Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China

Modeling epidemic spread under containment efforts

On a very basic level, an outbreak such as the one in Hubei is captured by SIR dynamics where the population is divided into three compartments that differentiate the state of individuals with respect to the contagion process: infected (I), susceptible (S) to infection, and removed (R) (i.e., not taking part in the transmission process) (18, 19). The corresponding variables S, I, and R quantify the respective compartment’s fraction of the total population such that S + I + R = 1. The temporal evolution of the number of cases is governed by two processes: the infection that describes the transmission from an infectious to a susceptible individual with basic reproduction number R₀ and the recovery of an infected after an infectious period of average length T_I. The basic reproduction number R₀ captures the average number of secondary infections an infected will cause before he or she recovers or is effectively removed from the population.

Initially, a small fraction of infecteds yields an exponential growth if the basic reproduction number is larger than unity. A simple reduction in the number of contacts caused by quarantine policies without additional shielding of susceptibles could be associated with a reduction in the effective reproduction number, which would, however, still yield an exponential growth in I(t) if R₀ > 1, which is inconsistent with the observed transient scaling law t^µ discussed above. To test the hypothesis that the observed growth behavior can be caused by mitigation policies that apply to both infected and susceptible individuals, we extend the SIR model by two additional mechanisms, one of which can be interpreted as a process of removing susceptibles from the transmission process: First, we assume that general public containment efforts or individual behavioral changes in response to the epidemic effectively remove individuals from the interaction dynamics or substantially reduce their participation in the transmission dynamics. We will refer to this mechanism as “containment” in the following. Second, we account for the removal of symptomatic infected individuals, which we will refer to as “quarantine” procedures. The dynamics are governed by the system of ordinary differential equations

a generalization of the standard SIR model, henceforth referred to as the SIR-X model. The rate parameters α and β quantify the transmission rate and the recovery rate of the standard SIR model, respectively. Additionally, the impact of containment efforts is captured by the terms proportional to the containment rate κ₀ that is effective in both I and S populations, because measures such as social distancing and curfews affect the whole population alike. Infected individuals are removed at rate κ corresponding to quarantine measures that only affect symptomatic infecteds. The new compartment X quantifies symptomatic, quarantined infecteds. Here we assume that the fraction X(t) is proportional to the empirically confirmed and reported cases C(t) and that the time period between sampling and test results can be neglected. The case κ₀ = 0 corresponds to a scenario in which the general population is unaffected by policies or does not commit behavioral changes in response to an epidemic. The case κ₀ = 0 corresponds to a scenario in which symptomatic infecteds are not isolated specifically.

In the basic SIR model that captures unconstrained, free spread of the disease, the basic reproduction number R₀ is related to transmission and recovery rate by $R_0\equiv R_{0,\mathrm{free}}=\alpha /\beta$ because β⁻¹ = T_I is the average time an infected individual remains infectious before recovery or removal. Here, the time period that an infected individual remains infectious is $T_{I,\mathrm{eff}}={(\beta +{\kappa }_0+\kappa )}^{-1}$ such that the effective, or “observed,” reproduction number $R_{0,\mathrm{eff}}=\alpha T_{I,\mathrm{eff}}$ is smaller than $R_{0,\mathrm{free}}$ because both κ₀ > 0 and κ > 0.

The key mechanism at work in the model defined by Eqs. 1 to 4 is the exponentially fast depletion of susceptibles in addition to isolation of infecteds. This effect is sufficient to account for the observed scaling law in the number of confirmed cases for a plausible range of model parameters as discussed below.

Effective protection of susceptibles leads to subexponential growth

We assume that a small number of infected individuals traveled from Hubei to each of the other affected provinces before traffic restrictions were effective but at a time when containment measures were just being implemented. Figure 2 illustrates the degree to which the case count for Hubei Province and the aggregated case count for all other provinces is captured by the SIR-X model as defined by Eqs. 1 to 4.

For a wide range of model parameters, the empirical case count is well reproduced, displaying the observed scaling law t^µ for a substantial period of time before saturating to a constant level. Notably, the model reproduces both growth behaviors observed in the data: It predicts the expected initial growth of case numbers in Hubei Province followed by an algebraic growth episode for ~11 days until the saturation sets in, a consequence of the decay of unidentified infected individuals after a peak time around 7 February (Fig. 2A). Furthermore, the model also captures the immediate subexponential growth observed in the remaining most affected provinces (Fig. 2, B and C). Again, saturation is induced by a decay of unidentified infecteds after peaks that occur several days before peak time in Hubei, ranging from 31 January to 4 February. For all provinces, following their respective peaks, the number of unidentified infecteds I(t) decays over a time period that is longer than the reported estimation of maximum incubation period of 14 days (4, 20). It is important to note that the numerical value of unidentified infecteds is sensitive to parameter variations—the general shape of I(t), however, is robust for a wide choice of parameters, as discussed in the materials and methods.

Parameter choices for best fits were a fixed basic reproduction number of R_0,free = 6.2 (note that this reproduction number corresponds to an unconstrained epidemic) and a fixed mean infection duration of T_I = 8 days consistent with previous reports concerning the incubation period of COVID-19 (4, 20). The remaining fit parameters are shown in table S1. For these values, the effective basic reproduction number is found to range between 1.4 ≤ R_0,eff ≤ 3.3 for the discussed provinces, consistent with estimates found in previous early assessment studies (4, 8, 21, 22). We discuss these parameters and their possible range in the materials and methods.

On 12 February, case counting procedures were altered regarding the outbreak in Hubei to include cases that were clinically diagnosed and not laboratory confirmed. Consequently, about 15,000 new cases were added on a single day, compared to about 1500 new cases on the day before. We therefore only use data predating 12 February to estimate model parameters for the most affected provinces and compare the obtained predictions with subsequent cases numbers. We find that for all provinces other than Hubei, our predictions accurately reflect the empirical observations (Fig. 2, B and C). When we omit the aforementioned discontinuity that arises in the empirical data, the saturating behavior of cases in Hubei is consistent with the prediction, as well. When the clinically diagnosed cases of 12 February are subtracted from the number of cases in Hubei, our model underestimates the final count of laboratory-confirmed cases in Hubei by 4%. In the remaining part of mainland China, we underestimate the final case count by 7% (as of 28 March).

A detailed analysis of the model parameters indicates that a wide range of values can account for similar shapes of the respective case counts (see materials and methods). Consequently, the model is structurally stable with respect to these parameters, and the numerical value is of less importance than the quality of the mechanism they control. In particular, we find that an exponential decay of available susceptibles is responsible for the observed subexponential growth behavior, i.e., a nonzero containment rate κ₀ > 0. We present model analyses for all affected Chinese provinces in the supplementary materials (compare figs. S1 and S2, and table S2). We further provide analytical arguments for the emergence of subexponential growth of confirmed cases and derive an approximate expression that relates the scaling-law exponent µ to model parameters, finding reasonable agreement with the empirical values shown in Fig. 1 (compare table S5).

Additionally, we analyzed two model variants in which (i) containment strategies affect the whole population equally (κ₀ > 0 and κ = 0) and (ii) only infecteds are quarantined (κ = 0 and κ₀ > 0). The first model captures the case number growth slightly less accurately compared to the complete model (compare figs. S3 and S4, table S3). The second model requires the assumption that shielding susceptibles from transmission was sufficiently effective that only a very small number was at risk of infection, i.e., the effective population size $N_{\mathrm{eff}}\ll N$ becomes an additional model parameter (compare figs. S5 and S6, and table S4). In this case, the immediate exponential depletion of susceptibles because of the transmission process itself causes the empirically observed growth in case numbers. Both of these limiting cases strengthen the point that the fast removal of susceptibles from the population is responsible for the observed subexponential growth.

The described saturation behavior of confirmed cases requires that eventually all susceptibles will effectively be removed from the transmission process. In reality, not every susceptible person can be shielded or shield themselves for such an extended period of time as the model suggests. One might expect instead that the number of unidentified infecteds will decay more slowly and saturate to a small, yet nonzero level, which is why we expect systematic but small underestimations regarding the final empirical case count of epidemic outbreaks.

Discussion and conclusion

In this study, we find that one of the key features of the dynamics of the COVID-19 epidemic in Hubei Province but also in all other provinces is the robust subexponential rise in the number of confirmed cases according to a scaling law t^µ during the transient episode of the epidemic before assuming saturating behavior. This general shape of growth suggests that fundamental principles are at work associated with this particular outbreak that are dominated by the interplay of the contagion process with endogenous behavioral changes in the susceptible population and external containment policies. Although the explicit shape of the growth curves discussed here can be influenced by factors such as seasonal effects, systematic delay in reporting, or heterogeneities in demographic structure and population mixing, the total case numbers eventually reached a stable value, which suggests that containment strategies that shielded the susceptible population from the transmission process were rather effective—compared to potential case numbers of an unmitigated outbreak, only a small fraction of the Chinese population that was at risk has been infected to date (29 March). Nevertheless, we cannot rule out that other factors contributed to the growth behavior displayed in the data that were collected over a short period of time during a tense situation.

The model defined by Eqs. 1 to 4 and discussed here indicates that the type of observed growth behavior can generally be expected if the supply of susceptible individuals is systematically decreased by means of implemented containment strategies or behavioral changes in response to information about the ongoing epidemic. Unlike contagion processes that develop without external interference at all or processes that merely lead to parametric changes in the dynamics, our analysis suggests that nonexponential growth is expected when the supply of susceptibles is depleted on a time scale comparable to that of the infectious period of a disease.

The model reproduces the empirical case counts in all provinces well for plausible parameter values. The quality of the reproduction of the case counts in all 29 affected provinces can be used to estimate the peak time of the number of asymptomatic or oligosymptomatic infected individuals in the population, which is the key quantity for estimating the time when an outbreak will wane. The current analysis indicates that this peak time was reached around 7 February for Hubei and within the first days of February in the remaining affected provinces.

The model further suggests that the public response to the epidemic and the containment measures put in place were effective despite the increase in confirmed cases. That this behavior was observed in all provinces also indicates that containment strategies were universally effective. On the basis of our analysis, such strategies would have to stay in effect for a longer time than the maximum incubation period after the saturation in confirmed cases sets in.

Our analysis shows that mitigation strategies that target the susceptible population and induce behavioral changes at this “end” of the transmission process can be very effective in containing an epidemic—especially in situations when asymptomatic or mildly symptomatic infectious periods are long or their duration unknown. Although standard containment strategies such as contact tracing may become infeasible during large-scale outbreaks of such diseases, the implementation of stricter measures can aid in the fast reduction of the number of new infections, thereby quickly increasing the feasibility of interventions that do not affect the general public as drastically. This may be of importance for developing containment strategies for currently developing large-scale secondary outbreaks of COVID-19 in several regions of the world or future outbreaks of other infectious diseases.

We stress that our model describes the general effects of containment mechanisms, effectively averaged over many applied strategies or individual changes of behavior. Our analysis therefore cannot identify the efficacy of specific actions. As the implementation of drastic measures such as mandatory curfews can have severe consequences for both individuals as well as a country’s society and economy, decisions about their application should not be made lightly.