Numerical simulations on scale-free and random networks for the spread of COVID-19 in Pakistan☆

COVID-19 multiwaves as multiphase percolation: a general N-sigmoidal equation to model the spread

Abstract

The aim of the current study is to investigate the spread of the COVID-19 pandemic as a multiphase percolation process. Mathematical equations have been developed to describe the time dependence of the number of cumulative infected individuals, I t , and the velocity of the pandemic, V p t , as well as to calculate epidemiological characteristics. The study focuses on the use of sigmoidal growth models to investigate multiwave COVID-19. Hill, logistic dose response and sigmoid Boltzmann models fitted successfully a pandemic wave. The sigmoid Boltzmann model and the dose response model were found to be effective in fitting the cumulative number of COVID-19 cases over time 2 waves spread (N = 2). However, for multiwave spread (N > 2), the dose response model was found to be more suitable due to its ability to overcome convergence issues. The spread of N successive waves has also been described as multiphase percolation with a period of pandemic relaxation between two successive waves.

Graphical abstract

Implementation of computationally efficient numerical approach to analyze a Covid-19 pandemic model

Corona virus disease (Covid-19) which has caused frustration in the human community remains the concern of the globe as every government struggles to defeat the pandemic. To deal with the situation, we have extensively studied a deadly Covid-19 model to provide a deep insight into the disease dynamics. A mathematical analysis of the model utilizing preventive measures is performed with the aim to reduce the disease burden. Some comprehensive mathematical techniques are employed to demonstrate several essential properties of solutions. To start with, we proved the existence and uniqueness of solutions. Equilibrium points are stated both in the absence and presence of the pandemic. Biologically important quantity known as threshold parameter is computed to handle the future disease dynamics and analyzed for its sensitivity. We proved the stability of the proposed model at equilibrium points by employing necessary conditions on threshold parameter. A reliable and competitive numerical analysis is conducted to observe the effectiveness of implemented strategies and to verify obtained analytical results. The most sensitive parameters are determined through sensitivity analysis. An important feature of this study is to employ Non-Standard Finite Difference (NSFD) numerical scheme to solve the system instead of other standard methods like Runge–Kutta method of order 4 (RK4). Finally, several numerical simulations are performed to validate our former theoretical analysis. Numerical results exhibiting dynamical behavior of Covid-19 system under the influence of involved parameters suggest that both the implemented strategies, especially quarantine of exposed individuals, are effective for the substantial reduction in the diseased population and to achieve the herd immunity.