Dynamic analysis of a three-strain COVID-19 SEIR epidemic model with general incidence rates

This work deals with the global stability analysis of three-strain COVID-19 SEIR epidemic model with general incidence rates. The problem is modelled by a system of eight nonlinear ordinary differential equations describing the evolution of susceptible, exposed, infected and removed individuals. The global stability of the disease-free equilibrium is proved depending on the basic reproduction number R0. Furthermore, using an appropriate Lyapunov functionals, the global stability results of the endemic equilibria are established depending on the strain 1 reproduction number R 0, the strain 2 reproduction number R 0 and the strain 3 reproduction number R 3 0. Numerical simulations are presented in order to investigate a comparison between the model results and COVID-19 clinical data. Introduction The late COVID-19 caused by the severe acute respiratory syndrome-related coronavirus SARS-Cov-2 is classified as a strain of SARS-CoV-1 [1]. The classical susceptible-infected-recovered (SIR) epidemic model was first introduced in [2]. The multi-strain SEIR epidemic models, which better describes the evolution of the COVID-19 pandemic within populations, present an important tool to study this serious pandemic because they include a long incubation period and also various infection strains. The relevance of studying multi-strain models is to find out the different conditions permitting the coexistence of all acting strains. Recently, a multistrain SEIR model with two general incidence rates are studied [3]. In this paper, we develop the study for a three-strain COVID-19 SEIR epidemic model with general incidence rates. To this end, we will consider the following three strains generalized epidemic model:  dS dt = Λ − f(S, I1)I1 − g(S, I2)I2 − h(S, I3)I3 − δS , dE1 dt = f(S, I1)I1 − (γ1 + δ)E1 , dE2 dt = g(S, I2)I2 − (γ2 + δ)E2 , dE3 dt = h(S, I3)I3 − (γ3 + δ)E3 , dI1 dt = γ1E1 − (μ1 + δ)I1 , dI2 dt = γ2E2 − (μ2 + δ)I2 , dI3 dt = γ3E1 − (μ3 + δ)I3 , dR dt = μ1I1 + μ2I2 + μ3I3 − δR , Where (S) is the number of susceptible individuals, (E1), (E2) and (E3) are, respectively, the numbers of each latent individuals class, (I1), (I2) and (I3) are, respectively, the numbers of each infectious individuals class and (R) is the number of removed individuals. The parameter Λ is the recruitment rate, δ is the death rate of the population, γ1, γ2 and γ3 are, respectively, the latency rates of strain 1, strain 2 and strain 3, μ1, μ2 and μ3 are, respectively, the three-strain transfer rates from infected to recovered. The general incidence functions f(S, I1), g(S, I2) and h(S, I3) stand for the infection transmission rates for strain 1, strain 2 and strain 3, respectively. Our main contribution centers around the global stability of a three-strain COVID-19 SEIR epidemic model with general incidence rates. Results and discussion The main interest of this section is to compare the numerical simulations from our three-strain COVID-19 SEIR epidemic model with the COVID-19 clinical data. We have chosen to make this comparison concerning Moroccan clinical data during the year 2020 in the period between March 31 and June 20. We observe a good fit between the curve representing the COVID-19 clinical data and the numerical simulations resulting from our mathematical model. The impact of the quarantine strategy on controlling the infection spread is also discussed. References [1] Gobalenya, A.E. , Baker, S.C. , Baric, R.S. , de Groot, R.J., Drosten, C. , Gulyaeva, A.A. , and al.: The species Severe acute respiratory syndrome-related coronavirus: classifying 2019-nCoV and naming it SARS-CoV-2. Nat. Microbiology. 5, 536-544 (2020) [2] Kermack, W.O. , McKendrick, A.G. : A contribution to the mathematical theory of epidemics. Proc. R Soc. Lond. A. 115, 700721(1927) [3] Khyar, O. , Allali, K. Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: application to COVID19 pandemic. Nonlinear Dyn (2020). https://doi.org/10.1007/s11071-020-05929-4.