A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions

Abstract

The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ($$\textsc {S}$$ S ), exposed ($$\textsc {E}$$ E ), asymptomatic infected ($$\textsc {I}_1$$ I 1 ), symptomatic infected ($$\textsc {I}_2$$ I 2 ), and recovered ($$\textsc {R}$$ R ) classes named $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model, using the Caputo fractional derivative. Here, $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model describes the effect of asymptomatic and symptomatic transmissions on coronavirus disease outbreak. The existence and uniqueness of the solution are studied with the help of Schaefer- and Banach-type fixed point theorems. Sensitivity analysis of the model in terms of the variance of each parameter is examined, and the basic reproduction number $$(R_{0})$$ ( R 0 ) to discuss the local stability at two equilibrium points is proposed. Using the Routh–Hurwitz criterion of stability, it is found that the disease-free equilibrium will be stable for $$R_{0} < 1$$ R 0 < 1 whereas the endemic equilibrium becomes stable for $$R_{0} > 1$$ R 0 > 1 and unstable otherwise. Moreover, the numerical simulations for various values of fractional-order are carried out with the help of the fractional Euler method. The numerical results show that asymptomatic transmission has a lower impact on the disease outbreak rather than symptomatic transmission. Finally, the simulated graph of total infected population by proposed model here is compared with the real data of second-wave infected population of COVID-19 outbreak in India.

Influence of Co-morbidities During SARS-CoV-2 Infection in an Indian Population

“…For this reason, during the SARS-CoV-2 pandemic, many researchers have proposed numerous mathematical models based on various correlations. The mathematical models took into account, inter alia , the method of transmission, the impact of quarantine and isolation, pharmaceutical and non-pharmaceutical interventions ( 50 , 51 ). Most of the models developed during the SARS-CoV-2 pandemic were based on the Susceptible-infected-recovered (SIR) model and the Susceptible-Exposed-Infected-Removed (SEIR) model.…”

A new approach to modeling pre-symptomatic incidence and transmission time of imported COVID-19 cases evolving with SARS-CoV-2 variants

“…Third, the separation of pre-symptomatic cases from asymptomatic case is to avoid the confusion of conventional use in defining asymptomatic cases in static property that actually consist of pre-symptomatic cases and asymptomatic cases at certain time point of infection axis. To this end, the previous compartment SEIR models (Batista et al 2020 ; Ahmed et al 2021 ; Chatterjee et al 2021 ; Ali et al 2022b ; Ma and Lin 2022 ; Massard et al 2022 ; Yi et al 2022 ) have been accommodated to estimate the asymptomatic growth and the fraction of asymptomatic infectives was parameterized in the model. Our proposed four-state stochastic process for estimating the proportion (incidence) of asymptomatic cases is tailored for such a purpose in a similar fashion.…”

“…To distinguish the infected persons before onset of symptoms, namely pre-symptomatic or non-persistent asymptomatic cases, from persistent asymptomatic cases, the modified SIR models was also developed to estimate the dynamics of the asymptomatically infected people (Sarkar, 2020; Gevertz, 2020). Similar but in-depth SEIR-based models were also adjusted to fit in with the context of asymptomatic infection (Batista et al 2020 ; Ahmed et al 2021 ; Chatterjee et al 2021 ; Ali et al 2022b ; Ma and Lin. 2022 ; Massard et al 2022 ; Yi et al 2022 ).…”

Modeling the impact of the vaccine on the COVID-19 epidemic transmission via fractional derivative

“…A fractional order COVID-19 model using different techniques and analysis was presented in [ 40 ] for the proper identification of the individuals in community of Saudi Arabia to reduce the intensity of the virus. In [ 41 ], author proposed a fractional order model to examine influence of symptomatic and asymptomatic communications on COVID-19 outburst where the analysis of the model is conducted using the Schaefer- and Banach-type fixed point theorems.…”