# Dynamics of a Two Serotype Disease with Antibody Dependent Enhancement

5-2006

Thesis

## Degree Name

Master of Science (MS)

## College/School

College of Science and Mathematics

## Department/Program

Mathematical Sciences

Lora Billings

Mark Korlie

Arup Mukherjee

## Abstract

The dengue virus is a serious infectious disease that can be found in many regions of Southeast Asia. There exist four serotypes of the virus. Recovery from one serotype produces a natural immunity from that serotype. However, it also creates complexes with a second infection and will increase viral production. This process is know as antibody dependent enhancement (ADE). As a result, it is very difficult to vaccinate against the disease. An optimal vaccination would have to cover all four serotypes at once. To understand the dynamics of the disease, we will study a mathematical model for two coexisting serotypes of the dengue virus. This is done using compartmental models based on a system of differential equations. After analyzing the system, we will consider various vaccination strategies for single serotypes.

The system that we study has four steady state solutions. We use various techniques of analyzing a dynamical system such as linearization about the fixed point and using the next generation matrix to determine the stability of the fixed points. The disease free equilibrium (DFE) is when both serotypes die out. Stability is determined by the basic reproduction number, R0, which is the number of secondary infections brought on by one infective in a susceptible population. When R0 is less than one, the DFE is stable, and it is unstable when R0 is greater than one. The system also has two boundary equilibrium where only one serotype will persist at a time. We use similar methods to find regions of stability for these fixed points.

The fourth steady state solution is the endemic equilibrium where both of the serotypes persist. This fixed point cannot be written in a succinct closed form, so other methods are used to analyze its stability. Using symmetry, we reduce the system to four equations. Using asymptotics and numerical techniques, we approximate the stability of the endemic equilibrium and show that it goes through a Hopf bifurcation. Using the parameters for Dengue, it can been shown that the endemic equilibrium is stable and then after the Hopf bifurcation experiences oscillations. Thus, the disease is always persisting.

We then model vaccination strategies to find if we can make the DFE stable. For example, if we vaccinate one hundred percent of the susceptible population against serotype one, then the system will behave as if only serotype two exists. We find a new reproduction number to determine when the existing serotype will persist and when it will die out. Another vaccination strategy is to partially vaccinate against one of the serotypes. Due to the ADE factors in the system, vaccinating partially against one serotype will never allow for the second serotype to die out. The final vaccination strategy that we consider is partial vaccination against both serotypes. We find conditions on the percentages of the population that should be vaccinated against each serotype to have the disease die out.

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