Abstract
Everyone is talking about coronavirus from the last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed, and as many as 180 countries have been so far affected with 928,287 (14 September 2020) deaths within a couple of months. Ironically, 29,185,779 are still active cases. Having seen such a drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. The major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 1 April to 15 March 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam–Hyers and its generalized version. The best parameters have been obtained via the nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than the classical version of the SIR model (Brauer et al. in Mathematical Models in Epidemiology 2019). Average absolute relative error under the Caputo operator is about 48% smaller than the one obtained in the classical case (nu =1). Time series and 3D contour plots offer social distancing to be the most effective measure to control the epidemic.
Highlights
Literature includes mathematical models for pathways for the transmission of infectious ailments
Having been inspired by a plethora of research works carried out in fractional mathematical epidemiology, we have introduced the following Caputo-type SIR model wherein its distinguished feature is that the dimensional inconsistency of the model has been removed by carrying the fractional order ν in the power of biological parameters β and γ : CDν0,t S(t) =
We have investigated the effects of fractional order ν on the dynamical behavior of Caputo model (5) for all three compartments
Summary
Literature includes mathematical models for pathways for the transmission of infectious ailments. The removed category includes people that no longer can catch the disease either because they have recovered or died, and this is going to increase at the constant rate (recovery rate = γ ) depending on how many infectives there are. There are only two biological parameters best fitted via the leastsquares fitting method thereby yielded best fit of the model’s solution to the real cases chosen from Pakistan as depicted by Fig. 1 under the classical situation, that is, when ν = 1 wherein the best parameters are as follows: β = 3.0918 and γ = 3.0190 for the transmission and recovery rate, respectively. Similar analysis for the Caputo fractional-order model is carried out wherein the best fitted parameters are found as β = 3.9405 and γ = 3.8010 for the transmission and recovery rate, respectively. We have shown the dynamics of the basic reproduction number R0 under different effects of the transmission rate β and the recovery rate γ in Fig. 11, wherein even a slightly larger value of β brings the reproduction number near to 1, which is clearly an alarming situation for policy makers to devise an effective approach to prevent R0 to be greater than 1
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