Abstract
In this article, a model representing the spread of Hepatitis B disease is constructed as a nonlinear autonomous system. The model divides the considered human population into three classes, namely susceptible, infected, and recovered class. The dynamical analysis shows that there are two equilibrium points in the model, namely a disease-free equilibrium point and an endemic equilibrium point. The existence and stability of the equilibrium points depend on the basic reproduction number (R_0). The disease-free equilibrium point is local asymptotically stable when R_0<1Â while the endemic equilibrium point exists and is local asymptotically stable if R_0>1. The five parameters of the model are estimated by applying Downhill Simplex (Nelder-Mead) Algorithm and by using the infected data cases taken from such a hospital in Malang. The estimated parameters are the transmission of infection rate, the saturation rate, the vaccination rate, the recovery rate, and the immunity loss rate. The resulting parameter estimation supports the analytical result and is used to illustrate the analytical results numerically. Based on the considered model and the result of the parameters estimation, it can be concluded that the Hepatitis B spread in Malang is controllable.Keywords: downhill simplex (Nelder-mead) algorithm, dynamical analysis, hepatitis B model, parameter estimation.Â
Highlights
Hepatitis B disease is a type of infectious disease caused by the Hepatitis B Virus (HBV)
HBV is more spread than other viruses, like Hepatitis C Virus (HCV) or Human Immunodeficiency Virus (HIV)
A SIRS model representing the spread of the Hepatitis B disease model has been reconstructed in the form of a three-dimensional nonlinear autonomous differential equation system with eight parameters
Summary
A Hepatitis B disease model was reconstructed and represented as a SIRS model by assuming that the recovered individuals can lose their immunity. Parameter Estimation The estimation process determines the value of the parameter in the model by minimizing the difference between the estimated solution of the model and the real data iteratively. 5. If Of1 ≤ Ofr < Ofi, the reflection point is accepted. 6. If Ofr < Of1, calculate the expansion point θ⃗e = θ⃗̅ + χ (θ⃗r − θ⃗̅). If Ofe < Ofr, accepted the expansion point θ⃗e and terminated the iteration. If Ofe ≥ Ofr, accepted the reflection point θ⃗r and terminated the iteration. If Ofi ≤ Ofr < Ofi+1calculate the outside contraction point θ⃗c = θ⃗̅ + γ (θ⃗r − θ⃗̅). Ofr , θ⃗c is accepted and terminated the iteration otherwise go to the step. If Ofr ≥ Ofi+1calculate the inside contraction point θ⃗cc = θ⃗̅ − γ (θ⃗̅ − θ⃗i+1). January February March April May June July August September October November December
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