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A complex SIR epidemic dynamical model using nonlinear incidence rate and nonlinear recovery rate is established to consider the impact of available hospital beds and interventions reduction on the spread of infectious disease. Rigorous mathematical results have been established for the model from the point of view of stability and bifurcation. The model has two equilibrium points when the basic reproduction number R 0 > 1 ; a disease-free equilibrium E 0 and a disease endemic equilibrium E 1 . We use LaSalle's invariance principle and Lyapunov's direct method to prove that E 0 is globally asymptotically stable if the basic reproduction number R 0 < 1 , and E 1 is globally asymptotically stable if R 0 > 1 , under some conditions on the model parameters. The existence and nonexistence of limit cycles are investigated under certain conditions on model parameters. The model exhibits Hopf bifurcation near the disease endemic equilibrium. We further show the occurring of backward bifurcation for the model when there is limited number of hospital beds. Finally, some numerical results are represented to validate the analytical results. [ABSTRACT FROM AUTHOR]

The new emerged infectious disease that is known the coronavirus disease (COVID-19), which is a high contagious viral infection that started in December 2019 in China city Wuhan and spread very fast to the rest of the world. This infection caused million of infected cases globally and still pose an alarming situation for human lives. Pakistan in Asian countries is considered the third country with higher number of cases of coronavirus with more than 200,000. Recently, many mathematical models have been considered to better understand the coronavirus infection. Most of these models are based on classical integer-order derivative which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using the non-integer Caputo derivative. In the absence of vaccine or therapy, the role of non-pharmaceutical interventions (NPIs) is examined on the dynamics of theCOVID-19 outbreak in Pakistan. First, we construct the model in integer sense and then apply the fractional operator to have a generalized model. The generalized model is then used to present the detailed theoretical results. We investigate the stability of the model for the case of fractional model using a nonlinear fractional Lyapunov function of Goh-Voltera type. Furthermore, we estimate the values of parameters with the help of least square curve fitting tool for the COVID-19 data recorded in Pakistan since March 1 till June 30, 2020 and show that our considered model give an accurate prediction to the real COVID-19 statistical cases. Finally, numerical simulations are presented using estimated parameters for various values of the fractional order of the Caputo derivative. From the simulation results it is found that the fractional order provides more insights about the disease dynamics. [ABSTRACT FROM AUTHOR]

The Coronavirus disease or COVID-19 is an infectious disease caused by a newly discovered coronavirus. The COVID-19 pandemic is an inciting panic for human health and economy as there is no vaccine or effective treatment so far. Different mathematical modeling approaches have been suggested to analyze the transmission patterns of this novel infection. this paper, we investigate the dynamics of COVID-19 using the classical Caputo fractional derivative. Initially, we formulate the mathematical model and then explore some the basic and necessary analysis including the stability results of the model for the case when R 0 < 1. Despite the basic analysis, we consider the real cases of coronavirus in China from January 11, 2020 to April 9, 2020 and estimated the basic reproduction number as R 0 ≈ 4.95. The present findings show that the reported data is accurately fit the proposed model and consequently, we obtain more realistic and suitable parameters. Finally, the fractional model is solved numerically using a numerical approach and depicts many graphical results for the fractional order of Caputo operator. Furthermore, some key parameters and their impact on the disease dynamics are shown graphically. [ABSTRACT FROM AUTHOR]