzed the stability to its equilibrium points, developed a model with control, and described the results numerically.

SIR Model Formulation without Control
As previously explained, based on the problem of the cholera spread and the assumptions given, a transfer diagram can be drawn in Figure 2.  Figure 2, the SIR model obtained is shown in the following nonlinear system of ordinary differential equations. ( System of ordinary differential equation (3) is a mathematical model of spreading cholera. The following section will be discussed the equilibrium points of the model (3).

Equilibrium Points
There are two kinds of equilibrium points, namely disease-free and endemic equilibrium point, which are expressed in Lemma 1 and Lemma 2.
Proof. For each equation in system (3) which taking in the right-hand side is zero, we obtain from the second equation From equation (4), we have ( ) 0 If ( ) 0 I t = is substituted to the third equation in system (3) which has zero in the right-hand side, then we obtain ( ) 0 R t = . If ( ) 0 I t = is substituted to the first equation in system (3) which has zero in the right-hand side, then we obtain ( ) In other word, we have an equilibrium point ( , 0, 0) N . □ Lemma 2. If ( ) 0 I t ≠ , then system (3) has equilibrium point which is called an endemic equilibrium point. Proof. Notice for equation (4) in this case. If this equation is substituted with the first and third equation in system (3), we have an endemic equilibrium point (6). □

Basic Reproduction Number
Basic reproduction numbers, 0 R , represent the number of cases of secondary infection in the population, then it sees the presence or reduction of infection at the disease-free equilibrium point (van den Driessche, 2017). Next 0 R from system (3) will be sought using the infected class equation, and the following is defined , as next generation matrix, so that the basic reproduction number is obtained from the largest eigenvalue of the matrix. Furthermore, it is obtained

Local Stability of Disease-Free Equilibrium Point
From Lemma 1, we have a disease-free equilibrium point. Now, analysis of stability around the disease-free equilibrium point 0 E is stated in the Theorem 1.
Proof. The Jacobian matrix of 0 E from system (3) is notated by Then we have λ is positive, so there is a positive eigenvalue. Therefore, 0 E is not stable. □

Local Stability of Endemic Equilibrium Point
The endemic equilibrium point is obtained based on Lemma 2 that is 1 The following is the Theorem 2 about the local stability of this endemic equilibrium point. Theorem 2. If 0 1 R > , then 1 E is stable asymptotic local.
Proof. Jacobian matrix in 1 E is denotated by To obtain the eigenvalue, λ , then it should be solved by Then We obtained the characteristic equation = + + Furthermore, the stability analysis of the equilibrium point can be conducted by using the Routh Hurwitz criteria (Niu & Wang, 2008) through the coefficient of the characteristic equation. Routh Hurwitz table is obtained as follows.
Thus, the first column is marked the same. Based on the Routh Hurwitz criteria, the eigenvalues in the real part are negative. Furthermore, 1 E is locally stable asymptotic. □

SIR Model Formulation with Control
The goal to be achieved is to minimize the class population S and I . Individual class population S and I become optimum by minimizing the objective function (Zaman et al., 2008) as follows where 1 C is the positive constant of the weight of ( )

S t , 2
C is the positive constant of the weight of ( ) I t , ( ) u t is proportion of susceptible population given vaccine,τ is a positive parameter corresponding with ( ) u t . Furthermore, it is assumed 0 N τ < < and f t is maximal time to infection. We defined the boundary condition of ( ) The population given vaccines is the susceptible individual class population, so system (3) is obtained by substituting ( ) u t into the system. After the vaccine is given, the individual class S is included in the individual class R to prevent transmission of cholera in that class, while the class I is given antibodies. Figure 3 is a transfer diagram to this model.   . The Theorem 3 is the optimal control to model (9). I t , and ( ) * R t are the constraint variables of system (9), then the optimal control for system (9) is

dS t I t S t N u t S t dt N dI t S t I t I t I t dt N dR t I t R t u t S t dt
and Hamiltonian function that The optimal control at the solutions ( ) ( ) ( ) S t I t R t of the corresponding state of system (9) is derived by

Simulation
In line with the analytical discussion, this section discusses numerically as an illustration to support the analytical results. In this section, we present the numerical simulation to describe the optimal solution of cholera spread. The parameter value is shown in Table 1.  Hendrix (1984) µ 0.14285714285 Chiang (1979) If 0.2 β = means 0 1 R < , then S toward to 9446001. Apparently, with enlarged t, the individual class of subpopulation S is always constant towards 9446001 while the individual class population I and R for enlarged t, the value goes to 0. It means the subpopulation I decreases, and the transmission of disease will gradually disappear. Figure 4 (a), (b), dan (c) show us the illustration of this condition. If 0.6 β = means 0 1 R > , then infected class will reach the highest number before decreasing slowly (see Figure 5). It can be interpreted that the disease will exist. This is in line with the analysis result described in Theorem 1 and Theorem 2.  Figure 6 (a) concludes that the smaller the value is given, the reduction in the population of individual class I will be longer. As a result, at a certain time t, the individual class population is still sufficient, so the opportunity to transmit cholera to the population the individual class S is larger than before. Figure 6 (b) provides the conclusion that the population of individual class I without control increased dramatically at time 13 t = . Meanwhile, the population of individual class I with control tends to decrease at time 13 t = . Figure 7 (a) shows that the optimal u is 0.9; there is a decrease at time 40 t = . The vaccine production function value in the current dose 40 t = based on equation (8) is K (0.9) = 12335.850. If we do simulation for three months which is approximately 100 days, then we can see from Figure 7 (b) that the graph decreases significantly.

DISCUSSION
The classical mathematical model related to the spread of disease is in the form of SIR (Rodrigues, 2016). Because in general, the human population is divided into susceptible subpopulations, infected subpopulations, and recovered subpopulations. Through this classic model, it was developed into SEIR. In this case, E stands for "exposed" meaning someone who is in the incubation period. The emergence of compartment E can be used to describe the spread of Singapore flu (Sari, 2012) and the COVID-19 (Feng et al., 2021). In the case of cholera, the suitable model to apply is SIR. Based on other researchers, the model can be divided into the human and bacteria populations (Emvudu & Kokomo, 2012;Hntsa & Kahsay, 2020;Liao & Wang, 2011;Panja, 2019). In our model, we focus on the human population to spread cholera disease. We interest in studying the dynamical behavior of the human population caused by bacteria infection. We have presented the mathematical model of cholera with and without control. The discussion of stability analysis based on basic reproduction number can be interpreted as an effort and prevention to reduce the value below one, so the cholera spread can be controlled. Based on Theorem 2, cholera spread becomes endemic if the value of basic reproduction number is above one. On the other hand, based on Theorem 1, if the value of basic reproduction number is below one, then there is a unique disease-free which is locally asymptotically stable. It is essential to know the behavior of the solution if the value of basic reproduction number is equal to one. In the dynamical analysis, the possibility of the emergence of bifurcation is huge when the value of basic reproduction number is one. The existence of bifurcation will help to interpret the prognosis of cholera in more detail. Nevertheless, we leave this concern to the subsequent research. Since Vibrio cholerae contaminated water, then water treatment is one of the effective ways to control the concentration of bacteria (Andam et al., 2015). In the susceptible and infectious individuals population, strong control, weak control, and no control can be done by looking at the value of basic reproduction number (Ayoade et al., 2018). If the value of basic reproduction number is less than one, the population has carried out strong control of the spread. If basic reproduction number is more than one, it means that the disease-free equilibrium point is unstable. In line with this, the endemic equilibrium point is globally asymptotically stable, which means that the control is weak. Meanwhile, when the value of basic reproduction number is increased from before, the infected population is greater than the susceptible, which means a significant cholera tragedy occurs (Ayoade et al., 2018). The other measurements of control are treatment, vaccination, water sanitation, and education campaign. All the parameter is included in the model as constant control (Edward & Nyerere, 2015). Generally, the control parameter is better as time-dependent since it may change all the time. Time-dependent characterization of control can be attained from several interventions, such as mobilization, drug administration, and sanitation improvement. The optimal control strategy is obtained by combining the three controls (Bakare & Hoskova-Mayerova, 2021).
In our model, the control parameter is a function of time. We concentrate on the vaccination control of the susceptible individual. Therefore, our goal is to minimize the objective function (8). The population given vaccines is the susceptible individual class population. After the vaccine is given, the susceptible population is included in the recovered population to prevent transmission of cholera in the susceptible class, while the infected population is given antibodies. To analyze the control problem, we use the Pontryagin Minimum Principle (PMP). The optimal value of this control is used in anticipation to resist disease (Njagarah & Nyabadza, 2015).
The vaccine dose is given to the individual class S with the minimum, and the individual class I is also the minimum so that the class individual R becomes increased. However, to enlarge t resulted in the vaccine is no longer effective, it is shown in Figure 9. It can be seen from Figure 9 concluded that at t = 100, the population of the individual class S who were given the vaccine has decreased, as well as when t = 100 vaccine given is no longer used effectively. This result is consistent with the reports that there was a decrease of 80% of approximately three months in the population of individual class I, after the class S individual population was given the vaccine (CDC, 2018). Our results also align with the other paper that although infection can still occur even after controls have been administered, it may be up to 8 times less damaging than cases in the absence of controls (Njagarah & Nyabadza, 2015).

CONCLUSION
This paper has discussed the formulation of a mathematical model for the cholera spread based on SIR. Initially, a model of the spread of cholera without vaccination is presented in system (3). Based on model (3), we presented in Lemma 1 and Lemma 2 that two types of equilibrium points are obtained, i.e., disease-free and endemic. The stability of the model around the equilibrium point has been shown in Theorem 1 and Theorem 2. If the contact rate is lower than the sum of the birth rate and the recovery rate, then the cholera disease disappears over time. If the contact rate is greater than the sum of the birth rate and the recovery rate, then cholera is still present, meaning that the disease can still spread.
Due to the emergence of disease endemicity, it is necessary to form a model of the spread of cholera with vaccination control, as shown in system (9). Using the Pontryagin Minimum Principle (PMP) analytically, there is an optimal control as described in Theorem 3. We also proved numerically that with a control strategy, we could decrease the population of infectious class. For future research can discuss global stability to find out the solution behavior in infinite time. Study about the value of basic reproduction number if equal with one is out of our concern. So, it is more interesting to discuss further. Moreover, since vaccine doses need to be repeated, it is necessary to divide class S according to age in administering the vaccine.