Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data

1.   Introduction

Many mathematical models of the COVID-19 coronavirus epidemic in China have been developed, and some of these are listed in our references ^{[1,2,3,4,5,6,7,8,9]}. We develop here a model describing this epidemic, focused on the effects of the Chinese government imposed public policies designed to contain this epidemic, and the number of reported and unreported cases that have occurred. Our model here is based on our model of this epidemic in ^[10], which was focused on the early phase of this epidemic (January 20 through January 29) in the city of Wuhan, the epicenter of the early outbreak. During this early phase, the cumulative number of daily reported cases grew exponentially. In ^[10], we identified a constant transmission rate corresponding to this exponential growth rate of the cumulative reported cases, during this early phase in Wuhan.

On January 23, 2020, the Chinese government imposed major public restrictions on the population of Wuhan. Soon after, the epidemic in Wuhan passed beyond the early exponential growth phase, to a phase with slowing growth. In this work, we assume that these major government measures caused the transmission rate to change from a constant rate to a time dependent exponentially decreasing rate. We identify this exponentially decreasing transmission rate based on reported case data after January 29. We then extend our model of the epidemic to the central region of China, where most cases occurred. Within just a few days after January 29, our model can be used to project the time-line of the model forward in time, with increasing accuracy, to a final size.

2.   Model

We consider the following system of ordinary differential equations to describe the propagation of the disease

The initial conditions of system (2.1) are the following

The onset of the epidemic is $t_0$ in days. The population is compartmentalized at each time $t\geq t_0$ into susceptible $S(t)$, asymptomatic infectious $I(t)$, symptomatic infectious who are unreported $U(t)$, and finally symptomatic infectious individuals reported by the public health service $R(t)$. We assume that reported people will non longer participate into the infections because they are isolated. Therefore only infectious individuals belonging to $I(t)$ or $U(t)$ spread the disease. The average time spent in the asymptomatic infectious class is $1/\nu$ days. We assume that once an individual becomes symptomatic (reported or unreported), he will be infectious during an average period of $1/\eta$ days.

Remark 2.1. The transmission rate of asymptomatic infectious individuals is not known ^[11]. It has been confirmed that asymptomatic transmission occurs ^[12]. We assume that the asymptomatic transmission rate is the same as the rate for symptomatic individuals for simplicity. For the model simulations it does not change the outcomes very much, as long as the two rates are not significantly different. In future work we will address this issue.

The parameters of model (2.1) are listed in Table 1 and Figure 1 is a flow diagram describing the model.

3.   Data

We use cumulative reported data from the National Health Commission of the People's Republic of China and the Chinese CDC for mainland China. Before February 11, the data was based on confirmed testing. From February 11 to February 15, the data included cases that were not tested for the virus, but were clinically diagnosed based on medical imaging showing signs of pneumonia. There were 17, 409 such cases from February 10 to February 15. The data from February 10 to February 15 specified both types of reported cases. From February 16, the data did not separate the two types of reporting, but reported the sum of both types. We subtracted 17, 409 cases from the cumulative reported cases after February 15 to obtain the cumulative reported cases based only on confirmed testing after February 15. The data is given in Table 2 with this adjustment.

We plot the data for daily reported cases and the cumulative reported cases in Figure 2.

4.   Model parameters

In the following we assume that $f = 0.8$. This means that $80\%$ of symptomatic infectious cases are reported while $20 \%$ are unreported and are still involved in the spread of the disease. We assume that the incubation period is an average $7$ days that is $\nu = 1/7$. We also assume that symptomatic individuals (reported or unreported) will be infectious during an average period of $7$ days that is $\eta = 1/7$. The values of these parameters can be readjusted according to the studies that will be carried out during the pandemic.

As in our previous work ^[10], we start from the assumption that the cumulative number of cases has an exponential growth in the early phase of the epidemic and satisfy

with values $\chi_1 = 0.15$, $\chi_2 = 0.38$, $\chi_3 = 1.0$. These values of $\chi_1$, $\chi_2$, and $\chi_3$ are fitted to reported case data from January 19 to January 28. We assumed the initial value $S_0 = 11, 000, 000$, the population of the city Wuhan, which was the epicenter of the epidemic outbreak where almost all cases in China occurred in this time period. The other initial conditions are

The time-dependent transmission rate $\tau(t)$ is supposed to be constant during the exponential growth phase of the cumulative number of reported cases and is given by

The initial time is

The value of the basic reproductive number is

These parameter formulas were derived in ^[10].

After January 23, strong government measures in all of China, such as isolation, quarantine, and public closings, strongly impacted the transmission of new cases. The actual effects of these measures were complex, and we use an exponential decrease for the transmission rate $\tau(t)$ to incorporate these effects after the early exponentially increasing phase. The formula for $\tau(t)$ during the exponential decreasing phase was derived by a fitting procedure. The formula for $\tau(t)$ is

The date $N$ and the value of $\mu$ are chosen so that the cumulative reported cases in the numerical simulation of the epidemic aligns with the cumulative reported case data during a period of time after January 19. We choose $N = 25$ (January 25) for our simulations. We illustrate $\tau(t)$ in Figure 3, with $\mu = 0.16$. In this way we are able to project forward the time-path of the epidemic after the government imposed public restrictions, as it unfolds.

5.   Model simulation

We assume that exponentially increasing phase of the epidemic (as incorporated in $\tau_0$) is intrinsic to the population of any subregion of China, after it is has been established in the epidemic epicenter Wuhan. We also assume that the susceptible population $S(t)$ is not significantly reduced over the course of the epidemic. We set $\tau_0 = 4.51 \times 10^{-8}$, $t_0 = 5.0$, $I(t_0) = 3.3$, and $U(t_0) = 0.18$, as in Section 4. We set $S(t_0)$ in (2.2) to $1, 400, 050, 000$ (the population of mainland China). We set $\tau(t)$ in (2.1) to

where $S_0 = 11, 000, 000$ (the population of Wuhan). We thus assume that the government imposed restriction measures became effective in reducing transmission on January 25.

In Figure 4, we plot the graphs of $CR(t)$, $CU(t)$, $R(t)$, and $U(t)$ from the numerical simulation for simulations based on six time intervals for known values of the cumulative reported case data. For each of these time intervals, a value of $\mu$ is chosen so that the simulation for that time interval aligns with the cumulative reported case data in that interval. In this way, we are able to predict the future values of the epidemic from early cumulative reported case data.

In Figure 5, we plot the graphs of the reported cases $R(t)$, the unreported cases $U(t)$, and the infectious pre-symptomatic cases $I(t)$. The blue dots are obtained from the reported cases data (Table 2) for each day beginning on January 26, by subtracting from each day, the value of the reported cases one week earlier.

Our model transmission rate $\tau(t)$ can be modified to illustrate the effects of an earlier or later implementation of the major public policy interventions that occurred in this epidemic. The implementation one week earlier (25 is replaced by 18 in (4.1)) is graphed in Figure 6A. All other parameters and the initial conditions remain the same. The total reported cases is approximately $4500$ and the total unreported cases is approximately $1100$. The implementation one week later (25 is replaced by 32 in (4.1)) is graphed in Figure 6B. The total reported cases is approximately $820, 000$ and the total unreported cases is approximately $200, 000$. The timing of the institution of major social restrictions is critically important in mitigating the epidemic.

The number of unreported cases is of major importance in understanding the evolution of an epidemic, and involves great difficulty in their estimation. The data from January 19 to February 15 for reported cases in Table 2, was only for confirmed tested cases. Between February 11 and February 15, additional clinically diagnosed case data, based on medical imaging showing signs of pneumonia, was also reported by the Chinese CDC. Since February 16, only tested case data has been reported by the Chinese CDC, because new NHC guidelines removed the clinically diagnosed category. Thus, after February 15, there is a gap in the reported case data that we used up to February 15. The uncertainty of the number of unreported cases for this epidemic includes this gap, but goes even further to include additional unreported cases.

We assumed previously that the fraction $f$ of reported cases was $f = 0.8$ and the fraction of unreported cases was $1- f = 0.2$. Our model formulation can be applied with varying values for the fraction $f$. In Figure 7, we provide illustrations with the fraction $f = 0.4$ (Figure 7A) and $f = 0.6$ (Figure 7B). For $f = 0.4$, the formula for the time dependent transmission rate $\tau(t)$ in (4.1) involves new values for $\tau_{0} = 4.08 \times 10^{-8}$ and $\mu = 0.148$. The initial conditions $I_0 = 6.65$ and $U_0 = 1.09$ also have new values. The other parameters and initial conditions remain the same. For $f = 0.4$, the total reported cases is approximately $63, 100$ and the total unreported cases is approximately $94, 700$. For $f = 0.6$, the formula for the time dependent transmission rate $\tau(t)$ in (4.1) involves new values for $\tau_{0} = 4.28 \times 10^{-8}$ and $\mu = 0.144$. The initial conditions $I_0 = 4.43$ and $U_0 = 0.485$ also have new values. The other parameters and initial conditions remain the same. For $f = 0.6$, the total reported cases is approximately $63, 300$ and the total unreported cases is approximately $42, 200$. From these simulations, we see that estimation of the number of unreported cases has major importance in understanding the severity of this epidemic.

The number of days an asymptomatic infected individual is infectious is uncertain. We simulate in Figure 8 the model with $\nu = 1/3$ (asymptomatic infected individuals are infectious on average 3 days before becoming symptomatic), and $\nu = 1/5$ (asymptomatic infected individuals are infectious on average 5 days before becoming symptomatic). For $\nu = 1/3$, $\tau_0 = 5.75 \, \times \, 10^{-8}$, $I_0 = 1.48$, $U_0 = 0.18$, ${\cal R}_0 = 2.78$, $\mu = 0.0765$, $N = 24$. For $\nu = 1/5$, $\tau_0 = 4.90 \, \times \, 10^{-8}$, $I_0 = 2.38$, $U_0 = 0.18$, ${\cal R}_0 = 3.45$, $\mu = 0.098$, $N = 24$. Because the asymptomatic infectious periods are shorter, The date $N$ of the effective reduction of the transmission rate due to restriction measures, is one day earlier.

In Figure 9, we illustrate the importance of the level of government imposed public restrictions by altering the value of $\mu$ in formula (4.1). All other parameters and initial conditions are the same as in Figure 4. In Figure 9A we set $\mu = 0.0$, corresponding to no restrictions. The final size of cumulative reported cases after 100 days is approximately 1, 080, 000, 000 cases, approximately 270, 000, 000 unreported cases, and approximately 1, 350, 000, 000 total cases. The turning point is approximately day 65 = March 6. In Figure 9B we set $\mu = 0.17$, corresponding to a higher level of restrictions than in Figure 4. The final size of cumulative reported cases after 70 days is approximately 45, 300 cases, approximately 11, 300 unreported cases, and approximately 56, 600 total cases. The turning point is approximately day 38 = February 7. The level and timing of government restrictions on social distancing is very important in controlling the epidemic.

6.   Discussion

We have developed a model of the COVID-19 epidemic in China that incorporates key features of this epidemic: (1) the importance of the timing and magnitude of the implementation of major government public restrictions designed to mitigate the severity of the epidemic; (2) the importance of both reported and unreported cases in interpreting the number of reported cases; and (3) the importance of asymptomatic infectious cases in the disease transmission. In our model formulation, we divide infectious individuals into asymptomatic and symptomatic infectious individuals. The symptomatic infectious phase is also divided into reported and unreported cases. Our model formulation is based on our work ^[10], in which we developed a method to estimate epidemic parameters at an early stage of an epidemic, when the number of cumulative cases grows exponentially. The general method in ^[10], was applied to the COVID-19 epidemic in Wuhan, China, to identify the constant transmission rate corresponding to the early exponential growth phase.

In this work, we use the constant transmission rate in the early exponential growth phase of the COVID-19 epidemic identified in ^[10]. We model the effects of the major government imposed public restrictions in China, beginning on January 23, as a time-dependent exponentially decaying transmission rate after January 24. With this time dependent exponentially decreasing transmission rate, we are able to fit with increasing accuracy, our model simulations to the Chinese CDC reported case data for all of China, forward in time from February 15, 2020.

Our model demonstrates the effects of implementing major government public policy measures. By varying the date of the implementation of these measures in our model, we show that had implementation occurred one week earlier, then a significant reduction in the total number of cases would have resulted. We show that if these measures had occurred one week later, then a significant increase in the total number of cases would have occurred. We also show that if these measures had been less restrictive on public movement, then a significant increase in the total size of the epidemic would have occurred. It is evident, that control of a COVID-19 epidemic is very dependent on an early implementation and a high level of restrictions on public functions.

We varied the fraction $1 - f$ of unreported cases involved in the transmission dynamics. We showed that if this fraction is higher, then a significant increase in the number of total cases results. If it is lower, then a significant reduction occurs. It is evident, that control of a COVID-19 epidemic is very dependent on identifying and isolating symptomatic unreported infectious cases. We also decreased the parameter $\nu$ (the reciprocal of the average period of asymptomatic infectiousness), and showed that the total number of cases in smaller. It is also possible to decrease $\eta$ (the reciprocal of the average period of unreported symptomatic infectiousness), to obtain a similar result. It is evident that understanding of these periods of infectiousness is important in understanding the total number of epidemic cases.

Our model was specified to the COVID-19 outbreak in China, but it is applicable to any outbreak location for a COVID-19 epidemic.

Acknowledgments

This research was partially supported by NSFC and CNRS (Grant Nos. 11871007 and 11811530272) and the Fundamental Research Funds for the Central Universities. Research was also partially supported by CNRS and National Natural Science Foundation of China (Grant No.11811530272).

Conflict of interest

The authors declare there is no conflicts of interest.