The Effect of Susceptible Immigrants in a System Dynamic on the Spread of Malaria in Indonesia

ABSTRACT

the susceptible human population ( ℎ ) and the infected human population ( ℎ ), assuming that humans who are infected can become susceptible again. The mosquito population is also divided into two groups, namely the susceptible mosquito population ( ) and the infected mosquito population ( ) (Wedajo et al., 2018). The SIR (Susceptible, Infectious, and Recovered) model to describe the spread of malaria was first introduced by Kermack and McKendrick in 1927 (Brauer & Castillo-Chavez, 2012). Various mathematical models have been developed to describe the dynamics of the spread of malaria. (Tumwiine et al., 2007) studied the malaria spread model, which consisted of the SIR compartment for the human population and the SI compartment for the mosquito population, assuming that recovered individuals could be re-infected. In this model, population migration factors are ignored, while in (Martens & Hall, 2000), it is stated that population migration factors have the potential to have an impact on the spread of malaria. Such as Geleta Wedajo et al. (2018) investigated the model of malaria spread that allow migration factor. Therefore, this article aims to determine the influence of susceptible immigrants on the spread of malaria in Indonesia using the SIR-SI model from Wedajo et al. (2018) modified by adding the effect of vaccination referring to (Putri et al., 2014) and the effect of using the longlasting insecticidal net, the use of long-lasting insecticidal nets while sleeping is effective in avoiding mosquito bites because Anopheles mosquitoes actively bite and suck human blood at night (Rahmawati et al., 2014). This research was carried out by determining the fixed point and basic reproduction number for the model, then stability and bifurcation analysis. Then perform a numerical simulation using data on malaria cases in Indonesia.

B. METHODS
Suppose the human population is divided into three groups, namely: the susceptible human population is denoted ℎ , the infected human population is denoted ℎ , and the human population recovered from infection is denoted ℎ . The mosquito population is divided into two groups, namely the susceptible mosquito population, denoted , and the infected mosquito population, denoted . In the formulation of the model, the following assumptions are made: On human: 1. Every individual who is born is susceptible to infection. 2. There are immigrants who enter into the susceptible population.
3. The effectiveness of the malaria vaccine can provide permanent immunity to susceptible individuals (who have been vaccinated) against the disease and recovery. 4. Susceptible individuals can become infected by being bitten by an infected mosquito. 5. Infected individuals may recover from the infection or die from the disease. 6. Individuals who have recovered from infection have permanent immunity, so recovered humans cannot become susceptible again. 7. Any individual can leave any group due to natural death.
On mosquitoes: 1. Every mosquito that is born is susceptible to infection. 2. The mosquito birth rate is assumed to be the same as the mosquito death rate.
3. Susceptible mosquitoes can die by natural death or can become infected by biting infected individuals. 4. Infected mosquitoes end in natural death due to the relatively short life cycle of mosquitoes. The period of spread of mosquitoes ending in death means never recovering from infection.
From the assumptions above, the spread of malaria can be illustrated in the compartment diagram in Figure 1.  Figure 1, the mathematical model that describes the spread of malaria takes the form: With positive constans ℎ , , ℎ , ℎ , ℎ , ℎ , ℎ , , , , . The total population of humans and mosquitoes is expressed by ℎ = ℎ + ℎ + ℎ and = + . The descriptions of variables and parameters used in equations (1) − (5) are given in the, as shown in Table 1 and Table 2. The natural death rate of mosquito 1/time The effectiveness of the long-lasting insecticidal nets 1/time

C. RESULT AND DISCUSSION 1. Dynamic Characteristics
In the following, the dynamic characteristics of the malaria distribution model are discussed in equations (1) -(5). For this purpose, the following dimensionless variables are introduced: If these dimensionless variables are substituted into equations (1) -(5), the following mathematical model of the spread of malaria is obtained, where the role of the notation * is replaced by .
with the positive constants , , , , , , , , which are stated as follows: The stability of the disease-free fixed point and the endemic fixed point was determined based on the basic reproduction number. The basic reproduction number is defined as the average number of susceptible individuals that can be infected by one infected individual during the transmission period.This number can provide information about a condition that indicates whether a disease has spread or not (Giesecke, 2017). The disease-free fixed point of equations (6) -(8) is: 0 ( ℎ , ℎ , ) = ( 1+ + , 0, 0), while the endemic fixed point is: .
Then the basic reproduction number is derived using the next-generation matrix (van den Driessche & Watmough, 2002). Based on equations (7) − (8), the rate of emergence of new infective individuals in compartment is: and the disease movement rate in compartment is: so that it is obtained .
The basic reproduction number, 0 is the largest non-negative eigenvalue of the nextgeneration matrix, which is: The following is the stability theorem of the disease-free fixed point and the endemic fixed point based on the value of the basic reproduction number.

Bifurcation Analysis
The following is an analysis of the bifurcation that occurs when ℛ 0 = 1. For that, suppose = is selected as the bifurcation parameter and 1 = ℎ ; 2 = ℎ ; 3 = . The Jacobian matrix of equations (6) -(8) is evaluated at 0 to find the eigenvalues and obtain equation (9). When ℛ 0 = 1, three eigenvalues are obtained, namely 1 = − − , 2 = − ( + + + ), and 3 = 0. So that the first assumption of the Castillo-Chaves and Song Theorem is fulfilled, that is, it has one simple zero eigenvalue and the other has negative real values. In addition, the zero eigenvalues are related to the 0 matrix, which has a right eigenvector and a left eigenvector. The right eigenvector, which corresponds to 3 = 0 is obtained as follows. The value of and obtained correspond to case 4 of the Castillo-Chaves and Song Theorem (Castillo-Chavez & Song, 2004). This shows that the bifurcation that occurs when ℛ 0 = 1 is a trans-critical bifurcation (Ndii, 2022). Next, the bifurcation simulation will be carried out to see the compatibility between the results of the analysis and the numerical results.

Bifurcation Simulation
In this section, a numerical simulation is carried out to see the bifurcation that occurs with the parameter values used, which are presented in Table 3.  (Nurmaliani & Arisanti, 2021) Based on the parameter values in Table 3, the following bifurcation curve is obtained. As shown in Figure 2. Based on Figure 2, The red curve is not an endemic fixed point because it gives a value of ℎ < 0 meaning that the endemic fixed point is unstable and the disease-free fixed point is stable, while the blue curve gives a value of ℎ > 0 meaning the endemic fixed point is stable and the disease-free fixed point is unstable. The simulation results show that there is a change in the stability of the disease-free fixed point and the endemic fixed point when ℛ 0 = 1, each changing from stable to unstable and vice versa, from unstable to stable.
So the results of the analysis and numerical simulation give the same result, namely bifurcation occurs when ℛ 0 = 1, so that changes in the value of a parameter will result in changes in the stability of the fixed point, one of which is a change in the value of the human migration rate. Furthermore, numerical simulations will be carried out to see the effect of changes in the value of the migration rate on the dynamycs of the spread of malaria.

Simulation of the Effect of Susceptible Immigrants on Population Dynamics
Furthermore, to determine the effect of the migration rate on the dynamics of the spread of malaria, a simulation was carried out by changing the parameter value because the parameter depends on the migration rate of humans ( ). Simulations were carried out using Mathematica 11.0 software.
In this case, it will be shown that there will be a decrease in the value of ℛ 0 if the value of the parameter is decreased by the value of another parameter being fixed. The simulation is carried out with the parameter values presented in Table 3 by assuming the initial values are ℎ0 = 0.9, ℎ0 = 0.1, and 0 = 0.1. The changes in the parameter value used are presented in Table 4. The following is an image that illustrates the dynamics of each population due to changes in the value of . As shown in Figure 3. is lowered, the proportion of the susceptible human population will decrease due to a decrease in immigrants entering the susceptible population, and the proportion of the infected human population will also decrease and even become extinct when = 5 and = 3. The proportion of the recovering human population has increased. Figure 3d-e shows that if the value of is lowered, the proportion of the infected mosquito population decreases and even experiences extinction when = 5 and = 3 so that the remaining population of susceptible mosquitoes.
The simulation results show that if the value of is reduced, the proportion of infected human populations and infected mosquitoes will decrease. Since the value of is directly proportional to the rate of human migration ( ), a decrease in the value of can reduce the proportion of infected human populations and infected mosquitoes and even cause them to become extinct. Thus, efforts that can be made to suppress the spread of malaria is to reduce the rate of human migration.

D. CONCLUSION AND SUGGESTIONS
The SIR-SI model was applied to the spread of malaria to predict the dynamics of malaria outbreaks. This model consists of a SIR compartment for the human population and an SI compartment for the mosquito population. The assumption made in this model is that it allows the recruitment of individuals through migration. Before performing the analysis, the model was simplified by non-dimensionalizing the variables in a system of differential equations that described the dynamics of the spread of malaria. The results of the analysis show that the model has two fixed points, namely the disease-free fixed point and the endemic fixed point. The disease-free fixed point is locally asymptotically stable, if the basic reproduction number is less than one, it means that the disease will die within a certain time. The endemic fixed point is locally asymptotically stable, if the basic reproduction number is more than one, it means that the disease will spread in the population. The result of further analysis shows that bifurcation occurs in the system when the basic reproduction number is equal to one, meaning that there is a change in the stability of the fixed point due to changes in parameters, one of which is a change in migration parameters.
Based on the simulation results, changes in the migration rate have a directly proportional effect on changes in the basic reproduction number. If the migration rate is reduced, the value of the basic reproduction number will decrease, so that it can reduce the rate of malaria spread. Therefore, the best thing that can be done to minimize the malaria outbreak caused by the human population and infected mosquitoes is to reduce the rate of population migration. Suggestions for further research are to consider the possibility of recovered individuals being infected with malaria again so that the SIR-SI model becomes SIRS-SI model.