Predictive Power of Stochastic Modelling on COVID-19

Athokpam Langlen Chanu *

I am sure most of us have heard the frequent words "mathematical modelling" and "SIR model" since December 2019 when the dreadful SARS-CoV-2 virus slowly set out from Hubei province in Central China to conquer the globe.

According to WHO, there have been around 481 million confirmed Covid-19 cases with over 6 million deaths worldwide as of 30 March 2022. The situation is still worrying with the virus continually evolving to new variants.

Scientists and Government policy makers have been closely monitoring "mathematical models" to understand the time evolution or dynamics of the Covid-19 pandemic to make possible future predictions about its transmission.

When there is a new outbreak of an epidemic and no treatment or cure is immediately available, we take the help of mathematical models that incorporate important control parameters to study the dynamics and suggest to the policy makers essential intervention strategies to mitigate the disease transmission.

Most of these mathematical models of Covid-19 are studied with deterministic equations, which give graphs/trajectories that are smooth and continuous. No matter how many times we solve the deterministic equations, we will get the same trajectories, given some initial conditions/situations.

But are these results given by deterministic equations realistic ? Imagine visiting a usually crowded vegetable market in Imphal in the morning, and it starts raining heavily.

Few people visit the market, so the population density in the market is less. After some time, the rain suddenly stops, and many buyers and sellers immediately come to the market, including a Covid-19 infected person. Now the Covid-19 transmission in the market has suddenly increased.

No one knows precisely when the rainfall will happen or stop and which person will contact whom, implying a non-deterministic or stochastic sense. Thus, rainfall is an environmental fluctuation that influences the Covid-19 dynamics, while random contact of people provides local intrinsic fluctuation in that demographic region. While deterministic models fail to incorporate this important aspect of fluctuations, stochastic models can capture it.

Even if the same initial conditions are given, their trajectories can differ each time we solve the equations. Stochastic methods have gained huge popularity nowadays due to the increasing computing performance of modern-day computers and their results being quite close to real situations.

Stochasticity becomes significant at the beginning or end of a disease outbreak when fewer infective individuals are present. Will the outbreak die out because there are fewer infective individuals ? Or will these few infections initiate a new outbreak again ?

Using publicly available data in the early phase of the pandemic, we have investigated the dynamics of the Covid-19 pandemic in India and five of its densely populated states, namely Delhi, West Bengal, Kerala, Maharashtra and Uttar Pradesh.

Using a set of ordinary differential equations and following the Next Generation Matrix method due to Diekmann et al. (1990), the values of basic reproduction number Ro are calculated for these States and India, which are found to be less than unity.

Since Ro<1 suggests a far-from-a-grim-situation, should we be less concerned ? A close observation of the data gives less than a hundred infected populations in these five Indian states and less than three hundred for the country. In such a situation, stochasticity is non-negligible and plays a crucial role.

During this early pandemic phase, most Indian States imposed home/institutional quarantine policies of the exposed population and social distancing measures. So we have extended the classic SIR model with two more compartments/sub-populations, namely Exposed (E) and Quarantine (Q), to predict Covid-19 dynamics in the country.

The SEQIR is then modelled with a partial differential equation whose solution gives the time evolution of the pandemic. With the help of computers, this equation is numerically solved with the Stochastic Simulation Algorithm by D.T. Gillespie (1976).

For a fixed total population (N) of a geographical area, the population density (D) decreases if the volume (V) increases as per relation D=N/V. Stochastic simulation results show that increasing the volume V (or decreasing the population density D) reduces the infected population I during the early stage of the pandemic. Strategies to decrease D can include measures such as social distancing and closing crowded places like offices, schools, colleges, tourist spots, markets, temples, etc.

Another critical parameter is the quarantine rate b 2 . No quarantine implies b 2 =0, while b 2 = 0.5 means quarantine of the exposed individuals in less than two days. Simulation results show that as b 2 increases, the peak values of I decreases, implying the significance of quarantine measures.

They also suggest that imposing social distancing among the populations above quarantining the exposed individuals is essential to control infections effectively in a densely populated state like Delhi. Similar results are observed for the other four states as well.

So, what is the importance of this study? Although deterministic analysis predicts a Ro<1 in the early phase of the Covid-19 pandemic in India, stochastic modelling predicts that the pandemic has disastrous consequences in the country and its states unless strict measures like social distancing and quarantine are put into effect as early as possible.

Another important consequence of this study is the timely prediction from data available during the early phase of the pandemic that only social distancing and quarantine are not sufficient enough, but other stringent policies such as mass testing of susceptible populations, lockdown and vaccination are necessary to bring an end to the pandemic in India and its states.

Indeed, we can now clearly see the positive impacts of mass Covid-19 testing, lockdown and vaccination in controlling the pandemic.

Stochastic modelling is not just limited to Covid-19, but it is also applicable to other infectious diseases caused by viruses like HIV, H1N1, Ebola, etc. In mathematical modelling of diseases, stochastic modelling is a more realistic and powerful approach than deterministic modelling to make predictions and recommend necessary control strategies to the policymakers.

The policymakers can then effectively impose these strategies which will keep the disease outbreak in control and prevent many consequences including the overburden of health care sectors and economic crises.