The spread of the Covid-19 has become a universal challenge across the globe. As there is no proper treatment available for this virus the use of a mathematical modelling, to predict the scale of Covid-19 outspread, has become very important. Modelling and simulations have a major role in scientific research and are being widely used in every field of science and technology. This includes the prediction of epidemics. Mathematical models forecast the size of the spread of the epidemic and provide guidelines for its control.

The World Health Organisation (WHO) employs modeling and simulation in order to decide the best policies to pursue in determining timely and effective measures to be taken in mitigating effects of the virus. Modeling predictions also help understand the spread of the disease, which in turn is used to estimate the potential burden on health care systems. Modeling also aids in measuring the impact of various intervention policies on the spread of the virus. These policies refer to decisions like start-end of lockdown, restrictions on international travel, quarantine strategies, case isolation, social distance measures to be taken and the effects of wearing masks. In addition, Covid-19 modelling could also provide valuable material to decision-makers in revising the strategies to take timely precautionary measures and in setting standard operating procedures (SOPs).

Computational models are developed using well-known equations. Various epidemiological models are being used by experts to predict the size of the Covid-19 pandemic locally and across the globe. Most popular of these models are: SIR (susceptible, infected and recovered), SEIR (Susceptible–Exposed–Infectious-Recovered), modified SEIR, ANN (Artificial Neural Networks), Stochastic Transmission Model, and Agent-Based Network models. We have used two well-known models to simulate the impact of Covid-19 for the first and second wave in Pakistan. One is a modified time-dependent Susceptible- Exposed-Infected- Recovered (SEIR) meta-population transmission model, used in the Global Epidemic and Mobility Model (GLEaM) and the second is the FB-Prophet model.

The SEIR is an elementary model used for the modeling of epidemics. The SEIR model splits the whole population into four apartments: the susceptible, equal to the general global population with an assumption of no existing immunity to infection; the exposed, people who might have the virus but have not developed symptoms; the infectious, the population infected by SARS-COV-2 with symptoms; and the recovered. For predicting the Covid-19 spread in Pakistan, the SEIR model implemented in GLEAMviz is used. In this model, seven population compartments are used. GLEAMviz is a simulator that uses actual world data of population and mobility networks on the server side. It integrates this data with the model developed by a user on the client-side.

FB-Prophet is a publicly available generalised additive framework developed for the forecasting of time series data by Facebook. It is vastly used to solve data science and forecasting problems in business and stocks. However, in the current Covid-19 pandemic situation researchers are employing FB-Prophet for the forecasting of coronavirus cases globally. The main strength of Prophet lies in its capability to predict seasonality behaviours and its robustness to missing data and anomalies. We employed the Prophet framework in Python for modelling and prediction of Covid-19 cases in Pakistan. We used Prophet’s logistic growth model for Covid-19 cumulative cases and predicted the behaviour for the next 60 days. Prophet needs a pre-determined cap (maximum value) for logistic growth. This value is derived by fitting a logistic function to actual data and optimising the parameters by using SciPy methods. The death cases are then modeled in a similar way using Prophet and SciPy.

The two models are compared with the actual data, as of December 30. We evaluated the models on performance metrics. From there, we concluded that FB-Prophet appeared to provide a better model (having a low error rate). Therefore, we used FB-Prophet for modeling and prediction at the regional scale in Pakistan. We see good agreement in actual and predicted values and finally, we predicted peak dates, 95 percent decay dates and corresponding cumulative diagnosed and death cases for Pakistan.

For the first wave, the simulation using GLEAMviz assumes the index case in Wuhan, China and models the global spread of SARS-COV-2 with reasonable results for several countries within the 95 percent confidence interval. It has been observed that in the current scenario, the epidemic trend of Covid-19 spread in Pakistan has attained a first peak in the second week of June 2020 with 3,600-4,200 daily cases, 210,000–226,000 cumulative cases and 4,400-4,750 cumulative lost lives by the end of August 2020 when the epidemic is reduced by 99 percent. Our predictions were in reasonable agreement with the actual data. The disease is controllable in the likely future if inclusive and strict control measures are taken.

For the second wave, the simulation started by the end of September 2020 when the actual daily cases reported in Pakistan were around 532. The actual data for this study was taken from the official portal of the Government of Pakistan. Relevant graphs show cumulative and daily diagnosed cases in Pakistan for the selected duration. The chosen parameters R0 and for GLEAMviz that best describe the epidemic are 1.45 and 7.0, respectively. The number of daily and cumulative reported death cases significantly varies from the first wave of Covid-19 in Pakistan. It was found that under the current scenario, using FB-Prophet model, the peak occurred in the second week of December 2020. The second wave of SARS-COV-2 is estimated to cause more than 55,000 cumulative cases by the end of February 2021 in Pakistan.

We also used FB-Prophet for modelling the provincial cases. The provisional governments are autonomous in managing their health care systems. Thus, it is very important to report the provisional forecast of the epidemic. The actual reported data for the Capital Territory and provinces along with the calculations using FB-Prophet were also compared.

It has been found that careful selection of parameters in GLEAMviz, based on the local conditions, and the best fitting strategies for FB-Prophet could model the epidemic in Pakistan. The recent selection of parameters and epidemic prediction is based on the number of diagnosed cases in Pakistan. An increase in the number of tests per million may improve estimates of the epidemic. We are not forecasting a future, but rather a range of outcomes that we believe are more probable given the scenarios tested, based on the data observed so far. These predictions are best considered as helpful guides, rather than definitive maps.

The writer is an Associate Professor in the Physics Department at Government College University, Lahore, Pakistan. He can be reached at dr.mazharhussain@gcu.edu.pk