STRATEGIES ON EPIDEMIC DYNAMICS: DOES IT PAY TO QUARANTINE?

A simple stochastic example using Monte Carlo Simulation

Under the current Corona Virus world crisis, I set myself to build a very simple model on the dynamics of how epidemics get to be born, grow and decay. After reading some material on epidemic dynamics and, very importantly, claiming that I am no expert whatsoever on this field, I constructed a simple Excel model using @RISK as its Monte Carlo simulation engine in order to validate what is going in many countries; namely, the self-imposition of strict quarantine protocols to mitigate the potential health disasters that this pandemic might turn into.

As in any model, which is a simplification of reality, assumptions are almost all what matters. You assume weak assumptions on your model and evidently, your results will be questionable, frail and dubious. Evidently, making good assumptions mostly depends on good expert opinion and/or historic information. This last element is still in the making as there are many unknowns about this novel virus.

Thus, I am not pretending to simulate the precise dynamics of a global pandemic, or country epidemic, by any means. I just want to show the mathematical principles operating here and demonstrate, at the end, whether it is consistent or not to validate national strategies that impose strict quarantines. The objective of national authorities will be, at the end, to minimize the risk of human deaths on their populations.

SIMPLE DYNAMICS OF PANDEMICS

World Health Organization explains that the number of people who die from a pandemic depends upon four factors:

1.      The number of people who become infected.

2.      The severity of disease caused by the virus.

3.      The vulnerability of affected populations.

4.      The effectiveness of preventive steps.

In the very short term of a disease epidemic, national authorities do not control either #2 or #3: the severity of the disease caused by the virus is clearly out of their control; and the vulnerability of affected populations can clearly not be controlled on a scale of weeks or months during the outbreak.

The number of people who become infected is more of a social and economic variable than anything else. It depends, in turn, on the degree of connectedness of a society, its geographic mobility, the degree of interaction with other national and international communities, the physical or spatial closeness where actual human contacts take place, the pervasiveness of multitudes against individual humans working, studying, servicing each other, having leisure, and in general, interacting with each other.

Regarding the last factor of the effectiveness of preventive steps; we have all been exposed to the specific tactics we should practice such as: washing hands, avoiding crowds, correctly coughing and sneezing, etc. Educating populations of these preventive steps are in place in the current case, under the absence of the huge missing mitigating strategy: a vaccine.

Therefore, this leaves national health authorities with little control over short-term strategies to mitigate the risk of this pandemic. Therefore, the obvious call becomes limiting social contacts and through it, minimizing the number of people who become infected.

MODEL OPERATION AND ASSUMPTIONS

I considered a time period of 30 weeks (209 days). On day 1, four infected individuals coming from another community or country enter a small country with a 5 million population. They are infected, but still they do not know about it since the incubation period for the virus is of 8 days. In other words, these individuals will show no sign of infection for 8 days and will continue asymptomatic, living normal leisure days as tourists, traveling all around the country. Social gathering will make them enter in contact with other not-yet infected individuals.

In this example, we assume a minimum number of daily social contacts of 1, a most likely number of contacts of 1.7, and a maximum number of contacts of 2.3. Consider that a social contact does not necessarily imply infection. We assume here that the probability of being infected by social contact is any number between 0% and 20%. In other words, on average only one out of ten social contacts will become a transmitted infection. After incubation period, we assume infected individuals enter an illness period of 10 days where they will show the symptoms of the disease and will also limit themselves to a quarantined or restricted contact with other people. After these 10 days of symptoms, each infected patient will survive the disease or will die, given the mortality rate considered for this disease.

Notice once again, that every single day, during the period of incubation, infected individuals keep spreading the virus unknowingly. This factor, as we will see later on, is critical for the outcome of the model.

MORTALITY RATES

On March 15, 2020, the Washington Post published[1] a mortality table per country for Covid-19, that I summarize here on the following graph:

A list of 46 countries out of total of 148, report mortalities larger than 0%, ranging from 100% in Sudan (1 case, 1 death) to Austria at 0.12% (860 cases, 1 death). We admit that this is yet a very tentative number that should become better known as time passes by. Countries reporting no deaths yet account for 102.

I used @RISK to fit a curve of mortality using these 148 cases. The best distribution function according to the Akaike Information Criterion was a LogLogistic distribution that looks like this:

Historic mean mortality of the data set accounts for 6.0%. A LogLogistic mean fits to 6.68%. The mean will tend to stretch too much to the right hand extreme of the distribution given its large asymmetry. Probably better to describe this fitted distribution with a median at 2.14%. In other words, assuming that this is a distribution that cannot be discarded and that the data has already become stable, 50% of the countries will experience a mortality rate smaller than 2.14%, whereas 50% of the countries will experience a larger mortality rate than that.

OTHER ASSUMPTIONS

As we have mentioned before, the model will be run with a Monte Carlo simulation. In other words, we will not assume static values for each one of the assumptions. The model will be iterated 20,000 times under different conditions. At each one of these iterations, random samples will be drawn from the distribution functions of each one of the stochastic variables.

As we have mentioned before, daily contacts will be modeled with three parameters using a Pert distribution. A minimum number of daily contacts per infected individual at 1, a most likely value of 1.7 and a maximum value of 2.3.

We assume a probability of infection per contact, as we have stated before of any given number between 0% to 20%. Given the ignorance that we display with respect to this variable, we use a continuous Uniform distribution; usually recommended where there is not too much information to display:

For Infection Days variable we recommend a Poisson distribution with one single parameter, its mean or lambda, at 10 days:

THE QUARANTINE

Remember that the objective of imposing a light, mild or very strong quarantine by national authorities is to limit the number of social contacts and by that, limiting the spread of the disease. In turn, in this example, there are two aspects of the quarantine that may be handled by authorities: the relative strength of the quarantine, meaning how strict you impose human contacts by relatively restricting or forbidding human gatherings, and the timeliness or number of days when you impose the quarantine.

Let us consider these scenarios for the first dimension: Relative reduction of contacts by quarantine.

This means that what we consider a strong quarantine reduction on social contacts by imposing restrictions means that all parameters on the daily contacts per person are reduced by 50%.

Observe, for example in this case, that the three parameters that define a Pert distribution, the minimum, the mode or most likely and the maximum, have been reduced in half.

The other controlling factor is the delay or relative timeliness with which the imposition of quarantine is being executed. Consider the following scenarios:

For example, a late establishment of quarantine would be executed until day 60. A very late scenario would only be executed until day 90 of the simulated epidemic.

Thus, 5 analyzed scenarios of the relative reduction of contacts by quarantine exposition, combined with 5 alternative scenarios of the day at which the quarantine is established allow us to examine 25 different alternative scenarios.

This is the table detailing those 25 scenarios:

SIMULATING RESULTS

Now we proceed to simulate results. Simulation essentially means generating random numbers carefully chosen with distribution functions for each one of the assumptions of the model and doing this many times, as to create a full curved or distribution of the output that we pretend to analyze. In this case, we will run 20,000 iterations per scenario. In other words, we will sample 20,000 times each one of the 844 variables present in the model. Remember we are evaluating 209 days and for each day we generate iterations for all of the assumptions. At the end, 844 variables times 20,000 times each, times 25 alternative scenarios, we end up collecting 422 million statistical samples[2].

Every single time, we iterate each one of these alternative scenarios, we obtain different results which are summarized by @RISK, the simulation engine on top of Excel. For example, on the following screen we see one particular scenario of deaths (392 on cell F2) under conditions of “Intermediate quarantine reduction on contacts (25% reduction) and medium establishment of quarantine (on day 45)”.

The next iteration, when @RISK redraws samples for each one of the distributions used in the model, we will get the following outcome that will be collected once again for each one of the 25 scenarios:

In this scenario, a total of 232 deaths was reported.

This process will be repeated, as we stated before, 20,000 times for each one of the 25 alternative combinations of public health policy. We just show a third and final scenario with 867 deaths under stated conditions:

INTERPRETATION OF SIMULATION RESULTS

The results of policy handling are staggeringly demonstrative:

Take a look at the first row, where policy consists of no quarantine imposition or effectiveness. In other words, there is no reduction on social contacts either enforced or voluntary. Regardless of whether a no-effect quarantine will be put in place on 15, 30 45, 60 or 90 days, the effect on mortality is the same. The mortality would be disastrous with an average of slightly less than 0.8 million people. This is just the calculated mean. The whole potential curve of deaths would look as follows:

With a mean of 777 thousand deaths and a median of 178 thousand, there would be 71% probability that deaths would be less 500 thousand. There would be an 11.5% probability of deaths from 500 thousand to 1 million. There is a 17% probability that the mortality outcome would be over 1 million, one every six possible outcomes.

Evidently, these numbers are dooming and unacceptable. They represent health officials not doing anything with respect to this pandemic crisis.

Now, observe the effect that causes a reduction of 10% in the number of social contacts, provoked by the existence of a quarantine. If the quarantine were to be imposed relatively early on, on day 15, total deaths would average 35,000 people. A quarantine established after 45 days would trigger an average death toll of 48,000 people, where as a late quarantine on day 90 would account for a mean death toll of over 140,000 people.

Conditions change dramatically if social contacts were to be decreased by 25% of the original conditions. Average deaths would be kept under 500 with a very early -15 days- or early -30 days- imposition of quarantine. The average number of deaths would almost double (932) if quarantine imposition were to be delayed until day 45. Relaxing the establishment of a quarantine until day 90 would power the average numbers to become close to 12,000 deaths in total.

By the same token, analyzing strong - 50% - or very strong - 75%- reductions on social contact by imposing quarantines, would render results of less than 350 average deaths given that the imposition of restrictions are imposed before 60 days. Even after a delayed imposition on day 90, the reduction of social contact from 25% to 50% would mean a reduction on the number of average deaths from 12,000 to 2,600, 4.5 times smaller by the effect of restricting social contact.

The following table compares numbers of times each one of the first four strategies on reduction of social contacts is with respect to the tightest scenario of severely limiting social contact to a 25% of its original figure.

The two red numbers on top to the left side of the table clearly indicate that it is close to 42 thousand times worse, on average, not doing anything as opposed to imposing a very tight control over social contact.

Another way to look at the power that quarantines might have upon the dissemination of the epidemic is by viewing comparable simulated scenarios. For example, the next trend line charts show the comparative simulated bands along 30 weeks, on a logarithmic scale, for scenarios under the premise of establishing quarantine on day 45. The five charts show respectively the trend under no reduction on social contacts, and then under slight (10%), intermediate (25%), strong (50%) and very strong (75%) reductions on social contacts. It is clear the dramatic vanishing of the epidemic under conditions where social contacts are enforced, viewed on a week by week basis:

In order to limit the uncontrollable growth of the epidemic, health officers need to enforce, at least, a reduction of 25% on social contacts.

This would stabilize eventually the epidemic on around 25 to 60 deaths as a mean per week. A strategy imposing a strong quarantine reduction of 50% of social contacts would give a median death peak around week 9, eventually extinguishing the epidemic:

Any more lenient restriction on social contacts would make the epidemic uncontrollable. For example, the next chart shows what happens under just a slight quarantine that reduces social contacts on 10%.

In other words, such a slight reduction of 10% on social contacts will not be enough to control the exponential growth of the epidemic. Therefore, it seems as though, on this example, some reduction from 25% to 50% on social contacts will be required in order to extinguish the epidemic.

CONCLUSION

Extrapolating to the current Corona virus crisis, under the absence of a vaccine and very little historic and scientific knowledge yet, national health authorities are at an important decision point with very little control over mitigation tools. Essentially, what is left beyond the effectiveness of preventive health steps (public education on healthy practices and sanitization), is control of social contacts to prevent a potentially hugely destructive event. This can be implemented through national quarantines. Both, the timeliness and the effectiveness of nationally imposed quarantines are demonstrated to be key elements on controlling the sheer size of a risk event that has not fully materialized itself yet.

[1] https://www.washingtonpost.com/world/2020/01/22/mapping-spread-new-coronavirus/?arc404=true

[2] With our HP laptop possessing a multi-core i7 processor, it took around 31 minutes to simulate.