Nonlinear dynamics for the spread of pathogenesis of COVID-19 pandemic

Abstract

Coronaviruses did not invite attention at a global level and responsiveness until the series of 2003-SARS contagion followed by year-2012 MERS plus, most recently, 2019-nCoV eruptions. SARS-CoV & MERS-CoV are painstaking, extremely pathogenic. Also, very evidently, both have been communicated from bats to palm-civets & dromedary camels and further transferred ultimately to humans. No country has been deprived of this viral genomic contamination wherever populaces reside and are interconnected. This study aimed to develop a mathematical model for calculating the transmissibility of this viral genome. The analysis aids the study of the outbreak of this Virus towards the other parts of the continent and the world. The parameters such as population mobility, natural history, epidemiological characteristics, and the transmission mechanism towards viral spread when considered into crowd dynamism result in improved estimation. This article studies the impact of time on the amount of susceptible, exposed, the infected person taking into account asymptomatic and symptomatic ones; recovered i.e., removed from this model and the virus particles existing in the open surfaces. The transition from stable phase to attractor phase happens after 13 days i.e.; it takes nearly a fortnight for the spread to randomize among people.Further, the pandemic transmission remains in the attractor phase for a very long time if no control measures are taken up. The attractor-source phase continues up to 385 days i.e., more than a year, and perhaps stabilizes on 386th day as per the Lyapunov exponent's analysis. The time series helps to know the period of the Virus's survival in the open sources i.e. markets, open spaces and various other carriers of the Virus if not quarantined or sanitized. The Virus cease to exist in around 60 days if it does not find any carrier or infect more places, people etc. The changes in LCEs of all variables as time progresses for around 400 days have been forecasted. It can be observed that phase trajectories indicate how the two variables interact with each other and affect the overall system's dynamics. It has been observed that for exposed and asymptomatically infected (y–z), as exposed ones (y) change from 0 to 100 the value of asymptomatically infected (z) increased upto around 58, at exposed ones (y) = 100, asymptomatically infected (z) has two values as 58 and 10 i.e. follows bifurcation and as exposed ones (y) changes values upto 180, the value of asymptomatically infected (z) decreases to 25 so for exposed ones (y) from 100 to 180, asymptomatically infected (z) varies from 58 to 25 to 10 follows bifurcation. Also, phase structures of exposed-symptomatically infected (y–u), exposed-removed (y–v), exposed-virus in the reservoir (y–w), asymptomatically infected-removed (z–v), symptomatically infected-removed (u–v) specifically depict bifurcations in various forms at different points. In case of asymptomatically infected-virus in the reservoir (z–w), at asymptomatically infected (z) = 10, the value of viruses in the reservoir (w) = 50, then as asymptomatically infected (z) increases to upto around 60. At this point, removed ones (v) increase from 50 to 70 and asymptomatically infected (z) decrease to 20 i.e., crosses the same value twice, which shows its limiting is known as limit cycle behavior and both the values tend to decrease towards zero. It shows a closed-loop limit cycle. Today, there has been no scientific revolution in the development of vaccination, nor has any antiviral treatment been successful, resulting in lack of its medication. Based on the phases, time series, and complexity analysis of the model's various parameters, it is studied to understand the variation in this pandemic's scenario.

Keywords

• COVID-19
• RNA genome
• Phase space
• Time domainic analysis
• Lyapunov exponents