Discrete-time staged progression epidemic models

Abstract

In the Staged Progression (SP) epidemic models, infected individuals are classified into a suitable number of states. The goal of these models is to describe as closely as possible the effect of differences in infectiousness exhibited by individuals going through the different stages. The main objective of this work is to study, from the methodological point of view, the behavior of solutions of the discrete time SP models without reinfection and with a general incidence function. Besides calculating $\mathcal{R}_{0}$, we find bounds for the epidemic final size, characterize the asymptotic behavior of the infected classes, give results about the final monotonicity of the infected classes, and obtain results regarding the initial dynamics of the prevalence of the disease. Moreover, we incorporate into the model the probability distribution of the number of contacts in order to make the model amenable to study its effect in the dynamics of the disease.

Figures (3)

• Figure 1: Disease Flowchart of the staged progression epidemic model (\ref{['modn1']}-\ref{['modn4']}).
• Figure 2: Two solutions of model \ref{['modn']} for $n=3$ and the incidence function (\ref{['finc1']}) in which $\mathcal{R}_{0}<1$ and however $Z(t)$ is not monotonically decreasing. The parameter values are: Left. $N=1,\ \beta_{1}=\beta_{2}=0.2,\ \beta_{3}=0.1,\ \gamma_{1}=0.6,\ \gamma_{2}=0.7,\ \gamma_{3}=0.3,\ \mathbf{I}(0)=(0.01,0,0),\ R(0)=0$. Right. $N=1,\ \beta_{1}=0.4,\ \beta_{2}=0.2,\ \beta_{3}=0.1,\ \gamma_{1}=\gamma_{3}=0.95,\ \gamma_{2}=0.9,\ \mathbf{I}(0)=(0,0,0.01),\ R(0)=0$.
• Figure 3: Three solutions of model \ref{['modn']} for $n=3$ and the incidence function (\ref{['finc1']}) in which $\mathcal{R}_{0}>1$ and however $Z(t)$ is not initially increasing and then decreasing. The parameter values are: Top Left. $N=1,\ \beta_{1}=0.8,\ \beta_{2}=\beta_{3}=0.1,\ \gamma_{1}=0.6,\ \gamma_{2}=\gamma_{2}=0.9,\ \mathbf{I}(0)=(0,0,0.01),\ R(0)=0$. Top Right. $N=1,\ \beta_{1}=0.8,\ \beta_{2}=\beta_{3}=0.1,\ \gamma_{1}=0.6,\ \gamma _{2}=\gamma_{2}=0.9,\ \mathbf{I}(0)=(0.01,0,0),\ R(0)=0$. Bottom. $N=1,\ \beta_{1}=0.4,\ \beta_{2}=0.01,\ \beta_{3}=0.5,\ \gamma_{1}=\gamma_{2}=\gamma_{3}=0.9,\ \mathbf{I}(0)=(0,0,0.01),\ R(0)=0$.

Theorems & Definitions (13)

• Proposition 2.1
• Proposition 2.2
• Proposition 3.1
• Theorem 3.2
• Proposition 3.3
• Proposition 3.4
• Lemma 3.5
• Proposition 3.6
• Proposition 4.1
• Proposition 4.2