Modeling the effect of screening in the propagation of an epidemic

Mi-Young Kim, Purdue University

Abstract

We study a model for the dynamics of an epidemic of S-I-R type in which the active population is randomly screened. Because of the variable infectivity and recovery rates, the governing dynamical system consists of integro-differential equations and it has a nonlinear, nonlocal feature. Physically meaningful screening terms create the possibility that the solution blow up in a finite time. This system is analyzed by reformulating it as an equivalent system of integral equations. Thanks to fact that the equation for the force of infection is a linear Volterra integral equation, local existence in time is obtained by a fixed point argument. Once we have non-negative solutions locally in time, we derive a priori estimates which we use in order to establish global existence. Next we investigate the asymptotic behavior of steady state solutions. By introducing dimensionless quantities, we derive an explicit threshold condition. A condition for the existence of a unique endemic state is proposed. When the threshold parameter satisfies the relation $R\sb0 <$ 1, the trivial equilibrium, which is then the only steady state solution, is globally asymptotically stable. In general, when $R\sb0 >$ 1, it is not possible to obtain the global stability. We use a perturbation method to treat the stability of the endemic equilibrium. The system for the perturbations reduces to a linear system of integral equations which is solved by taking Laplace transforms. With the aid of the Laplace transform we prove that the endemic equilibrium (if it exists) is locally exponentially stable when the characteristic equation $\Delta$(s) = 0 (which is defined in chapter 3) has no solution whose real part is non-negative. However instability must be treated in a different way. It is known that the endemic equilibrium might lose stability if the threshold parameter $R\sb0$ increases. We describe the characteristic equation of the linearized problem. Instability of equilibrium solutions will be investigated in future work. We propose a numerical algorithm of second order of convergence for the approximation of the solution based on the centered finite difference method. Nonlinearities are dealt with by a Crank-Nicolson type method and second order accuracy is preserved. We carry out the numerical analysis of the algorithm proposed. Finally, we present numerical experiments to see the effect of various screening programs on the spread of infection.

Degree

Ph.D.

Advisors

Milner, Purdue University.

Subject Area

Mathematics

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