BAYESIAN MARKOV CHAIN MONTE CARLO SIMULATION OF NONLIENAR MODEL FOR INFECTIOUS DISEASES WITH QUARANTINE

The SIQS (Susceptible, Infective, Quarantine


INTRODUCTION
In general, the characteristics of infectious diseases have different clinical symptoms according to the causative factors of the disease (Mukhsar et al, 2016;Mukhsar et al, 2013;Mukhsar et al 2022). Based on clinical manifestations, the characteristics of infectious diseases consist of: 1) The spectrum of infectious diseases, 2) Covert infections (without clinical symptoms), 3) Sources of transmission. The phenomenon of the spread of infectious diseases can be described through modeling to find out the physical interpretation. This physical interpretation is used to describe phenomena that occur in everyday life (Allen, 2003;Hethcote, 2000;Mukhsar, 2018).
One of the models used in the spread of infectious diseases is the SIQS (Susceptible, Infective, Quarantine, and Susceptible) model with the characteristic that every individual is susceptible to being infected with a disease. Individuals in the infection class can recover through quarantine, but are not immune, so they may be re infected and enter the infection class (Bain and Engelhardt, 2000). Bayesian Markov Chain Monte Carlo is used for numerical simulations in obtaining the estimation of parameter model.

MATERIALS AND METHODS
Modeling uses Bayesian concepts, for example ′ = ( 1 , 2 , … , ) is a vector of n cases that has distribution ( | ) and parameter requirements ′ = ( 1 , 2 , … , ), defined probability distribution of ( ) as The likelihood function of the n random variables 1 , 2 , … , is defined as the joint density function. The joint density function ( 1 , 2 , … , ) considers of the θ. For example, we have n random samples 1 , 2 , … , and probability density function of ( , ), then the likelihood function is defined ( ) = ( 1 ; ) ( 2 ; ) … ( ; ) The Bayesian concept is then used to estimate the parameters of the SIQS model in Figure 1. The physical meaning of each symbol in equation (1) is described in Table 1.  (1) is linearized using the Runge-Kutta principle [Box and Tiao, 1973;Nakamura, 1991], with +1 ( ) is the k-th iteration approximation for +1 , and +1 (0) is an initial number of +1 . Iterations are declared convergent when | +1 ( ) − +1 ( −1) | less than the tolerance value set. The decritization of model (1), respectively, is In the term of +1 , let 1 = 1 in Poisson distribution 1~( 1 ), then probability density function of 1 is written as ( 2 ), then probability density function of 2 is written as For +1 , let 3 = 3 , we have 3~( 3 ) and probability density function for 3 written as The likelihood of ( 1 ; 1 ) is written as The likelihood of ( 2 ; 2 ) written as The likelihood of ( 3 ; 3 )written as

RESULTS AND DISCUSSION
The prior distribution used is a non-informative prior. In the SIQS model there are several parameters used to describe the model, namely α which describes the natural death rate in individuals, β which describes the level of transmission from individuals infected with the disease to susceptible individuals, δ which describes the level of infection from infected individuals to quarantined individuals, γ which describes the recovery rate from infected individuals to susceptible individuals without being quarantined and ε which describes the level of prevention and control of the spread of disease to susceptible individuals.

CONCLUSION
Discretization of the nonlinear model (1) for parameter estimation is purposed. The physical interpretation of the nonlinear model (1) is very useful if the mode parameters can be estimated. In this study, discretization of the nonlinear model (1) uses the Runge-Kutta method. Bayesian Markov Chain Monte Carlo for its numerical simulation. We use monthly measles data from of Kendari from 10 subdistricts for 2016-2018. After 10,000 iterations, convergent and significant parameters were obtained, namely beta = 94.37, beta0 = -10.19, mu = -0.23 and b = 0.5.