On exact integrability of a Covid‐19 model: SIRV

Abstract

In this study, the integrability conditions and the exact analytical solutions of the initial‐value problem defined for the prominent SIRV model used for the pandemic Covid‐19 are investigated by using the partial Hamiltonian approach based on the theory of Lie groups. Two different cases are considered with respect to the model parameters. In addition, the integrability properties and the associated approximate and exact analytical solutions to the SIRV model are analyzed and investigated by considering two different phase spaces. Furthermore, the graphical representations of susceptible, infected, recovered, and vaccinated population fractions evolving with time for subcases are introduced and discussed.

Integrability properties and invariant solutions of some biological models

“…In addition, the Lie groups theory can be applied to the nonlinear dynamical systems as well by considering the approach called the artificial Hamiltonian, a method that provides a gateway by which to compute the exact solution of a variety of coupled nonlinear systems of ordinary differential equations (ODEs) 13 . In addition, it is a fact that every first‐order system of ODEs can be written in the form of an artificial Hamiltonian system, and then the concept of artificial Hamiltonian presents a novel approach for finding the solutions to dynamical systems of first‐order ODEs, and this approach is very applicable to determining first integrals of the aforementioned systems 14–17 . The study of obtaining the first integrals of ODEs is very crucial since they enable us to get closed‐form solutions.…”

“…13 In addition, it is a fact that every first-order system of ODEs can be written in the form of an artificial Hamiltonian system, and then the concept of artificial Hamiltonian presents a novel approach for finding the solutions to dynamical systems of first-order ODEs, and this approach is very applicable to determining first integrals of the aforementioned systems. [14][15][16][17] The study of obtaining the first integrals of ODEs is very crucial since they enable us to get closed-form solutions. However, there is no systematic way to determine the first integrals of ODEs; furthermore, the nonlinear dynamical systems lack the existence of closed-form solutions.…”

Key Epidemic Parameters of the SIRV-Model Determined from Past COVID-19 Mutant Waves

“…The susceptible-infected-recovered/removed-vaccinated (SIRV) epidemics model [1][2][3][4][5] is an important generalization of the simpler susceptible-infected-recovered/removed-(SIR) epidemics model, developed originally by Kermack and McKendrick [6] and refined by Kendall [7], as it accounts for the effects of vaccination campaigns on a considered population. Both models are realistic compartment models where persons from the considered population are assigned to the three (SIR) and four (SIRV) compartments S (susceptible), I (infectious), R (recovered/removed) and (V) vaccinated, respectively.…”

“…Two important key parameters of the SIRV pandemics model are the ratios k(t) = µ(t)/a(t) of the recovery to infection rate and b(t) = v(t)/a(t) of the vaccination to infection rate. The recently derived analytical solutions to the SIRV equations [1,2] have adopted originally stationary values of the ratios k(t) = k 0 and b(t) = b 0 allowing for arbitrary time-dependent infection rates a(t). This implies that the recovery and vaccination rates have the same time dependence as the infection rate.…”

Editorial of the special issue on modelling, analysis, and applications

“…A. Tenreiro Machado, who unexpectedly passed away on October 4, 2021, at his 64th birthday, proposed, in co‐authorship, a new deterministic model for SARS‐CoV‐2 propagation in distinct countries [1, 2] and one paper on variation, in option pricing [3]. Other works predicting COVID‐19 patterns focus on fractional and other integer order models [4] and the application of integral operators [5]. The influence of media coverage is measured in [6], and the effect of shelter hospitals in [7], which contributed to scale the capacity of health‐care systems in Fangcang, during the COVID‐19 pandemic.…”