Abstract
In this study, a deterministic SEQIR model with standard incidence and the corresponding stochastic epidemic model are explored. In the deterministic model, the reproduction number is given, and the local asymptotic stability of the equilibria is proved. When the reproduction number is less than unity, the disease-free equilibrium is locally asymptotically stable, whereas the endemic equilibrium is locally asymptotically stable in the case of a reproduction number greater than unity. A stochastic expansion based on a deterministic model is studied to explore the uncertainty of the spread of infectious diseases. Using the Lyapunov function method, the existence and uniqueness of a global positive solution are considered. Then, the extinction conditions of the epidemic and its asymptotic property around the endemic equilibrium are obtained. To demonstrate the application of this model, a case study based on COVID-19 epidemic data from France, Italy, and the UK is presented, together with numerical simulations using given parameters.
Highlights
At the end of 2019, COVID-19 was reported in Wuhan, China [1], setting the beginning of a global epidemic
It was found that COVID-19 was caused by the SARS-CoV-2 coronavirus [2, 3]. is coronavirus mainly spreads in three ways: (i) direct transmission, which refers to the infection caused by patient’s sneezing, coughing, and talking droplets, and the direct inhalation of exhaled gas at a short distance; (ii) aerosol transmission, which refers to droplet mixing in the air to form an aerosol that can lead to infection after inhalation; (iii) contact transmission, which refers to the droplet deposited on the surface of an object, contact with contaminated hands, and contact with the oral cavity, nasal cavity, eyes, and other mucous membranes, resulting in infection
Novel coronavirus pneumonia models were reported by scholars who have been involved in this task since the outbreak of the epidemic
Summary
At the end of 2019, COVID-19 was reported in Wuhan, China [1], setting the beginning of a global epidemic. In [25], the authors studied an SEQIR model with Markovian switching, established a random threshold of disease extinction and persistence, and used the data from Indian states to confirm their conclusions. Based on the above studies, an SEQIR model with standard incidence is analyzed in this study, together with deterministic and stochastic models. Brahim et al considered a model with Markovian switching and generalized incidence in [25] They considered that people who are exposed and asymptomatic are infectious. E standard incidence rate is more suitable for infectious disease models with a large population. Based on these results, we consider an SEQIR model with standard incidence as follows: dS(t) βS(t)E(t). We declare that the standard Brownian motion in the system is defined on a complete probability space (Ω, F, P) with a filtration Ftt∈R+ satisfying the usual conditions (i.e., it is right continuous and increasing while F0 contains all P-null sets)
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