The geographic speed of an epidemic can be modeled by incorporating the non-linear interaction between susceptible and infectives into one or more diffusion equations. For example, restricting to one spatial dimension, the speed of spread of rabies in a fox population has been modeled by J. Murray (Murray, J., Mathematical Biology, 2nd Ed., Springer-Verlag, Berlin, 2002).

Nobel (Noble, J.V., Geographic and temporal development of plagues. Nature, 250:726, 1974) has modeled the plague geographic spread in one dimension by including the diffusion term in the susceptible equation (the 2-nd derivative of the susceptible population density over the spatial dimension). The author assumed: population density of susceptible in Europe in 1347 was 50/mile sq, the transmission coefficient r=0.4 mile sq/year, the mortality rate 15/year, diffusion coefficient D=10^4 miles sq/year.

Mendez V., Epidemic models with an infected-infectious period. Phys Rev. E., 57:3622, 1998 has shown that introducing a reasonable delay (incubation time) for the appearance of the infectious members bring the minimum speed of propagation into the experimental range ~340 miles/year (Langer, W. L., The black death, Scientific American, 210:114, 1964).

Overall, predictions of all these and related models depend on the assumed numerical values of its parameters. In most cases these parameters are not known, or they are difficult to estimate accurately. Therefore, all these models are nice academic exercises with limited practical prediction value.

Thank you for the answer. I understood it very well. Hence, what do you suggest to improve the current spatial prediction model? Different infectious diseases have different parameters/variables, demographic and geographic backgrounds. Hence, it affects the model. How do you think the spatial prediction can be fit into real world problem?

Prediction (or better to say forecasting) the spread of the number of cases itself is of little practical value. Increasing the sophistication of the mathematical models by including more terms will inevitably require more parameters with unknown numerical values.

In order to make the models practically useful, the factors/instruments should be included in the models that can predict the effect of their leveraging by policymakers and health authorities on the spread of the disease. Factors that affect the spread can include both individual susceptibility/immunity to the infection (mostly out of control of the policy decision-makers) and some external epidemiologic measures that could be controlled and developed by the policy decision-makers.

So, the problem, as I see it, is development models that include external factors that can be leveraged to limit and finally extinguish the spread of the disease as soon as possible, such as, e.g., vaccination rate, the volume of needed vaccine doses and their distribution policy, efficiency of vaccines, required healthcare resources (number of beds, nurse staffing, equipment, financial resources, etc).

This is a challenge not yet met as it is illustrated by the current COVID-19 state of affair. Indeed, numerous papers were already published aimed mainly at predicting the growth of the COVID-19 cases based on the past cases. However, almost none of them included major uncertainties that come from multiple different information/database sources, various demographics, comorbidities that can be lung- or heart-related, diabetes and obesity, effect of the asymptomatic people, hospital resource settings/capacity, testing rate and testing reliability and types, social distancing/habits, and income levels. No developed forecasting models of COVID-19 spread of various types included the aforementioned factors. And no models have been really helpful in developing the effective policies for limiting COVID-19, except some old empirically known factors such as mask policy, social distancing, multiple vaccinations, etc. This is a challenge that should be overcome to make modeling practically useful and fit the real world rather than remain just an academic area of research.

Alexander Kolker, as you mentioned before that the real problem is the limitation of external factors, why did this happen? Is it because of the limitation of data sharing by the Health Authority Department causing difficulties to build the model?

The limitation of data sharing by the Health Authority is only one part of the problem. Indeed, despite a lot of efforts worldwide, there are significant knowledge gaps. Coming back to COVID-19 modelling, different countries and regions collect data in different ways. There’s no single standard form to fill in that can easily allow comparing cases around the world. The same inconsistencies apply to who gets tested. Some countries are giving tests to anyone who wants one. Others are not. That affects how much is known about the number of people who have actually contracted COVID-19, versus how many people have tested positive depending on the test nature.

And the COVID-19 virus itself, being an RNA type prone to random mutations, is an unpredictable contagion, hurting some groups more than others — meaning that local demographics and health care access are going to be big determinants when it comes to the virus’s impact on communities. All these factors are difficult to formalize to include in the model.

But there is another problem in unpredictability of models that was not widely addressed yet. All realistic infective disease models are non-linear ones. This means that many of these models are highly sensitive to initial conditions. On top of that, the question arises that some epidemics, such as measles and chickenpox, are chaotic. The best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics. See, e.g., Rand, D., Wilson, H., Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. 1991, Nov 22;246(1316):179-84. Proc Biol Sci., doi: 10.1098/rspb.1991.0142.

Spatial and nonlinear dynamics have only recently been brought together. This indicates that (social and/or geographical) spatial heterogeneity is needed in the models. That spatial heterogeneity can help to increase the realism of models. However, more refinements of the models (particularly in representing the impact of human demographic changes on infection dynamics) are required. It is a challenge that was not addressed yet. See, e.g., Complexity of COVID-19 Dynamics by Sivakumar, B., Deepthi, B., 2021, Dec 27;24(1):50. Entropy (Basel), doi: 10.3390/e24010050