#### Date of Award

2013

#### Degree Name

Mathematics

#### College

College of Science

#### Type of Degree

M.A.

#### Document Type

Thesis

#### First Advisor

Anna Mummert

#### Second Advisor

Bonita Lawrence

#### Third Advisor

Scott Sarra

#### Abstract

The rate at which susceptible individuals become infected is called the transmission rate. It is important to know this rate in order to study the spread and the effect of an infectious disease in a population. This study aims at providing an understanding of estimating the transmission rate from mathematical models representing the population dynamics of an infectious diseases using two different methods. Throughout, it is assumed that the number of infected individuals is known. In the first chapter, it includes historical background for infectious diseases and epidemic models and some terminology needed to understand the problems. Specifically, the partial differential equations SIR model is presented which represents a disease assuming that it varies with respect to time and a one dimensional space. Later, in the second chapter, it presents some processes for recovering the transmission rate from some different SIR models in the ordinary differential equation case, and from the PDE-*SIR* model using some similar techniques. Later, in the third chapter, it includes some terminology needed to understand "inverse problems" and Tikhonov regularization, and the process followed to recover the transmission rate using the Tikhonov regularization in the non-linear case. And finally, in the fourth chapter, it has an introduction to an optimal control method followed to use Tikhonov regularization to recover the transmission rate.

#### Subject(s)

Epidemics -- Transmission - Mathematics.

#### Recommended Citation

Elkadry, Alaa, "Transmission Rate in Partial Differential Equation in Epidemic Models" (2013). *Theses, Dissertations and Capstones*. 542.

http://mds.marshall.edu/etd/542