OPTIMAL CONTROL OF INFLUENZA A DYNAMICS IN THE EMERGENCE OF A TWO STRAIN

. This paper examines the influenza spread model by considering subpopulation, vaccination, and resistance to analgesic/antipyretic drugs + nasal decongestants Based on the studied model, the non-endemic, endemic stability points and the basic reproduction number are determined. In the model studied, control is given in an effort to prevent contact between individuals infected with influenza and susceptible (u 1 ), and control treatment for infected individuals in an effort to accelerate the recovery of infected individuals (u 2 ). In the numerical simulation, using the control u 1 the number of infected individuals in the subpopulation decreased compared to that without control. The number of individual recovered subpopulations using the u 2 control increased more than that without the control.


INTRODUCTION
Flu or influenza sufferers will experience a fever, headache, runny nose, stuffy nose, and cough. A person can catch the flu if he accidentally inhales the droplets of saliva in the air, which are released by the sufferer when he sneezes or coughs. In addition, touching the mouth or nose after handling objects that have been splashed by the patient's saliva can also be a means of transmitting the flu virus [1].
Mathematics can play a role in controlling, analyzing, evaluating, and optimizing control of the spread of influenza. Mathematical models can be used to analyze the dynamics of flu transmission by including aspects of drug resistance [2]. Post-coinfection mathematical system with flu and a Gram-positive, used a Grampositive, and COPD risk appraisal [3]. Study of a good system of avian flu with fifty percent-impregnated prevalence, developed a formula of avian flu for both raspberry and mortal poeples. The factor of halfimpregnated prevalence on the transmission dynamics of the complaint is delved into [4]. The study of the epidemiological consequences of viral interferences: a mathematical model of two interacting viruses investigates the spread of two contagions that come into contact with repression of a single contagious if the two contagions do in these hosts [5]. The formulation system studied flu and other spreadable diseases of the respiratory tract. Influenza and infectious disease transmission in vivo, regardless of the effect of prolixity and advection on disease kinetics and sites [6].Recent timing and estimation of initial conditions of Zika transmission were also studied [7].
Modeling studies of drug-resistant antimicrobials in patients who have not recovered for a long time also discuss resistance to new pathogens [8]. E-commerce analysis between the ingrain and the adaptive susceptibility response to COVID-19 and the aftermath of growing pathogenesis [9]. Influenza infection models in the host or cell culture need to be investigated for the future [6]. Multiscale studies for different spatial scales of flu spread in individuals ranging from small to large to population scale. [10]. The study aimed to develop a model for calculating the transmissibility of the infection [11].
Influenza vaccine is a vaccine to prevent flu. Influenza vaccination is recommended to be carried out regularly every year to maintain optimal vaccine protection. Research has also been done to control flu, by giving vaccinations, paying attention to priority groups, and individuals who are resistant to flu drugs [12]. Control of treatment and prevention is given in an effort to optimize the prevention of the transmission of flu [13], [14]. This paper will examine the optimal control of overcoming the spread of influenza by amount consideration to vaccination and drug resistance. The controls given are: prevention control and treatment control.

RESEARCH METHODS
This research method is a literature review on the influenza spread model. The population in this paper is divided into five subpopulations. The susceptible subpopulation, that is, individuals who are still healthy but susceptible to infection with influenza, is denoted by S. Vaccination subpopulations are individuals who are vaccinated, denoted by V. Subpopulation infected with type A-strain, that is individuals who are still sensitive to analgesic/antipyretic drugs + nasal decongestants and are denoted by IA. Subpopulation infected with type B-strain, that is individuals infected with influenza virus who are already resistant to analgesic/antipyretic drugs + nasal decongestants and are denoted by IB. The recovered subpopulation is individuals who have recovered from influenza or are immune from vaccination and are denoted by R.
The assumption of the model studied in this paper is that recruitment individuals enter the S subpopulation at a rate of . S subpopulation enter to V subpopulation at a rate of . Individuals of the S subpopulation in contact with the IA individuals enter to the IA subpopulation at a rate of 1. Individuals of the S subpopulation in contact with IB individuals entered to the IB subpopulation at a rate of 2. Individuals in IA subpopulation treated with analgesics/antipyretics + nasal decongestants and recovered into the R subpopulation at a rate of t1, Individuals who are resistant to analgesics/antipyretics + nasal decongestants enter the IB subpopulation, with the rate of . Individuals in the IB subpopulation were treated with analgesics/antipyretics + nasal decongestants + antihistamines + antitussives/expectorants and recovered into subpopulation R, with a cure rate of t2. Infected individuals with influenza do not cause death. Individuals who have recovered or who are temporarily immune from vaccination may be susceptible to influenza re-infection with the rate of . Individuals in each population can die naturally with the rate of . The schematic diagram of the mathematical model of the spread of influenza can be expressed as Figure 1 below.  Based on the model assumptions, the study model in this paper can be expressed as of differential equations system: Based on equation (1)-(6), we get = − We assume that all parameters in model are positive and the initial conditions of system (1)-(6) are given: Parameters, parameter explanations and source of parameter values are stated in Table 1 below: with 1 = + , 2 = + , dan 3 = + .
Endemic equilibrium point, based on Eq. (1)- (6), equilibrium point of influenza disease-free provided that each subpopulation change per unit time is equal to zero and the number of infected individuals is not equal to zero, in other words From Eq. (1) by substituting * * , * * , * * , * * to = 0, then obtained * * .

Reproduction number (R0)
Reproduction number is a parameter to determine whether the number of spread of the disease is increasing or decreasing. If R0 < 1, then the disease has decreased or disappeared in the population. If R0 =1, then the number of infected people in the population is monotonous. If R0 > 1, then the number of infected in the population increases. reproduction number obtained by using the next generation matrix method [16].
To determine R0 can be determined by involving Eq. (3)-(5). The first step is to determine the Jacobian matrix by substituting the non-endemic equilibrium point of Eq. (3)-(5) in the subpopulation whose contacts are between susceptible and infected, the matrix F is obtained. .
RA is the expectation of individuals infected with influenza caused by one individual subpopulation IA entering a susceptible subpopulation. RB is the expectation of individuals infected with influenza caused by one individual IB subpopulation entering a susceptible subpopulation.

Optimal Control of Influenza
Optimal control of countermeasures to combat the spread of influenza, given control of prevention by providing campaigns in the form of counseling to susceptible subpopulation individuals which is denoted by u1. Treatment control is given to individuals in the IA and IB subpopulation in an effort to accelerate healing by providing vitamins which are denoted by u2.
Based on Eq. (1)-(5), given the control u1 and u2, the system of differential equations is obtained as follows: Control function u1(t) dan u2(t) are bounded, Lebesgue integrable functions. Control (1-u1) given in an effort to reduce contact between infected individuals with susceptible individuals and between infected individuals with vaccination individuals. Control (1 + u2) given in an effort to accelerate the healing of infected subpopulation individuals. The objective function is defined as follows: with tf is the end time of influenza control, A and B, is the balance weight of influenza treatment costs, C1 and C2 is the balancing weight cost of influenza control.
The costate functions obtained with use the Hamiltonian (16), with associated to S, V, IA, IB and R, we obtain

RESULTS AND DISCUSSION
Solution of a dynamic model with control intervention using an iterative method with a Runge-Kutta fourth orders. Solution dynamic model with control intervention with an initial guess forward in time next we solve the costate models backward in time. Starting with an initial guess for the costate variables, obtained solution the state equation a forward Runge-Kutta fourth order method in time [17]. In the numerical simulation, we will compare the spread of influenza for susceptible subpopulations, vaccination, infected and recovered with control and without control. Suppose the initial number of each subpopulation S(0) = 1000000, V(0) = 100000 , IA(0) = 50000 , IB(0) = 20000 , dan R(0) = 0, parameter value used on Table 1. Numerical simulation for suspected subpopulation with control and without control can be seen as Figure 2.  Based on Figure 3, the number of individuals in the vaccinated subpopulation with control increases from time t = 0 to t = 5 days, and decreases from time t = 5 days to t = 100 days. Whereas, with u1 control, it decreased from the time t = 0 to t = 100 days. Based on Figure 4, the number of individuals in the IA subpopulation with no control increased from time t = 0 to t = 65 days, and after time t = 65 days it decreases slowly until time t = 100 days. Whereas, with control, the number of individuals in the IA subpopulation decreased slowly from the time t = 0 to t = 100 days. Based on Figure 4, the number of individual IB subpopulations without u1 and u2 increased from time t = 0 to t = 100 days. Whereas, with u1 and u2, the number of individuals in the IA subpopulation decreased slowly from time t = 0 to t = 100 days. Based on Figure 5, the number of individuals subpopulation R with u2 increased quickly from time t = 0 to t = 100 days. Whereas, without control, the number of individuals in the R subpopulation continued to increase but very slowly from time t = 0 to t = 100 days. Based on Figure 6, optimal u1 from time t = 0 to t = 57 days, while from time t = 57, u1 decreased until t = 100 days. Optimal u2 from time t = 0 to t = 15 days after t = 15 days u2 decreased until time t = 40 days, after t = 40 days treatment control was no longer needed, as all individuals of the influenza-treated subpopulation had recovered.

CONCLUSIONS
Based on the results of the study and control measures in the model of the spread of influenza, the following conclusions were obtained: (1) Analysis of the influenza disease-free equilibrium point, the endemic equilibrium point, and the basic reproduction number. (2) In the model given control of influenza prevention (u1) and control of influenza disease treatment (u2). In the model of the spread of influenza that is given control, we obtain the theorem for the existence of adjoint variables from the state variables in the system. (3) The results of numerical simulations show that the individual subpopulations IA and IB with controls u1 and u2 decreased from time t = 0 to t = 100 days. Whereas, individuals in the IA subpopulation with no control increased from time t = 0 to t = 65 days, and after time t = 65 days it decreases slowly until time t = 100 days and individuals in the IA subpopulation with no control increased from time t = 0 to t = 65 days, and after time t = 65 days it decreases slowly until time t = 100 days. The number of individual recovered subpopulations increased more with the u2 control than without the control.