Covid-19 is currently a pandemic that is considered as the biggest global threat that classified as the human-to-human transmissionable disease. This pandemic has affected practically all world countries. The aim of this paper is to construct a mathematical model for the spread of the Covid-19 outbreak and analyze its stability. The proposed mathematical model is STQIR (Susceptible, Traced, Quarantine, Infectious, Recovered) model. The form of the model is a nonlinear differential system with five variables. The step of the method i.e., compute the variables positifity, boundedness of solutions, and the basic reproduction number that computed using next generation matrix. Then the basic reproduction number will be used for testing the local stability of the disease free equilibrium using Routh Hurwitz criteria to examine its epidemiological relevance. This work also investigate the sensivity of the model with respect to the variation of each one of its parameters and is tested in application to the recent data on Covid-19 outbreak in Central Java Province, Indonesia.

Covid-19 is a disease caused by a new corona virus that has spread throughout the world and become a pandemic. It is classified as an infectious disease that can be transmitted from human to human through droplets. So that we need controls to reduce the spread of Covid-19. The optimal control that will be carried out this work is self-precaution, treatment and quarantined that will be applied to the dynamical modelling of Covid-19 spread using the Pontryagin’s Maximum Principle (PMP) to find out the optimal solution for the control. According to this principle the optimal control, corresponding optimal state, and adjoint function must minimize the Hamiltonian function. PMP converts the optimal control problem into a multipoint boundary value problem. That is, the optimality condition results in control. The optimal control variable, corresponding state and adjoint can be computed by solving an Ordinary Differential Equation system. The control strategies is aimed to reduce covid-19 transmission. Numerical results show the effectiveness of the control strategies in reducing Covid-19 spread. It is found that self-precaution more effective than treatment and quarantined.