## Date of Award

8-1-2024

## Degree Name

Doctor of Philosophy

## Department

Mathematics

## First Advisor

Schurz, Henri

## Abstract

The stochastic SEIR(S) model with random total population is given by the system of stochastic differential equations:dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4+σ_5 S(K-N)dW_5\\ dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2+σ_5 E(K-N)dW_5 \\ dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3+σ_5 I(K-N)dW_5 \\ dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4+σ_5 R(K-N)dW_5, where σ_i>0 and constants α, β, η, γ, ζ, μ≥0. K represents the maximum carrying capacity for the total population and W_k=(W_k (t))_(t≥0) are independent, standard Wiener processes on a complete probability space (Ω,F,(F_t )_(t≥0),P). The SDE for the total population N=S+E+I+R has the form dN(t)=μ(K-N)dt+σ_5 N(K-N)dW_5 on D_0=(0,K). The goal of our study is to prove the existence of unique, Markovian, continuous time solutions on the 4D prism D={(S,E,I,R)∈R_+^4:0≤S, E,I,R≤K, S+E+I+R≤K}. Then using the method of Lyapunov functions we prove the asymptotic stochastic and moment stability of disease-free and endemic equilibria. Finally, we use numerical simulations to illustrate our results.

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