The effect of host population heterogeneity on epidemic outbreaks

TL;DR

The overall aim of the paper is to describe the state-of-the-art and to catalyse new work on epidemic outbreaks by offering a range of potential (quasi-mechanistic) submodels.

Abstract

In the first part of this paper, we review old and new results about the influence of host population heterogeneity on (various characteristics of) epidemic outbreaks. In the second part we highlight a modelling issue that so far has received little attention: how do contact patterns, and hence transmission opportunities, depend on the size and the composition of the host population? Without any claim on completeness, we offer a range of potential (quasi-mechanistic) submodels. The overall aim of the paper is to describe the state-of-the-art and to catalyse new work.

Figures (5)

• Figure 1: Incidence with heterogeneity in latent period. We model two groups of equal size and random mixing of both groups. The groups only differ in the duration of the latent period. Both are Gamma-distributed with shape-parameter 3 while the scale parameter is 1 for group 1 and 0.01 for group 2. The infectious period is exponentially distributed with mean 3 for both groups. $R_0$ equals 3. (a) Incidence of new infections. (b) Incidence of new infectious individuals. (c) I-compartment as fraction of the total population
• Figure 2: Incidence with heterogeneity due to weak coupling. We model two groups of equal size and with weak coupling between the two groups: The within group transmission parameter is 100 times higher than the between-group transmission parameter. The groups have the same parameters (latent period Gamma-distributed with shape-parameter 3 and scale parameter 1; infectious period exponentially distributed with mean 3 for both groups), $R_0$ equals 3. We introduce the disease one of the groups. (a) Incidence of new infections. (b) Incidence of new infectious individuals. (c) I-compartment as fraction of the total population
• Figure 3: HIT in case of a small 'isolated' community. We model two groups, comprising 5% and 95% of the population. 99% of contacts of individuals in the small group are with other individuals in the small group. The groups have the same parameters (the latent period is Gamma-distributed with shape-parameter 3 and scale parameter 1, the infection period is exponentially distributed with mean 3). In the initial phase, the expected number of secondary cases per primary case is 3, irrespective of the group. Initially there is no immunity. In (a), (b), and (c) a fraction 0.001 of the small population is infectious while there are initially no infectious individuals in the large population. In (d), (e), and (f) a fraction 0.001 of the large population is infectious while there are initially no infectious individuals in the small population. The shading changes at the HIT. (a) and (d): Incidence of new infections. (b) and (e): Incidence of new infectious individuals. (c) and (f): I-compartment as fraction of the total population
• Figure 4: HIT and the final size as a function of contact rate $c_2$ in a static heterogeneous setting ($\nu=0$) and in a homogeneous setting ($\nu=\infty$). $\theta=2$. $c_1$ is a function of $c_2$ such that the average contact rate equals $c=2$. The results for $\nu = 0$ follow from numerically solving equations \ref{['HIT_eq_w']} and \ref{['finalsize_eq_w']}. The results for $\nu = \infty$ follow directly from \ref{['HIT_SIR']} and from solving \ref{['finalsize_SIR']}.
• Figure 5: Final size as a function of $\nu$ in a homogeneous, a static heterogeneous and a dynamic heterogeneous setting. $\theta=2$, $c = 2$ , $c_1 = 3$, $c_2 = 1$ The results follow from numerically solving \ref{['combined_model_ode']}.

Theorems & Definitions (13)

• Lemma 5.1
• proof : Proof
• Theorem 5.2
• proof
• Lemma 5.3
• proof : Proof
• Theorem 5.4
• proof : Proof
• Theorem 5.5
• proof