THE EFFECT OF LONG-LASTING INSECTICIDAL NETS ON THE DYNAMICS OF MALARIA SPREAD IN INDONESIA

Article History: Malaria is an infectious disease that can lead to death. Deaths from malaria globally have increased in 2020. So the spread of this disease is still a serious problem for society. The mathematical model used is the SIR-SI model, assuming that recovered individuals can be re-infected with malaria. Analysis was carried out on the effectiveness parameters of long-lasting insecticidal nets to determine their effect on the dynamics of the spread of malaria. The sensitivity analysis results showed that changes in the parameters of the effectiveness of long-lasting insecticidal nets had an inverse effect on the rate of spread of malaria. These results follow numerical simulations conducted using malaria case data in Indonesia (some assumptions). Thus, efforts can be made to suppress the spread of malaria by increasing the effectiveness of long-lasting insecticidal nets.


INTRODUCTION
Malaria is an infectious disease caused by protozoa of the genus Plasmodium.Female Anopheles mosquitoes transmit this disease.When an Anopheles mosquito bites a human infected with malaria, the mosquito will suck Plasmodium along with blood.The malaria parasite then reproduces in the body of the Anopheles mosquito.When it bites another human not infected with malaria, the malaria parasite enters the victim's body along with the mosquito's saliva [1].The initial symptoms resemble influenza [2], but if left untreated, complications can occur, leading to death [3].WHO records that deaths from malaria in 2020 increased by 69 thousand from the previous year [4].So that the spread of this disease needs to be a concern of the government, and malaria case-control programs need to be encouraged so that the number of malaria cases or deaths from malaria does not increase in the following years.
Malaria research has been studied in various scientific fields, one of which is in mathematics, namely by using a mathematical model.A mathematical model was constructed to see the dynamics of the spread of malaria.The article [5] examines a model of the spread of malaria, which consists of the SIR compartment for the human population and the SI compartment for the mosquito population.The model considers vector control as using long-lasting insecticide nets, namely insecticide-treated mosquito nets, which are effective for at least three years [6].Long-lasting insecticidal nets not only prevent mosquitoes from biting humans but can also kill mosquitoes attached to mosquito nets because mosquito nets are equipped with insecticides that are harmless to humans [7].However, this model assumes that recovered individuals get permanent immunity, so they cannot be re-infected.At the same time, recovered individuals can be re-infected with malaria [8].
Therefore in this article, the authors modified the model [5] by assuming that recovered individuals could be re-infected with malaria.Furthermore, the model will be simulated and interpreted to determine the effect of parameters of long-lasting insecticidal nets on population dynamics.

RESEARCH METHODS
In this section, the model is formed by modifying the model [5] with the additional assumption that recovered individuals can be re-infected with malaria.In the model, there are two populations, namely the human population and the mosquito population, with the human population divided into three, namely susceptible, infected, and recovered.The mosquito population is divided into two populations, susceptible and infected because the mosquito distribution period cycle ends in death (never recovering from infection).Before the analysis, a simplification of the model with nondimensional will be carried out [9].So that dimensionless variables are defined as follows: If these variables are substituted into Equation (1) -Equation ( 5), a mathematical model is obtained, which will then be used for analysis (the role of the notation  * is replaced with ).

Fixed Point Stability Analysis
From Equation (6) -Equation ( 8), two fixed points are obtained, namely the disease-free and endemic fixed points.The stability of the two points is determined based on the basic reproduction number.This number can indicate that a disease is spreading in the population or experiencing extinction [10].The disease-free fixed point and endemic fixed point are respectively obtained as follows: ).
The largest non-negative eigenvalue of the next-generation matrix is called the basic reproduction number, so it is obtained The following section will prove the fixed point stability theorem for the two fixed points obtained from the model.
Theorem 1.The disease-free fixed point from Equations (6) -Equation ( 8) is locally asymptotically stable if   <  and unstable if   > .
Proof.Local stability analysis of point  0 can be proven using the Jacobian matrix.Based on Equation (6) -Equation ( 8), the Jacobian matrix at  0 is: .
The classification of fixed point stability is based on the eigenvalues (λ) typical of the Jacobian matrix.The eigenvalues of the Jacobian matrix  0 are obtained from the characteristic equation | −  0 | = 0 so that with  =  + .Based on Equation ( 9), three eigenvalues are obtained, namely: 2 .
The values of  1 and  2 are negative for all ℛ 0 values, while the value of  3 depends on ℛ 0 .If ℛ 0 < 1, then  3 is negative so that point  0 is locally asymptotically stable.Meanwhile, if ℛ 0 > 1, then  3 is positive; this shows that point  0 is unstable because there is one positive eigenvalue.∎ Theorem 2. The endemic fixed point in Equations ( 6) -Equation ( 8) is locally asymptotically stable if and only if   > .
Then, the  0 matrix has one right eigenvector  and one left eigenvector , corresponding to zero eigenvalues.The right eigenvector and left eigenvector are obtained as follows.Furthermore, the values of  and  are obtained as follows.
The values  and  obtained correspond to one Castillo-Chaves and Song Theorem case.
Consequently, when ℛ 0 changes from ℛ 0 < 1 to ℛ 0 > 1, the unstable  1 endemic fixed point changes from negative to positive and is locally asymptotically stable.So, it is proven that if ℛ 0 > 1, then the endemic fixed point  1 is locally asymptotic.∎ Based on the two theorems above, the stability of the fixed point depends on the basic reproduction number.The basic reproduction number depends on several parameters, so changes in the value of specific parameters will affect the value of ℛ 0 .However, each parameter has a different influence.The magnitude of the influence of parameters on changes in the value of ℛ 0 can be seen from the parameter sensitivity index values.

Sensitivity Analysis
The sensitivity analysis carried out in this study aims to see the effect of changes in the effectiveness parameters of long-lasting insecticidal nets on the basic reproduction number (ℛ 0 ).This analysis needs to be carried out because the basic reproduction number is a benchmark in predicting whether a disease will spread or not in a population.The sensitivity index measures sensitivity analysis [13].The following is the sensitivity index for the effectiveness parameter of long-lasting insecticidal nets.
The above equation is negative, meaning that there will be a decrease in the value of ℛ 0 if the value of parameter  is increased with the value of another parameter with a fixed value, and vice versa.This can be explained through the following simulation.

Numerical Simulation
In this section a numerical simulation will be carried out to determine the effect of the effectiveness of long-lasting insecticidal nets on the dynamics of the spread of malaria.A simulation will be carried out by changing the value of the parameter  because this parameter depends on the parameter of the effectiveness of long-lasting insecticidal nets (  ).The simulation was 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 increased by the value of another parameter having a fixed value.The simulation was carried out with the parameter values used, which are presented in Table 2 assuming initial values, namely  ℎ0 = 0,9;  ℎ0 = 0,1; and  0 = 0,1.The following is a picture of the solution field, which describes the dynamics of each population due to changes in the value of , with changes in the value of  shown in Table 3. Figure 2 shows that changes in the value of  affect the dynamics of each population.Figure 2e shows that there is a decrease in the proportion of the infected mosquito population because more and more mosquitoes die due to the use of long-lasting insecticidal nets and even experience extinction when  = 20 so that the remaining population of mosquitoes is susceptible, as can be seen from Figure 2d.The decrease in infected mosquitoes can affect interactions with humans, which can cause humans to become infected.As a result, the proportion of the infected human population decreases and even experiences extinction when  = 20, illustrated in Figure 2b, so the remaining susceptible and recovered humans in the population are shown in Figures 2 (a The numerical simulation results show that if the value of  is enlarged, the proportion of infected human populations and infected mosquitoes decreases.Because the value of θ is directly proportional to the effectiveness of mosquito nets, an increase in the value of   can reduce the proportion of infected human populations and infected mosquitoes and even become extinct.

CONCLUSIONS
a) Malaria still needs to be addressed in people's lives because deaths from malaria have increased globally in 2020.Research on malaria in the field of mathematics is by using a mathematical model.In this article, the model used is the SIR-SI model, with the possibility that recovered individuals could become infected again.
b) The analysis results show that in the model, there are two fixed points, namely the disease-free fixed point and the endemic fixed point, whose stability depends on the basic reproduction number.Numerical simulations show that the rate of spread of malaria will decrease or even become extinct within a certain period if the effectiveness of the long-lasting insecticidal net is increased.This is because the sensitivity index for the effectiveness parameter of long-lasting insecticidal nets is negative, which means that changes in this parameter have an inverse effect on changes in the basic reproduction number.
c) Therefore, efforts can be made to minimize malaria outbreaks caused by human populations and infected mosquitoes by increasing the effectiveness of long-lasting insecticidal nets, such as properly using and caring for mosquito nets.

Figure 1 .
Figure 1.Compartment diagram of disease spread Based on Figure 2, the mathematical model formed:

Figure 2 .
Figure 2. The dynamics of population proportions in (a) susceptible human, (b) infected human, (c) recovered human, (d) susceptible mosquito, and (e) infected mosquito Figure2shows that changes in the value of  affect the dynamics of each population.Figure2eshows that there is a decrease in the proportion of the infected mosquito population because more and more mosquitoes die due to the use of long-lasting insecticidal nets and even experience extinction when  = 20 so that the remaining population of mosquitoes is susceptible, as can be seen from Figure2d.The decrease in infected mosquitoes can affect interactions with humans, which can cause humans to become infected.As a result, the proportion of the infected human population decreases and even experiences extinction when  = 20, illustrated in Figure2b, so the remaining susceptible and recovered humans in the population are shown in Figures2 (a) and Equation 2 (c), respectively.

Table 1 .
For example, the total population of humans and mosquitoes is expressed by  ℎ and   , respectively, then  ℎ =  ℎ +  ℎ +  ℎ and   =   +   .

Table 1 . Description of Parameters
The contact rate between susceptible mosquitoes and infected humans with a chance of being infected is 1 1/(humans × time)