#### Document Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Mathematics

#### Faculty/School

Faculty of Science

#### First Advisor

Connell McCluskey

#### Advisor Role

Supervisor

#### Abstract

In this thesis we focus first on studying the susceptible, exposed, and infected ($SEI$) disease model without immigration. We determine the basic reproduction number $\mathcal{R}_0$, which can be interpreted as the expected number of new cases that can be produced by a single infection in a completely susceptible population. Further, by using the Jacobian matrix, we determine the local stability of the disease model. Then we have the result that when $\mathcal{R}_0<1$ the DFE point is locally asymptotically stable(L.A.S). In contrast, when $\mathcal{R}_0>1$ we find that the endemic equilibrium is L.A.S. After that, we analyze the $SEI$ model with immigration of infected individuals.

Furthermore, we investigate the direction that the disease-free equilibrium moves, as a function of $\mathcal{R}_0 $, when this immigration rate increases from zero. %Since immigration of infected individuals results in a unique endemic equilibrium for all values of $\mathcal{R}_0$,

There are implications for what must happen to the disease-free equilibrium as the immigration rate increases away from zero:

\begin{itemize}

\item If $\mathcal{R}_0<1$, then the disease-free equilibrium moves to the interior of $\mathbb{R}_{\geq 0}^n$

\item If $\mathcal{R}_0>1$, then the disease-free equilibrium moves away from $\mathbb{R}_{\geq 0}^n$.

\end{itemize}

This is an interesting phenomenon.

In fact, we also study the susceptible, infectious, vaccination, and recovered (SIYR) disease model with immigration of infection individuals, with the same mathematical procedure as for the $SEI$ model. Our study shows that the phenomenon is continuing. Then, we will consider the phenomenon for a general model, using matrix theory.

#### Recommended Citation

Almarashi, Reem, "Disease Models with Immigration" (2017). *Theses and Dissertations (Comprehensive)*. 1930.

https://scholars.wlu.ca/etd/1930

#### Convocation Year

2017

#### Convocation Season

Fall