[Paper Review] Analysis and Prediction of COVID-19 Pandemic in Pakistan using Time-dependent SIR Model

Muhammad Sohail Waqas, M. Zaeem Farooq+2 more|arXiv (Cornell University)|May 5, 2020

COVID-19 epidemiological studies4 references20 citations

TL;DR

This study applies a time-dependent SIR model to predict the COVID-19 pandemic trajectory in Pakistan, using dynamic transmission ($\beta(t)$) and recovery ($\gamma(t)$) rates fitted to real data. It forecasts a peak of 20,000–47,000 active infections between late May and 9 June 2020, with cumulative cases ranging from 57,651 to 153,149, depending on recovery rates, and predicts the pandemic will subside by late August–September 2020 with 97% recovery.

ABSTRACT

The current outbreak is known as Coronavirus Disease or COVID-19 caused by the virus SAR-COV-2 which continues to wreak havoc across the globe. The World Health Organization (WHO) has declared the outbreak a Public Health Emergency of International Concern. In Pakistan, the spread of the virus is on the rise with the number of infected people and causalities rapidly increasing. In the absence of proper vaccination and treatment, to reduce the number of infections and casualties, the only option so far is to educate people regarding preventive measures and to enforce countrywide lock-down. Any strategy about the preventive measures needs to be based upon detailed analysis of the COVID-19 outbreak and accurate scientific predictions. In this paper, we conduct mathematical and numerical analysis to come up with reliable and accurate predictions of the outbreak in Pakistan. The time-dependent Susceptible-Infected-Recovered (SIR) model is used to fit the data and provide future predictions. The turning point of the peak of the pandemic is defined as the day when the transmission rate becomes less than the recovering rate. We have predicted that the outbreak will reach its maximum peak occurring from late May to 9 June with unrecovered number of Infectives in the range 20000-47000 and the cumulative number of infected cases in the range of 57500-153100. The number of Infectives will remain at the lower end in the lock-down scenario but can rapidly double or triple if the spread of the epidemic is not curtailed and localized. The uncertainty on single day projection in our analysis after April 15 is found to be within 5\%.

Motivation & Objective

• To model and predict the dynamics of the COVID-19 outbreak in Pakistan using a time-dependent SIR framework.
• To estimate key epidemiological parameters ($\beta(t)$ and $\gamma(t)$) from real-time case data to improve forecast accuracy.
• To assess the impact of recovery rates and preventive measures on the pandemic peak and duration.
• To validate the model using data from Switzerland and European countries before applying it to Pakistan.
• To provide policymakers with data-driven projections for effective public health interventions.

Proposed method

• A time-dependent SIR model is used, with differential equations for susceptible (S), infected (I), and recovered (R) populations.
• The transmission rate $\beta(t)$ is modeled as a time-varying parameter estimated via exponential fitting to daily case data.
• The recovery rate $\gamma(t)$ is treated as a tunable parameter to reflect changes in healthcare capacity and response efficiency.
• The model is validated by applying it to Switzerland and other European countries, showing strong agreement with observed data.
• The reproduction number $R_0 = \beta(t)/\gamma(t)$ is computed to identify the pandemic turning point when $R_0 < 1$.
• Prediction intervals are generated by varying $\gamma(t)$ within a plausible range to assess uncertainty in peak timing and magnitude.

Experimental results

Research questions

• RQ1 What is the predicted peak timing and magnitude of active infections in Pakistan under current conditions?
• RQ2 How do variations in the recovery rate $\gamma(t)$ affect the cumulative number of infections and pandemic duration?
• RQ3 When will the pandemic turn around, defined by $R_0 < 1$, and how does this align with the SIR model’s peak?
• RQ4 How accurate is the time-dependent SIR model in predicting real-world case trends in Pakistan and other countries?
• RQ5 What role do preventive measures and lock-downs play in reducing the peak number of active infections?

Key findings

• The pandemic peak in Pakistan is predicted to occur between 29 May and 9 June 2020, with 20,094 to 47,043 active infections depending on the recovery rate.
• Cumulative infections are projected to range from 57,651 (for $\gamma = 0.046$) to 153,149 (for $\gamma = 0.034$), showing high sensitivity to recovery efficiency.
• The one-day prediction error remains within 5% uncertainty after April 15, indicating strong model reliability.
• The reproduction number $R_0$ falls below unity after the peak, confirming the turning point of the epidemic.
• With 97% recovery achieved by late August to September 2020, the pandemic is expected to fade out gradually.
• A 1% reduction in the recovery rate could triple the cumulative number of infections, highlighting the critical role of healthcare system performance.