Disclaimer : This assessment is non-prescriptive and not validated by healthcare experts or epidemiologists. This is meant as an academic exercise to understand and possibly extend certain methods used in infection analysis. Please exercise caution and understand the limitations and intent of the author.

Code reference – the notebook used for this analysis is now available here.

The slope rate of change for covid-19 spread is one of the ways to visualize the exponential growth, which remains elusive to comprehend by our populace and control by the governments. If you see the below plot, fig. 1, it is clear that the rate of growth is dependent on the current number of cases, which makes the right side plot slightly optimistic.

Fig. 1 Daily Growth of epidemic across nations is similar, a drop on the left curve shows nations which have to some degree been able to control. The right hand figure conveys some insight, but depends on the factor of time which is difficult to comprehend.

The underlying point being our minds do not comprehend exponential system intuitively and that results in behavior non-conducive to control, looking at benign numbers which soon turn ugly. There are however simplified medical models which can help communicate this as well.

Compartment models

Compartment models are a simplified way to mathematically simulate the spread of infectious diseases. The assumption underlying these models is that the population within a compartment has the same characteristics. There is transport across compartments with time, which is quite analogous to a diffusion process that is then modeled using simplified ordinary differential equations.

3Blue1Brown does a good job of explaining the SIR models using simulations, much better than I would be able to in detail here.

In a quick summary, there are three dynamic variables in SIR Model, which are eponymous with its name. You assume a rate of change which describes these three populations and then model it. The simple version of SIR models don’t use vital dynamics (things like birth rate and death rate which are presumed to have longer cycle than the disease)

Fig. 2 – Wikimedia Image of SIR model

The can of SIR models

Simplified models like this can be used for hypothetical scenarios. What if questions can be answered with such scenarios. If we control the spread, but don’t spend efforts in recovery what could happen relative to if we don’t control but actively improve recovery. Some of these reflect early efforts by few countries.

Fig. 3 – Simulations run with SIR

If however we start to look deeper and fit data, some of the assumptions begin to stand-out.

The cannot (or rather can with difficulty) of SIR models

The first assumption is aforehand knowledge of the number of susceptible population. In a spread in progress situation, this is often impossible to estimate which is why the curve for China and Korea look well fit, the curves for other countries might start to show signs of numerical inconsistency. Fig. 4 shows some of these results. You can see that while spread and recovery numbers in Korea and China are sensible.

Fig. 4 – SIR model results for Korea and China (infected population optimized for parameter estimation)

However, with country that haven’t yet reached the stable infection point, the impact of total susceptible population is important. With US for example, the assumption of susceptible population plays a significant role, see fig. 5 below.

Fig. 5 – See the change in spread based on total susceptible population. Also, the recovery variable hasn’t kicked in (this may be a numerical glitch as well)

This fore-knowledge can be achieved however through additional modeling and forecasting. Using rate of growths, a pessimistic and optimistic range can be arrived at, which becomes part of our future efforts.

The second assumption is the sad fact that fatalities in this case which amount to vital dynamic variables are non-significant. This means the assumption of confirmed cases and infection rates would change. There are ways to address this using additional terms in the differential equations which define SIR models.

Update 1 [3 April 2020] – SIRD models

The second assumption above plays a key role here and we have attempted to bring that into the system of differential equations using SIRD model with no re-infection at this point since there isn’t sufficient evidence of that (thankfully). Fig. 6 below shows the schematic of SIRD models.

Fig. 6 – additional variable on ‘r’ and ”f” added to the model

With this model as well, the upper limit of fatality rate is as assumption, which is currently bound as ratio of fatalities to the total number of recovered and passed away patients. With this constraint, a convergence is achieved and results are shown in fig. 7 below.

Fig. 7 – Impact of adding fatality variable is small on spread, but considerable on recovery rate

Empirical models such as these can be used for characterizing time domain behaviors to static variables which can be used in further advanced models, some of which will be published in this space in the future updates.

Leap of faith assumptions

SIR models, even with their limitations, could provide opportunities to extend diffusion based modeling into economics. The susceptible population could be replaced with currency, which is locked down in certain sectors and still moving in others. As the lock-downs intensify or ease out in certain geographic regions, could a diffusion based model (with more rigor of course) provide guidance on economic recovery? Let us know what you think (or know already) in comments.

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