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PureMathematics
n
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,2022,12(6),938-951
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126103
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5
TheThresholdDynamicBehaviorofa
StochasticInfectiousDiseaseModel
withVaryingTotalPopulationSize
RuyiRen
SchoolofScience,LanzhouUniversityofTechnology,LanzhouGansu
Received:May4
th
,2022;accepted:Jun.8
th
,2022;published:Jun.15
th
,2022
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[J].
n
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,2022,12(6):938-951.
DOI:10.12677/pm.2022.126103
?T
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Abstract
Tostudytheeffectsofmortality,infectionratesofrandomdisturbancesandrandom
perturbationofthesystemonthespreadofdisease,inthispaper,weconsidera
stochasticinfectiousdiseasemodelinwhich themortality,transmissionandsystemof
the diseasearedisturbed withchanging populationsize.The existenceanduniqueness
ofknowledgeisprovedbyconstructingappropriateLyapunovfunctions.Established
thresholds
R
s
0
fordeterminingdiseaseextinction,applyingMartingale’stheoremof
largenumbersand
It
ˆ
o
formulas,sufficientandalmostnecessaryconditionshavebeen
obtainedfortheextinctionofinfectiousdiseases.Morespecifically,if
R
s
0
<
1
,the
diseasewilldieout.Finally,byconstructingthreshold
R
s
0
,wefindthatthestochastic
perturbations ofthe deathrate and randomperturbationof the systemfor susceptible
populationcanenhancethespreadofdisease,whilethestochasticperturbationsof
thedeathrateandrandomperturbationofthesystemforinfectiouspopulation,as
wellasthetransmissionrateofthediseasecansuppressthespreadofthedisease.
Keywords
SIRSModel,ExtinctionofDisease,ExistenceandUniquenessofSolutions
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.126103944
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DOI:10.12677/pm.2022.126103945
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DOI:10.12677/pm.2022.126103946
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DOI:10.12677/pm.2022.126103947
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DOI:10.12677/pm.2022.126103948
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