One of the most distinguishing features of the coronavirus is that there are a significant number of asymptomatic cases and both pre-symptomatic and asymptomatic individuals appear to be capable of spreading a significant number of cases. The current best estimate of the percentage of asymptomatic cases is 40% although there is great uncertainty in this value. A simple model is built for a coronavirus outbreak and the role of symptomatic and asymptomatic cases is examined in different phases of the outbreak. A surprising result is found about the role of asymptomatic cases and controlling an outbreak of coronavirus.
A Python Mesa model is constructed for a coronavirus outbreak. Python Mesa is an agent-based modeling framework. The model can be run in diffusion mode and in Imperial College mode where home, workplace and community spaces are considered. Agents that become infected are randomly assigned as symptomatic or asymptomatic so that 40% of the new cases are expected to be asymptomatic. Symptomatic agents are assumed to be infectious for five days before they self-isolate or enter the hospital. Even those that do not will be forced to cut back their activities which will reduce the possibility of transmission. After five days they appear as recovered agents because they are immune and, since they are isolated, no longer much of a threat to transmit the disease. Since asymptomatic agents continue to circulate and they are difficult to discover they are assumed to be infectious for 14 days. This model has no efforts to mitigate the spread.
The model starts with one asymptomatic infected agent. Agents cycle through home, work and community. Susceptible agents (black) that come near an infected agent (red) are infected.
The output from one run is shown. Although the runs vary they all have similar characteristics. During the phase where the outbreak is rapidly spreading the symptomatic cases comprise the majority of active cases where active cases are agents that can transmit the disease. Although the number of asymptomatic cases is significant and can vary from run to run the outbreak is mostly driven by the symptomatic cases. As the number of new cases drop during the latter part of the outbreak the asymptomatic cases dominate and overwhelmingly drive the outbreak. Symptomatic cases often are no longer significant in driving the outbreak. Note that if only the symptomatic cases are visible then there is going to be a tendency to assume the outbreak is improving much faster than is the actual case.
The model can vary in each run but the theoretical values can be computed. This is dependent on the assumptions but even if the numbers are off the overall characteristics still hold. The active asymptomatic cases are the result of infections over 14 days while the symptomatic cases are only from infections over the last five days since once symptoms appear the agent is isolated. When new cases are rapidly increasing then infections from more than five days before are small compared to more recent infections. In this phase active asymptomatic cases will tend toward the overall percentage of asymptomatic cases (40%). When new cases are decreasing the new cases from the last five days are small relative to the nine preceding days. So the active cases will overwhelmingly be asymptomatic. It is difficult for the number of new cases to be steady but in such an event the active asymptomatic cases are expected to be two thirds of all active cases.
The current best estimate of asymptomatic cases is 40%. Naively, it may be thought that sampling all of the individuals are capable of transmitting the disease (and not isolated) will result in 40% of the active cases being asymptomatic. But the model of a coronavirus outbreak demonstrates that the percentage of asymptomatic cases amongst active cases may be very different and follows a consistent pattern. In the real world there may be multiple clusters of coronavirus outbreaks and the pattern may not be as clear but it is expected that it still is there.
These observations have implications for managing a coronavirus outbreak. In the early days of an outbreak identifying symptomatic cases should be the focus. Symptomatic cases are far easier to identify than asymptomatic cases and this approach seems to work although often with difficulty. But as the peak is reached a focus on symptomatic cases will no longer work since asymptomatic cases are driving the outbreak. It appears a plateau can occur where controlling symptomatic cases can keep new cases from increasing but asymptomatic cases keep new cases from decreasing. Often at this point testing resources are stretched and so testing is limited to symptomatic cases. Switching to a focus on asymptomatic cases is difficult. Random testing is unlikely to be efficient. Testing and tracing is more effective but probably is not sufficient because hidden clusters may quickly expand to large numbers. So both testing and tracing and randomly testing are probably going to be necessary. Social distancing and wearing masks is another approach but since the extent of the outbreak may be less visible and appear better than it is the public may not be willing to cooperate.