[Originally Posted on March 25, 2020. Updated on March 26, 2020]
Considering the enormity of the threat presented by the COVID-19 virus, it is imperative that policy makers have robust guidance based on the best that can be offered by the scientific community. One complicating factor with COVID-19 is that the dynamics of contagion are complex and highly nonlinear, with tipping points that are not necessarily understood by those in positions of power. This blog post summarizes insights I have gleaned from building, calibrating, and testing a System Dynamics model of the COVID-19 virus in the state of Colorado.
Caveat: I am not an epidemiologist and do not claim to understand the gory details of virus transmission. Rather, I am a generalist with dynamic modeling skills that can help shed light on some of the more pressing questions being asked about how to deal with this virus. Further, the model developed for this analysis is limited in scope due to the time and resources available. It should continue to be refined and re-calibrated as more infections are confirmed to enhance the robustness of these recommendations and to offer more quantifiable recommendations. I would welcome the opportunity to collaborate with policy makers and staff to refine and enhance this model and associated analysis, either in Colorado or in other states or cities.
For those who wish to cut to the chase, I will start with an executive summary of key insights from the modeling effort. I will then move onto the methodology in addition to providing graphical output supporting the key conclusions.
I created a classic SEIR(D) chain model to simulate the dynamics of COVID-19 infections. SEIR(D) stands for "Susceptible, Exposed, Infected, Recovering, and (Deaths)." The model is implemented in the Analytica software platform due to its ease of running sensitivity analysis and facility in creating dynamic models. The approach employs the System Dynamics methodology , which accounts for key stocks of people and the flow rates from one stock to another (e.g., from Susceptible to Exposed). Additionally, it allows capturing key feedback (e.g., the vicious cycle of infected people infecting more people, resulting in exponential growth of the infected population). A graphic of the model's stock and flow diagram is provided below.
In models such as these, it is important to conduct many sensitivity analyses and other tests to understand which factors most strongly influence the overall model behavior. To this end, I created a graphical user interface that can be used to test various policies (e.g., timing and duration of a potential "lock down," magnitude of transmission risk reduction due to behavioral measures). This user interface is provided below for illustration.
After developing the model and assigning plausible values for all key parameters based on the best information available at the time, I calibrated the model to historical confirmed infections by adjusting the assumed R0 values for infected people who are asymptomatic and symptomatic. This calibration process suggested the R0 value (or the reproduction rate of the virus at time zero) is 4.04 in Colorado. All other model parameters were left at their default values. The result of this calibration is shown below, which compares simulated infections with actual confirmed cases through March 24, 2020 (sourced from https://www.bing.com/covid).
As should be apparent, the model is able to replicate historic behavior quite well. Of course, forecasts of new infections are still highly uncertain and depend on many parameters whose values are still being researched as the virus progresses. However, the conclusions described above are likely to hold even considering the high degree of uncertainty of model parameters.
The next graphic is intended to illustrate the remarkable sensitivity to the assumed value of what is referred to in this model as the "Sustained Contact Risk Reduction" (i.e., the value of risk reduction due to social distancing and other measures assumed to be present over the long term, until a vaccine becomes available or an anti-viral is found that is effective in combating the virus). This graphic illustrates the simulated new confirmed cases per day under various assumptions regarding the contact risk reduction. What should be apparent is the large difference between different values of this parameter. A value of 80%, for instance (representing a 80% reduction in contacts that could result in a virus transmission) is barely visible on the chart, whereas a value of 60% results in a peak new confirmed cases per day of nearly 60,000, an astronomical number. This graphic assumes no additional policy interventions take place, an assumption that will be relaxed in other simulation scenarios provided below.
Zooming in on the above chart, as shown below, permits us to see more clearly the effect of higher risk reduction values. In the chart below, for instance, we see that a Sustained Contact Risk Reduction value of 80% results in being below the tipping point for an epidemic, with a maximum new confirmed cases per day of roughly 300. However, a value of 70% results in being over the tipping point for an epidemic and results in an estimated 4,500 new confirmed cases per by July, rising to over 10,000 per day by October, as illustrated in the graphic above. These two graphics assume that the current efforts to contain the virus to date are continued, without any further intervention or additional lock-downs. As will be discussed below with supporting graphics, it appears we are currently on a trajectory closer to the 70% value, which would suggest that we have not yet taken actions sufficient to turn the corner on the exponential growth of the virus. Time will tell, however, whether this estimation is correct.
Looking now at the estimated deaths from COVID-19 in Colorado for different values for the assumed Sustained Contact Risk Reduction, we see that this value is also of course quite sensitive to this assumption. A relatively small value of only 50% for the Sustained Contact Risk Reduction would likely result in roughly 180,000 deaths in the State of Colorado by August, 2020. Even with a 70% reduction in contact risk, we could still expect roughly 25,000 deaths by October, 2020. However, if we can reduce our contact risk by 80%, a line barely visible on this scale, we would expect less than 200 total deaths in the state.
With the above graphics in mind, an obvious question is on which trajectory does Colorado appear to be? That is a question with an uncertain answer, but initial model calibration suggests that we are closer to the 70% trajectory than the 80% trajectory, which means we are likely beyond the tipping point for a sustained epidemic that infects hundreds of thousands of people and kills tens of thousands. The graphic below, for instance, shows the actual confirmed cases with simulated confirmed cases for various assumptions regarding the sustained contact rate risk reduction, which is assumed to have started on March 15, 2020, since that is when businesses began to close as a result of guidelines from the Centers for Disease Control. As should be apparent, the data appear to better fit the 70% assumption than the 80% assumption, which would indicate that more drastic measures are needed in Colorado if we are to avoid continued exponential growth of COVID-19. Continued daily monitoring of new confirmed cases will help us to understand the trajectory more accurately.
The above graphics all assume that current social distancing measures are undertaken for the foreseeable future. However, many regions are taking this a step further in an attempt to turn the corner on new infections and to buy time. The graphic below shows the simulated effect of a hypothetical 3-week shelter-in-place lock-down, starting on April 1, 2020, that is assumed to temporarily increase the contact risk reduction to 95% during that 3-week period. Outside of that period, we still assume a 70% value for the contact risk reduction. As we can see below, the 3-week lock-down dramatically reduces the new confirmed cases per day. However, since the estimated contact risk reduction after the lock-down is relaxed is 70%, which is beyond the tipping point, we do see a resurgence of the virus over the next several months. However, the lock-down would buy quite a bit of time to implement other measures such as aggressive testing and isolation of both asymptomatic and symptomatic populations.
The graphic below further assumes that we not only implement a temporary 3-week lock-down, but that we also take measures that would sustain the contact risk reduction to less than 80%. In this scenario, we see that the virus is ultimately contained, with new confirmed cases dropping to less than 25 per day by June, 2020.
Another key item being discussed is the availability of mechanical ventilators, which are often required for the most serious cases. The chart below further illustrates that over-relaxation of behavioral risk reduction after a hypothesized 21-day lock down will quickly result in surpassing the estimated ventilator capacity of the state (assuming the median value of 19.7 ventilators per 100,000 people, resulting in an estimated 1,122 ventilators in the state, though this value is not actually known for Colorado at this time). See https://www.ncbi.nlm.nih.gov/pubmed/21149215.
As we know, mechanical ventilators are in short supply but are absolutely vital for many patients. Once the supply of ventilators is exhausted, we can expect to see the fatality rate rise considerably. The model assumes a "treated" fatality rate of 1% and an "untreated" fatality rate of 4% (i.e., the fatality rate if ventilator capacity is exceeded). As ventilators become exhausted, we can expect the fatality rate to be closer to the larger value than the smaller value, though both of these parameters are of course uncertain. Additionally, it is assumed that roughly 5% of cases will require mechanical ventilation.
Though considerable additional analysis can and should be performed regarding the likely outcome of our battle with the COVID-19 virus, the key conclusions presented at the beginning of this blog post are expected to hold even with additional analysis. In particular, the high sensitivity to our sustained behavior risk reduction illustrates that we ought to be erring on the conservative side of risk reduction, social distancing, etc. if we want to minimize loss of life and overwhelming our hospitals and their equipment. Likewise, we can expect considerable impact on our behavior (e.g., some degree of social distancing) over the long term if we want to prevent a resurgence of the virus after initially slowing the rate of new infections. Many other measures (e.g., increased testing, contact tracing) are also of course being considered to minimize the impact on our behavior and our economy. These measures could significantly reduce the behavioral social distancing requirements since they would identify early those who are infected and permit isolating them from the population before they could infect others.
And again, while this model is not economic in nature, one can reasonably conclude that long-term reductions in behavioral risk reduction may come at substantial impact to our economy. We will have to be very creative in re-designing our economic policies and systems to ensure that the most vulnerable among us are able to weather the storm until a vaccine or anti-viral treatment is made available to large swaths of the population.
In the interest of the greater public good, this model is provided below as open-source software that could be used or enhanced by others having a need and the ability to conduct this type of analysis.
Step 1: Download the Free 101 version of Analytica here.
Step 2: Download the "COVID-19 Colorado" model here.
I am available to conduct this type of analysis in Colorado or in other states and cities as desired. Please reach out to firstname.lastname@example.org, or call me at 720-984-5184.