Inferring R0 in emerging epidemics-the effect of common population structure is small

Abstract

When controlling an emerging outbreak of an infectious disease, it is essential to know the key epidemiological parameters, such as the basic reproduction number R0 and the control effort required to prevent a large outbreak. These parameters are estimated from the observed incidence of new cases and information about the infectious contact structures of the population in which the disease spreads. However, the relevant infectious contact structures for new, emerging infections are often unknown or hard to obtain. Here, we show that, for many common true underlying heterogeneous contact structures, the simplification to neglect such structures and instead assume that all contacts are made homogeneously in the whole population results in conservative estimates for R0 and the required control effort. This means that robust control policies can be planned during the early stages of an outbreak, using such conservative estimates of the required control effort.

Keywords: R0; emerging epidemics; infectious disease modelling; population structure; real-time spread.

Figure 1.

The four contact structures considered: individuals are represented by circles and possible contacts are denoted by lines between them. (a) A homogeneously mixing population, in which all individuals have the same frequency of contacting each other. (b) A network-structured population, in which, if contact between two individuals is possible, the contacts occur at the same frequency. (c) A multi-type structure with three types of individuals, in which individuals of the same type have the same colour and lines of different colour and width represent different contact frequencies. (d) A population partitioned into three households, in which members of the same households have the same colour and household contacts, represented by solid lines, have higher frequency than global contacts, represented by dotted lines.

Figure 2.

The factor by which estimators based on homogeneous mixing will overestimate (a) the basic reproduction number R₀ and (b) the required control effort v_c for the network case. Here, the epidemic growth rate α is measured in multiples of the mean infectious period 1/γ. The mean excess degree κ = 20. The infectious periods are assumed to follow a gamma distribution with mean 1 and standard deviation σ = 1.5, σ = 1, σ = 1/2 and σ = 0, as displayed from top to bottom. Note that the estimate of R₀ based on homogeneous mixing is 1 + α. Furthermore, note that σ = 1 corresponds to the special case of an exponentially distributed infectious period, whereas if σ = 0 the duration of the infectious period is not random. (Online version in colour.)

Figure 3.

The factor by which estimators based on homogeneous mixing overestimate key epidemiological variables in a population structured by households. The basic reproduction number R₀ for Markov SIR epidemics with expected infectious period equal to 1 (a,d), critical vaccination coverage v_c for Markov SIR epidemics (b,e) and v_c for Reed–Frost epidemics (c,f), as a function of the relative influence of within-household transmission, in a population partitioned into households. For (a–c), the household size distribution is taken from a 2003 health survey in Nigeria [29] and is given by for m_i is the fraction of the households with size i. For (d–f), the Swedish household size distribution in 2013 taken from [30] is used and is given by . The global infectivity is chosen, so that the epidemic growth rate α is kept constant while the within-household transmission varies. Homogeneous mixing corresponds to , in which case . Note that the order of the graphs is different in (b) and (e) from that in (a,c,d,f).

Figure 4.

The estimated basic reproduction number, R₀, for a Markov SEIR model in a multi-type population as described in [32], based on the real infection process (who infected whom) plotted against the computed R₀, assuming homogeneous mixing, based on the estimated epidemic growth rate, α, and given expected infectious period (5 days) and expected latent period (10 days). The infectivity is chosen at random, such that the theoretical R₀ is uniform between 1.5 and 3. The estimate of α is based on the times when individuals become infectious. In (b), a box plot of the ratios of the two R₀ estimates (the estimate based on the homogeneous mixing assumption divided by the estimate based on the real infection process for each of the 250 simulation runs) is given. (Online version in colour.)

Figure 5.

Estimates for the basic reproduction number R₀ of an SEIR epidemic on the collaboration network in condensed matter physics [33] based on 1000 simulated outbreaks. Each epidemic is started by 10 individuals chosen uniformly at random from the 23 133 individuals in the population. The infection rate is chosen such that . In (a), the black solid line provides the density of estimates based on full observation of who infected whom, the blue dashed line denotes the density of estimates based on the estimated epidemic growth rate α and the assumption that the network is a configuration model with known κ, whereas the red dotted line denotes the density of estimates based on α and the homogeneous mixing assumption. (The modes of these three densities are in increasing order.) The orange vertical line segment denotes the estimate of R₀ based only on the infection parameters and κ, assuming that the network is a configuration model (see equation (2.12) in the electronic supplementary material). We excluded the 50 simulations with highest estimated α and the 50 simulations with lowest estimated α. In (b), a box plot of the ratios of the two R₀ estimates (the estimate based on the homogeneous mixing assumption divided by the estimate based on the real infection process for each of the 250 simulation runs) is provided. (Online version in colour.)