Cansu Evcin, M.Sc.

Department of Scientific Computing

February 2013, 48 pages

*Supervisor*: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)

### Abstract

Threshold dynamics used to control the spread of the disease in infectious disease phenomena has an overwhelming importance and interest in mathematical epidemiology. One of the famous threshold quantity is known to be the basic reproduction ratio. Its formulation as well as computation is the main concern of infectious diseases.

The aim of this thesis is to analyze the basic reproduction ratio in both autonomous and periodic systems via defining \(R_0\) as the spectral radius of the next generation operator.

This thesis presents the vector host model for the diseases Dengue fever and avian influenza. As emerging of the diseases shows periodicity, systems of periodic ordinary differential equations are considered for both types of diseases. Simple implementation of the time-averaged systems gives rise to the comparison of these with the periodic systems. Thus, we investigate the occurrence of the existence of underestimation or overestimation of the basic reproduction ratio in time-averaged systems.

*Keywords*: Threshold dynamics, basic reproduction ratio, periodicity, compartmental models, time averaged systems