An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

Abstract

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.

Introduction

One of the fundamental questions of mathematical epidemiology is to find threshold conditions that determine whether an infectious disease will spread in a susceptible population when the disease is introduced into the population. The threshold conditions are characterized by the so-called reproductive number, the reproduction number, the reproductive ratio, basic reproductive value, basic reproductive rate, or contact number, commonly denoted by R₀ in mathematical epidemiology [5], [10], [17], [19], [20], [21], [23], [29], [35], [39], [43]. The concept of R₀, introduced by Ross in 1909 [39], is defined in epidemiological modeling such that if R₀<1, the modeled disease dies out, and if R₀>1, the disease spreads in the population.

There have been intensive studies in the literature to calculate R₀ for a wide class of epidemiological models of infectious diseases [6], [8], [9], [12], [17], [18], [25], [26], [28], [30], [32], [33], [34], [41]. In mathematical models, the reproductive number is determined by the spectral radius of the next-generation operator in continuous models and, in particular, is determined by the dominant eigenvalue of the Jacobian matrix at the infection-free equilibrium for models in a finite-dimensional space [8], [9], [24], [27]. It can also be obtained, in certain models, by suitable Lyapunov functions [28], [41].

The biological meaning of the reproductive number is the average number of secondary cases produced by one infected individual during the infected individual's entire infectious period when the disease is first introduced. Let r be the average number of contacts per unit of time per individual, β be the probability of transmitting the infection per contact, and τ be the mean duration of the infectious period. Then the reproductive number can be estimated by the following intuitive formula:

$$\text{R}{}_0\text{=rβτ.}$$

This formula can give insight into the transmission dynamics of infectious diseases for various relatively simple epidemiological models [3], [4], [5], [7], [40].

For simple homogeneous models, it is easy to define r, β, and τ. For example, consider a simple homogeneous AIDS model governed by the following system of ordinary differential equations:

$$\text{d}\text{S}\text{d}\text{t}\text{=}\text{μ}\text{S}{}^0\text{−S(t)}\text{−λ(t)S(t),}$$$$\text{d}\text{I}\text{d}\text{t}\text{=}\text{λ(t)S(t)−(μ+ν)I(t),}$$$$\text{d}\text{A}\text{d}\text{t}\text{=}\text{νI(t)−δA(t),}$$

where S, I, and A denote the individuals susceptible to infection, the infected individuals, and the AIDS cases, respectively; μS⁰ is the input flow into the susceptible group; μ the removal rate; ν the rate of contracting AIDS; δ the removal rate due to the death from AIDS or other reasons; and λ is the rate of infection given by

$$\text{λ(t)=βr}\text{I(t)}\text{S(t)+I(t)}\text{.}$$

Here, β is the transmission probability per contact and r is the average number of contacts per individual per unit of time. We assume here that transmission by the AIDS cases is neglected. To focus our attention on the issues we will address, we assume, for simplicity, that the mixing is proportional for this model and other models in this paper.

The system has the infection-free equilibrium (S⁰,0). The stability of (S⁰,0) determines the reproductive number as

$$\text{R}{}_0\text{=}\text{rβ}\text{μ+ν}\text{.}$$

Formula (1.1) then holds when we define the duration of infection as $\text{τ}\text{=}\text{def}\text{1/(μ+ν)}$. (We will use the symbol $\text{=}\text{def}$ to indicate that the equation is the definition of a quantity.)

However, as more heterogeneous structures or subgroups for the infected population are included in an epidemiological model, the calculation of R₀ becomes more complicated, and it is difficult to find an explicit formula for R₀. Even when an explicit formula can be obtained, it is not always clear whether it is appropriate to define a mean contact rate, a mean duration of infection, and a mean transmission probability so that the reproductive number can still be estimated by formula (1.1). Furthermore, even if it can be claimed that such an estimate is adequate, a deep understanding of the model is absolutely necessary so that those means can be well defined.

Moreover, for models of the diseases for which differentiation of the contact rates or the partner acquisition rates must be addressed, such as sexually transmitted disease (STD) models, not only the mean but also the second moment or the variance about the mean must be taken into account. Then, formula (1.1) can no longer be applied. For certain simple models, a more accurate formula for the reproductive number is

$$\text{R}{}_0\text{=}\text{r+}\text{σ}{}^2\text{r}\text{βτ,}$$

where r is the mean number of contacts per individual and σ is the variance or standard deviation of the mean number of contacts [1], [2], [3], [5], [8], [28], [36].

Formula (1.2) is an effective formulation for providing insight into the transmission dynamics of diseases. Unfortunately, as more heterogeneities are considered, it becomes impracticable to define the variance or the standard deviation, and expression (1.2) becomes inadequate.

For risk-group models, Hethcote and Yorke [23] first introduced and Jacquez et al. [28], [29] specified the idea of defining a mean reproductive number as the average number of infected individuals generated per infected individual over the duration of the infected state. They defined a reproductive number for each subgroup and then express the mean reproductive number as a weighted mean of those group reproductive numbers.

In this paper, we use the models in [26] as a basis and formulate new heterogeneous models to demonstrate how different cases can be treated so that an appropriate reproductive number can be estimated. We show that for models with no risk structure, that is, the models with a homogeneous susceptible population in the contact rates, it is still possible to define the mean quantities and to apply formula (1.1). We show, however, for susceptible populations with heterogeneous structure such as risk structure and age structure, it is more appropriate to define a reproductive number for each subgroup or each cohort and then express the reproductive number for the whole population as the weighted average of those reproductive numbers for the subgroups or cohorts.