Optimal control mathemathical SIR model of malaria spread in South Kalimantan

South Kalimantan is one of the provinces vulnerable to malaria because their work is in the vicinity of forests such as miners, and seekers living on the edge of the forests based on a report from South Kalimantan Health Office (2013). Malaria eradication has always been carried out by the government through the South Kalimantan Health Office (2016) and its achievements have increased, but it is still a problem and needs tougher efforts to achieve malaria-free targets 2020. One way to eradicate the disease is to control it through mathematical modeling. This study discussed the formation of modified malaria distribution model. Analyze infection rates in South Kalimantan using the malaria distribution model. Conducting Optimal Control to complete Mathematical Model SIR of Malaria Spread in South Kalimantan.


Introduction
Malaria is a contagious disease caused by a female Anopheles mosquito bite that contains plasmodium. This disease was first discovered by Charles Alphonse Laveran in 1880 in Algeria, in the form of plasmodium falciparum gametocytes (banana form). Plasmodium will multiply in human blood cells. The cause of the disease is natural (natural), through the bite of anopheles female mosquito. This disease attacks all men and women. Symptoms that occur after infection include fever, chills, sweating, headache, nausea, and even vomiting based on reports from the Ministry of Health of the Republic of Indonesia in Data and Information Center in 2016.
Ministry of Health of the Republic of Indonesia in the Bulletin of Data and Health Information Data in 2011 states that malaria is one of the diseases that must be eradicated (total destruction of the affected part of the disease to the root to eradicate the disease). Most areas of Indonesia are still endemic areas of malaria. The rate of malaria transmission in a given region is determined, Reservoir, reflected by the prevalence of the case, the species suitability vector or anopheles mosquito strain as vector, breeding rate, flying distance, resting habits, eating habits and the amount. The new human Hospes, which is meant is the presence of non-immune human groups entering endemic areas, local climatic conditions, geographical and hydrographic conditions, coupled with human activities and activities, affecting their exposure and access to developing places and mosquito habitats anopheles. One of the areas suitable for the place is the area of South Kalimantan. South Kalimantan is also vulnerable to malaria, due to the fact that the community's work is still in the vicinity of forests such as mining workers, gold miners, forest product seekers even living on the edge of the forests based on South Kalimantan Health Agency report in 2013. One of the local electronic media reported that there are two districts belonging to malaria endemic category namely Tanah Bumbu and Kotabaru. Throughout the year 2012, data from the Health Service of South Kalimantan recorded 9,385 cases of malaria or by 0.00248% of the population.
In a previous study "Optimal Control on Determination of Time Interval and Optimal Dosage of Malaria" Pardi Affandi et al. discussed the formation of malaria disease model then optimize the Determination of Time Interval and Optimal Dose of Malaria Disease, in the following study 'Optimal Control Mathematical Model of Malaria Spread in South Kalimantan' will be discussed modification of malaria spread model then model used to analyze the rate of invasion of malaria disease based on data obtained sourced from South Kalimantan Provincial Health Office will then be determined model solution by using control theory.

The set of convex and convex function
The concept of a convex function underlie some part in the discussion section. The following definitions and theorems related to the set and convex function. Definition 1 (Mangasarian).The set߁ ⊂ ℝ called convex set if for to any‫ݔ‬ ଵ , ‫ݔ‬ ଶ ∈ ߁ and for ߣ ∈ ℝsuch that 0 ≤ ߣ ≤ 1 ‫݈݈݅ݓ‬ ‫݈ܽ‬ ‫1(ݕ‬ − ‫ݔ()ߣ‬ ଵ ) + ‫ݔ(ߣ‬ ଶ ) ∈ ߁.

Non homogen linear differential equation and solution
Definition 2 (Ross S L, 1984). Differential equations are equations containing derivatives of one or more dependent variables for one or more independent variables. Definition 3 (Ross S L, 1984). Order linear differential equations-n, with the dependent variable, and the independent variablex, can be expressed as follows

Basic reproduction numbers
The basic reproduction number serves to inform the spread of disease and can be a parameter in providing strategies for disease control. Biologically the reproduction number can mean the average number of secondary infection cases that occur when the infected individual enters the population that are all Susceptible. ܴ 0 < 1 : The disease will disappear over time ܴ 0 = 1 : The disease will become spread and remain on a large scale epidemic ܴ 0 > 1 : There will be an epidemic with a very high degree of relevancewith death.

Epidemic model
Malaria is caused by parasites of the Plasmodium species. The parasite is transmitted to humans through the bite of Anopheles female mosquito. After biting the human parasite changes through a complex life cycle. Parasites multiply in the liver and bloodstream of humans. After 10 to 15 days mosquitoes carry parasites and can infect new people. Approximately 9 to 14 days after bitten, will show symptoms of malaria (Rollback, 2010). In Indonesia, the most commonly found species are 4, namely Plasmodium falcifarum and Plasmodium vivax, Plasmodium malariae, and Plasmodium ovale. According to Arsin (2012) ever found malaria caused by Plasmodium ovale reported from Flores.
The nature of the parasite varies with each species of malaria that affects its transmission. Plasmodium falciparum has the shortest period of infection, but it produces the highest parasitemia, the most severe symptoms and the shortest incubation period. Plasmodium falciparum gametocyte develops after 8-15 days after the entry of the parasite into the blood. Plasmodium falciparum gametocytes show the periodicity and infectivity associated with vector bite activity. Plasmodium vivax and Plasmodium ovale generally produce low parasitemia, milder symptoms but have a longer incubation period. Sporozoit Plasmodium vivax and Plasmodium ovale develop into primary and hypnozoit tissue scizons in the liver. This hypozoite results in relapse (Arsin, 2012).

Optimal control
In the following discussion, the problems given in the case of optimal control with state end and the end time is known. In other words, the target set S shaped{ܵ = ‫ݔ‬ ଵ ‫ݐ(‬ ଵ )}in the form ‫ݔ(‬ ଵ, ‫ݐ‬ ଵ )with‫ݔ‬ ଵ specially elementin R n and ‫ݐ‬ ଵ element at (T 1 ,T 2 ).Given the state system by the end and the end time unknown ‫)ݐ(ݔ‬ = ‫,)ݐ(ݔ[݂‬ ‫,)ݐ(ݑ‬ ‫]ݐ‬ with x(t) vector state sized ‫,1ݔ݊‬ u(t) input vector sized ‫,1ݔ݉‬ f a vector valued function. Initially given state is X 0 and initially time is t 0 . Target set S form ‫ݔ(‬ ଵ, ‫ݐ‬ ଵ ) with t ଵ ϵ (T ଵ , T ଶ ) known value and ‫ݐ‬ ଵ > ‫ݐ‬ . Optimal control problem is to find the admissible control u(t) with the initial value ‫ݔ(‬ , ‫ݐ‬ )and the final value ‫ݔ(‬ ଵ, ‫ݐ‬ ଵ ) that maximizes the objective function ‫)ݑ(ܬ‬ = ∫ ‫,)ݐ(ݔ(ܮ‬ ‫,)ݐ(ݑ‬ ‫)ݐ‬ Tosolve the problems mentioned above optimal control, first determined necessary condition for optimal control are met.

Discussion of problems
Patients with malaria with gametocytes become a source of transmission with an intermediary is a mosquito as a vector. Humans are an important reservoir, but the disease is transmitted through the bite of a female anopheles mosquito containing the plasmodium species parasite. Most occur naturally, through the bite of a female anopheles mosquito, rarely transmitted by blood transfusion and/or bone marrow transplantation. When female anopheles mosquitoes suck blood containing gametocytes, then in the body of the mosquito occurs fertilization that produces a zygote. The zygote develops into an ookinet then penetrates the gastric wall of the mosquito. On the outer wall of the ookinetic mosquito will become an oxista and then become sporozoit that is infective and ready to be transmitted to humans. When a mosquito sucks human blood, it will enter the bloodstream. Approximately 9 to 14 days after humans are bitten, the symptoms of malaria appear.

Mathematics modelling
Malaria is one of the epidemic diseases and can be modelled in mathematical models. The establishment of this malaria dispersion model follows a SIR model consisting of three compartments: 1. The susceptible (susceptible) group is further denoted by the number of individuals susceptible to malaria at the time. 2. Infected group (infected) further denoted by the number of individuals infected with malaria at the time. 3. The recovered group is further denoted by the number of individuals who recover from malaria at the time.
The assumptions used in the malaria distribution model are as follows: Population is assumed to be open so that there is an increasing (incoming) or decreasing (out) population of the population, Natural birth and death are constant, There is only malaria disease in the population Disease spreads through contact between individuals with mosquitoes, Every individual born directly into a susceptible group, Malaria-infected individuals can recover from illness and can experience death only because of illness and the malaria-infected individuals can recover because of short cycles of malaria and the presence of natural immunity. Every individual in doing Immigration as well as Emigration. The process of forming the model of malaria disease spread as follows obtained a system of nonlinear differential equations for the spread of malaria as follows is called the model of malaria transmission.

The endemic equilibrium point
The endemic equilibrium point is a condition in which the invading disease still exists and is still Next we substituted to equation then obtained, ‫ܫ‬ = ‫ܤ(‬ + ‫ߚ)ܭ‬ − ߤ(ߙ + ݀ + ‫)ܮ‬ ߚ(ߙ + ݀ + ‫)ܮ‬ and also we obtained, So as to obtain the endemic equilibrium point that is :

Basic reproduction numbers
Basic reproduction number (R ) is a certain parameter used to determine the dynamics of the spread of disease, R is used as a quantity to express the extent of disease spread.
There are two states that occur from equation that is as follows.  . So conditions ܴ 0 < 1so let it be After the calculation obtained value ܴ 0 can be seen from the table as follows.  Based on table 2 with the value of K given the smaller the value of ܴ obtained is also smaller from year to year every Regency/City. This is due to the fact that when the large total population, with small infected populations, has a large vulnerable population, so the value of ܴ obtained is small. Whereas if the total population is small, with a large infected population then the small vulnerable population so that the value of ܴ obtained is also large. In Figure 3 given the value of K = 0.75 from 2012 to 2016 can be seen the spread of malaria infection rate in South Kalimantan based on ܴ value. The lowest rate of malaria infections occurred in Banjarmasin, while the highest malaria infection rate occurred in Balangan Regency.
Based on the figure 3 with the value of K is getting smaller then the value of ܴ in each Regency/City is also getting smaller every year. Based on the value ܴ obtained gives information that the lowest level of malaria infection occurred in Banjarmasin City because the lowest ܴ value while the highest malaria infection rate occurred in Balangan regency because the value of ܴ is the greatest.

Linearization of the System at the Point of Equilibrium Free of Illness
The stability of an equilibrium point can be determined based on the eigenvalues. Since the system of this model is a system of nonlinear differential equations it can be determined by linearizing using a Jacobian matrix.Furthermore, the above equation is exemplified : Function ݂ ଵ ,݂ ଶ , dan ݂ ଷ derived toI obtained: based on the above scaling process, a Jacobian matrix of the following equation system is formed : Characteristikequations :  populations is smaller than the number of recovered individuals plus the individual who has suffered good death from the disease then the eigenvalue is negative. So in time the system will stabilize toward the point of disease-free equilibrium.

Optimal Control for Malaria Distribution Model.
Optimal control of the Malaria distribution model aims to maximize the number of healthy pink (reticulocyte) blood cells by controlling malarial drugs that inhibit reticulocyte invasion in the human body. The action of this drug can be controlled optimally by applying the maximum principle of Pontryagin. The maximum principle of Pontryagin is a condition such that it can be obtained by completion of optimal control in accordance with the objective of maximizing performance index. The state equation is: With t adalah first time, t is finishing time, x(t) is state variable and u(t) is control variable. Equations performance index expressed as the system from the initial state to the final state is: With T s the initial state of healthy reticulocyte cells, Aweighting parameters used in controls andu (t)is a drug control. Then, the maximum valueJ(u) = 10 administration of the drug works optimally so that the weight parameter can be assumedA = ଵ ଶ γwith γis a controlling weight in the form of malaria drugs. Furthermore, u is squared to become a quadratic function u 2 to maximize the action of the drug by doubled administration so that the performance index for the malaria dispersion model is as follows: Model of malaria spread before being given control (u(t)) that: In the equation above there are two equations that are influenced by the virus is the equation Ṡ andI. In the equation Ṡ and İ the mosquito-infected reticulocyte cell is symbolized byβSI. Control variable (u(t))n the form of malaria drugs given when the human body is attacked by mosquitoes and reticulocyte cells begin to become infected. In this case, drug efficiencyu ≡ 1which means the reticulocyte cell is not infected again. While, u ≡ 0 is a case where the drug does not provide changes to the disease. Then, to know the value of u (t) is optimal then the value of the controller is (1 − u(t)) so the model changes after being given the controller as follows: with boundary conditionst < ‫ݐ‬ < t dan 0 ≤ u(t) ≤ 1. With t is the initial time when malariainfected reticulocyte cells were administered malaria and drugst is the final time when reticulocyte cells infected with the malaria virus are given the drug. Then, based on the principle of maximum Pontryagin the first step is to determine the Hamiltonian function is as follows: The second step is assumed that at the optimum time, the control variables and state variables are respectively u * (t) dan x * (t)must : Because valueu(t) is0 ≤ u(t) ≤ 1so canu * (t) is 0 ≤ u * (t) ≤ 1so as to obtain the following possibilities: Based on the above three possibilities, the first possibility in the calculation of the mathematical value obtained u * (t) is0 < Furthermore, of the three possible values of u * (t) obtained then a value can be determinedu * (t) which is optimal in step three. Because u(t) ∈ U = {u(t)|0 ≤ u(t) ≤ 1, t < ‫ݐ‬ < t } so as to obtain the value of the smallest limit (supremum) as follows: and obtained the value of the largest lower limit (infimum) as follows: so the valueu * (t)is : Based on this caseu * (t)in equationu * (t) optimally declare the maximum of drug work with minimal medication.Then, the fourth step is to solve the state equation because of the controller formu * (t)contains state variables(S, I, R)that is ẋ= ∂H ∂λ so as to obtain the optimal state equation as follows: In addition to state variables there are also variable costate λ ଵ , λ ଶ and λ ଷ on the form of the controller u * (t)it is necessary to solve the costate equation to obtain the optimal costate equation in the fifth step is as follows: Then, the sixth step substitutes the equationu * (t)which has been obtained into the state equation to obtain the optimal form of the solution. Here are the results of the optimal equation system obtained: İ= βSI − (α + d)I ቂmin ቀmax ቀ0, Ṙ= αI − u * R Based on the above description, to obtain S, I and R from the optimal form u * (t) then it is necessary to solve the non-linear state and costate equations. Because the system of nonlinear equations is difficult to resolve analytically, it can be resolved and numerically illustrated.

Simulation results
The optimal control of the malaria distribution model obtained in the previous section can be illustrated in graphical form. The result of the optimal equation system that has been obtained is simulated using MATLAB application. In the simulation model of malaria distribution by defining with S = y 1 , I = y 2 and R = y 3 as well as some parameters.  Then, the above parameters are used in the simulation process to find out the results of the optimum equation system. Here is a picture of the simulation results obtained:  Based on the simulation model of malaria distribution before and after being given controller there is a difference in the two pictures above.

Simulation for a healthy reticulocyte cell population (T)
The graphic results obtained after the simulation are as follows: Whereas, in Figure 6 it is seen that after administration of the control of drugs the population of reticulocyte cells continues to increase every day. This suggests that the drug works to maximize the number of healthy reticulocyte cells for better immunity.

Simulation for population of infected cells reticulocytes
The second simulation is a simulation for infected cell populations of reticulocytes before being given a control of the formate. Then the next simulation is a simulation for the population of infected cells of the reticulocytes after being given a controlled drug. The graphic results obtained after the simulation are as follows:  Figure 8 it appears that after administration of a drug-controlled population the infected reticulocyte cell continues to decline every day. From the graph can be determined the optimum time interval for malaria treatment, and the optimal dose of treatment for malaria disease

Conclusions and Suggestions
From the discussion obtained some conclusions: Model of malaria distribution in South Kalimantan as follows: dS dt = B + K − βSI − μS dI dt = βSI − (α + d + L)I dR dt = αI − (μ + L)R The lowest level of malaria infection occurs in Banjarmasin City because the value of reproduction numbers is essentially the smallest. While the highest malaria infection rate occurred in Balangan Regency because the value of reproduction is essentially the largest number.