Quantifying heterogeneity in SARS-CoV-2 transmission during the lockdown in India

Abstract

The novel SARS-CoV-2 virus shows marked heterogeneity in its transmission. Here, we used data collected from contact tracing during the lockdown in Punjab, a major state in India, to quantify this heterogeneity, and to examine implications for transmission dynamics. We found evidence of heterogeneity acting at multiple levels: in the number of potentially infectious contacts per index case, and in the per-contact risk of infection. Incorporating these findings in simple mathematical models of disease transmission reveals that these heterogeneities act in combination to strongly influence transmission dynamics. Standard approaches, such as representing heterogeneity through secondary case distributions, could be biased by neglecting these underlying interactions between heterogeneities. We discuss implications for policy, and for more efficient contact tracing in resource-constrained settings such as India. Our results highlight how contact tracing, an important public health measure, can also provide important insights into epidemic spread and control.

Figure 1:

The data from Punjab. (A) Timeseries of reported cases in Punjab during the period of lockdown in the state (red bars) and those due to the Nanded event (black bars), and total cases from early March to the middle of June. (B) Visualisation of case clusters in the dataset, and their linkages from self-reported contacts. This network-type graph requires assumptions (see Materials and Methods). Most individuals infected only few others, while a few infected many: overall, 10% of cases accounted for 80% of infection events.

Figure 2:

Heterogeneity of the data in secondary cases, and in numbers of contacts. (A) The distribution of secondary cases amongst ‘seeds’ (i.e. first cases in each cluster shown in Fig. 1B). Also shown, for comparison, are the best-fitting Poisson distribution (with λ = 1.4)), and the best-fitting negative binomial distribution (with distribution parameters r = 0.067, k = 0.1). The difference between the latter two curves illustrates the strong extra-Poisson variation in the secondary case distribution. (B) Scatter plot of secondary cases vs degree, at the individual level. The secondary case and degree distributions are shown at the logarithmic scale, and adjusted by 1 to account for zeros, to address skewness of the distributions. Although both secondary case and degree distributions show a strong right-skew (panel A), this figure illustrates that the latter does not explain the former: despite a positive relationship between the two distributions, a substantial number of individuals with low degree generate some infections, while many with high degree generate zero onward infections. (C) Estimated marginal density of per-contact-infectiousness (PCI) that, alongside degree, is needed to explain the heterogeneity in secondary cases. Shaded intervals show 95% Bayesian credibility intervals. (D) Estimated PCI vs degree. The figure displays relationship between the logarithm of the odds (logit) of PCI and the logarithm of the degree. These transformations allow us to plausibly model the joint distribution of PCI and degree as a multivariate normal in section 4 (see Materials and Methods and Supporting Information). There is a discernible lower band due to a large number of cases with zero onward infections, which have very low estimated PCI. Among those with onward infections, there is a discernible negative association.

Figure 3:

Results of simple transmission models incorporating heterogeneity. Top panels show the average behaviour of an index case in a fully susceptible population of 3,000: (A) The proportion of individuals that cause no further infections. (B) Distributions for the mean number of secondary cases caused by an index case, when averaged over the whole population. In each panel, blue points show outcomes when simulating only secondary case distributions, while red points show outcomes when simulating from the joint degree/PCI distribution described in the main text. Model numbers are as listed in Table 1. Of all models, only the negative binomial secondary case distribution, and the joint degree/PCI models capture the high proportion of index cases who do not cause secondary cases (panel A). However, even amongst these models, there can be substantial variation in R₀ (panel B), owing to the role of correlation between degree and PCI. Middle panels (C,D) show epidemic outcomes over 500 time periods, assuming a 1% probability per time period, of exogenous introduction of an infectious case (here, an ‘epidemic’ is denoted as any simulation having a cumulative incidence > 500 cases (see Materials and Methods for rationale)) . Uncertainty intervals arise from repeating simulations 250 times, and reflect 95% simulation intervals. (E) Modelled timecourse of incidence, when aggregated over 250 simulations (with each simulation being interpreted here as an independent location). A Poisson secondary case distribution (in yellow) gives rise to a large surge in aggregate infection because of epidemics in multiple locations occurring in a synchronised way.

Figure 4:

An approach to efficient contact tracing. Figure shows simulated outcomes of a strategy to test all contacts of an index case, only if there is at least one positive individual in an initial ‘pilot’ sample of s contacts. (A) The proportion of infections found as a function of s (B) Overall contact tracing effort, as measured by the proportion of contacts that would be traced, again as a function of s. Owing to the right-skew of the PCI, the left-hand panel illustrates diminishing returns with increasing s, suggesting, for example, that it would be possible to identify 80% of the cases in this dataset, with <40% of the contact tracing effort.