The local stability of a modified multi-strain SIR model for emerging viral strains

Miguel Fudolig; Reka Howard

In the case of the influenza virus, it is possible to have multiple strains exist in a 72 population, but only have vaccine for a certain strain that will not be effective for 73 others. The fact that viruses undergo changes regularly indicates that people who have 74 recovered from the virus, as well as individuals who have been vaccinated for a specific 75 strain of the virus, can be susceptible again to a newly-emerged strain. It is important 76 to determine the conditions in which a newly emerged strain and a common strain that 77 has a means of immunity will coexist in a population. 78 Another apt example for emerging diseases is the emergence of the COVID- 19 virus 79 in 2019 [39, 40] . As of March 17, 2020, there have been 167,545 confirmed cases of 80 COVID-19 in 150 different countries that has led to 6,606 deaths since it was declared 81 as an outbreak in January 2020 according to the WHO situation report [41] . At the 82 time that this paper is being written, there are papers that have modeled the dynamics 83 of the virus using different modifications of the SIR model. Zhou et. al. [42] included 84 compartments corresponding to suspected cases, which consists of the individuals that 85 show similar symptoms but are not confirmed cases, and indirectly infected individuals. 86 Pan et. al. [43] used a modified SEIR model which included asymptomatic and given that one of these strains has a vaccine available. This paper introduces a model 97 that approaches the lack of cross-immunity across different viral strains by introducing 98 new compartments to the SIR with vaccination model. This enables us to introduce 99 acquired immunity through vaccination and cross-immunity between strains in a simple 100 compartmental model and investigate the existence and stability of the resulting 101 equilibrium points. We aim to model the dynamics of an epidemic where a new emergent strain of an 103 existing virus affects a closed population. The existing virus will be modeled using a 104 modified SIR model with vaccination, however we assume that the vaccine does not 105 provide immunity to the newer strain. The equilibrium points were determined for the 106 system based on the transition equations and local stability was investigated for each 107 point. Once the stability conditions have been established, the epidemic model was 108 simulated using R [46] to check the steady-state behavior of the surveillance data for next two subsections will explain the dynamics before and after the emergence of the 124 mutated strain.