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

Miguel Fudolig; Reka Howard

is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.03. 19.20039198 doi: medRxiv preprint population. After establishing the possible transitions between compartments, the 459 system was found to have four equilibrium points: the disease free equilibrium, the 460 existing strain equilibrium, the emergent strain equilibrium, and the endemic 461 equilibrium. Upon examining the conditions for existence and local stability, the 462 disease-free equilibrium was determined to be locally stable when both R v and R 0 are 463 less than one. This is consistent with the reproduction number for this multi-strain 464 model. The existing strain equilibrium surprisingly did not impose that R 0 < 1, which 465 is the condition for the DFE of a normal SIR model for a single-strain without 466 immunity. This implies that if the original strain is much more contagious than the 467 emergent strain, the emergent strain would still die out eventually regardless of the fact 468 that it would persist in the population if it was on its own. On a similar note, the local 469 stability condition for the emergent equilibrium condition also does not impose that 470 R v < 1. This means that the state of the system prior to the emergence of the new 471 strain does not matter as long as the emergent strain is more contagious than the 472 existing strain. This leaves a highly restrictive condition for endemic equilibrium to 473 exist: the emergent strain must be able to survive by itself and the original strain must 474 be contagious enough to infect enough people, which is given by Eq 44. Eq 44 also 475 implies that the two-strain endemic equilibrium will only be locally asymptotically . CC-BY 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.