TY - JOUR
T1 - Stability, Bifurcation, and a Pair of Conserved Quantities in a Simple Epidemic System with Reinfection for the Spread of Diseases Caused by Coronaviruses
AU - Camacho, Jorge Fernando
AU - Vargas-De-León, Cruz
N1 - Publisher Copyright:
© 2021 Jorge Fernando Camacho and Cruz Vargas-De-León.
PY - 2021
Y1 - 2021
N2 - In this paper, we study a modified SIRI model without vital dynamics, based on a system of nonlinear ordinary differential equations, for epidemics that exhibit partial immunity after infection, reinfection, and disease-induced death. This model can be applied to study epidemics caused by SARS-CoV, MERS-CoV, and SARS-CoV-2 coronaviruses, since there is the possibility that, in diseases caused by these pathogens, individuals recovered from the infection have a decrease in their immunity and can be reinfected. On the other hand, it is known that, in populations infected by these coronaviruses, individuals with comorbidities or older people have significant mortality rates or deaths induced by the disease. By means of qualitative methods, we prove that such system has an endemic equilibrium and an infinite line of nonhyperbolic disease-free equilibria, we determine the local and global stability of these equilibria, and we also show that it has no periodic orbits. Furthermore, we calculate the basic reproductive number R0 and find that the system exhibits a forward bifurcation: disease-free equilibria are stable when R0<1/σ and unstable when R0>1/σ, while the endemic equilibrium consist of an asymptotically stable upper branch that appears from R0>1/σ, σ being the rate that quantifies reinfection. We also show that this system has two conserved quantities. Additionally, we show some of the most representative numerical solutions of this system.
AB - In this paper, we study a modified SIRI model without vital dynamics, based on a system of nonlinear ordinary differential equations, for epidemics that exhibit partial immunity after infection, reinfection, and disease-induced death. This model can be applied to study epidemics caused by SARS-CoV, MERS-CoV, and SARS-CoV-2 coronaviruses, since there is the possibility that, in diseases caused by these pathogens, individuals recovered from the infection have a decrease in their immunity and can be reinfected. On the other hand, it is known that, in populations infected by these coronaviruses, individuals with comorbidities or older people have significant mortality rates or deaths induced by the disease. By means of qualitative methods, we prove that such system has an endemic equilibrium and an infinite line of nonhyperbolic disease-free equilibria, we determine the local and global stability of these equilibria, and we also show that it has no periodic orbits. Furthermore, we calculate the basic reproductive number R0 and find that the system exhibits a forward bifurcation: disease-free equilibria are stable when R0<1/σ and unstable when R0>1/σ, while the endemic equilibrium consist of an asymptotically stable upper branch that appears from R0>1/σ, σ being the rate that quantifies reinfection. We also show that this system has two conserved quantities. Additionally, we show some of the most representative numerical solutions of this system.
UR - http://www.scopus.com/inward/record.url?scp=85120896540&partnerID=8YFLogxK
U2 - 10.1155/2021/1570463
DO - 10.1155/2021/1570463
M3 - Artículo
AN - SCOPUS:85120896540
SN - 1026-0226
VL - 2021
JO - Discrete Dynamics in Nature and Society
JF - Discrete Dynamics in Nature and Society
M1 - 1570463
ER -