The extinction and persistence of a stochastic SIR model

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1 Ji and Jiang Advances in Difference Equaions 7 7:3 DOI.86/s z R E S E A R C H Open Access The exincion and persisence of a sochasic SIR model Chunyan Ji,* and Daqing Jiang 3,4 * Correspondence: chunyanji8@homail.com School of Mahemaics and Saisics, Changshu Insiue of Technology, Changshu, Jiangsu 55, P.R. China School of Mahemaics and Saisics, Norheas Normal Universiy, Changchun, Jilin 34, P.R. China Full lis of auhor informaion is available a he end of he aricle Absrac The objecive of his paper is o explore he long ime behavior of a sochasic SIR model. We esablish a hreshold condiion called he basic reproducion number under sochasic perurbaion for persisence or exincion of he disease. Especially, some numerical simulaions are applied o suppor our heoreical resuls. MSC: 49K5; 6H; 93E5 Keywords: a sochasic SIR model; exincion; persisence; hreshold; basic reproducion number Inroducion Epidemiology is he sudy of he spread of diseases wih he objecive o race facors ha are responsible for or conribue o heir occurrence. Significan progress has been made in he heory and applicaion by mahemaical research [ 6]. Recenly, we discussed he dynamic of a suscepible-infeced-recovered SIR epidemic model wih sochasic perurbaion [7], ha is, ds=[ S βsi] d σ SI db, di=[βsi + γ + ɛi] d + σ SI db, dr=[γi R] d,. where S, I, R, andb are he suscepible, he infecive, he recovered, and a sandard Brown moion, respecively. The parameers all posiive consans have he following meaning: is he birh rae, is he deah rae, β is he average number of conacs per infecive per day, γ is he recovery rae, ɛ is he deah rae of infecive caused by he disease, and σ is he inensiy of he whie noise. We invesigaed he persisence and exincion of he disease, and so we gave he hreshold of his sochasic model, which is affeced by whie noise and is less han he basic reproducion number of he corresponding deerminisic sysem. However, considering sochasic perurbaions o have differen paerns, i is ineresing o invesigae wheher he hreshold of his model wih oher sochasic perurbaions has a relaion wih whie noise [8 ]. The Auhors 7. This aricle is disribued under he erms of he Creaive Commons Aribuion 4. Inernaional License hp://creaivecommons.org/licenses/by/4./, which permis unresriced use, disribuion, and reproducion in any medium, provided you give appropriae credi o he original auhors and he source, provide a link o he Creaive Commons license, and indicae if changes were made.

2 Jiand JiangAdvances in Difference Equaions 7 7:3 Page of 8 We devoe his paper o he sudy of he following sochasic sysem: ds=[ S βsi] d + σ S db, di=[βsi + γ + ɛi] d + σ I db, dr=[γi R] d + σ 3 R db 3,. where B, B, B 3 are muually independen Brown moions wih inensiies σ, σ, σ 3, respecively. For any iniial value S, I, R R3 +,hereisasoluion S, I, R R 3 + of sysem. see[]. In his paper, we also inend o give he basic reproducion number of his sysem hrough discussing he long ime behavior of he model, which will be given in he nex secion. In Secion 3, we illusrae resuls hrough simulaions. A he end, a discussion is given o conclude his paper. Throughou his paper, unless oherwise specified, le, F, {F }, P beacomplee probabiliy space wih a filraion {F } saisfying he usual condiions i.e. i is righ coninuous and F conains all P-null ses, and le B be a scalar Brownian moion defined on he probabiliy space. The long ime behavior of sysem. In his secion, we deduce when he disease will die ou and when he disease will prevail. And so he basic reproducion number of sysem. isgiven.asheproofin[], we can conclude o he following lemmas. Lemma. Le S,I,R be he soluion of sysem. wih any iniial value S, I, R R 3 +. Then S+I+R =. Remark. In fac, ogeher wih he posiiviy of he soluion and., we have S I R =, =, =. Lemma. Assume > σ σ σ 3. Le S, I, R be he soluion of sysem. wih any iniial value S, I, R R 3 +, hen Sr db r =, Rr db 3r = Ir db r =,.3 Le β β R = = R + γ + ɛ + σ σ + γ + ɛ + γ + ɛ + σ, β where R = is he basic reproducion of he corresponding deerminisic sysem +γ +ɛ [3]. Based on hese wo lemmas, we give mainly he resuls in his secion.

3 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 3 of 8 Theorem. Assume > σ σ σ 3. Le S, I, R be he soluion of sysem. wih any iniial value S, I, R R 3 +. If R <,hen sup log I + γ + ɛ + σ R < namely, I ends o zero exponenially In oher words, he disease dies ou wih probabiliy one. Proof Noe ha S S I I R R = = β = γ Ss ds β SsIs ds Is ds SsIs ds + γ + ɛ Rs ds + σ 3 + σ Ss db s, Is ds + σ Is db s, Rs db 3s,.4 hen Ss ds Is ds + + γ + ɛ = S+I := + ϕ, + S + I where ϕhasheproperyha + σ Ss db s + σ Is db s ϕ=.5 according o.and.3. Then Ss ds + + γ + ɛ In addiion, log I=log I + β = log I + β + σ B [ β = + σ B [ β β + γ + ɛ Is ds =..6 Ss ds + γ + ɛ + σ + σ B Is ds + β ϕ + γ + ɛ + σ + γ + ɛ + σ + γ + ɛ + σ ] β + γ + ɛ Is ds + log I + β ϕ ] + log I + β ϕ + σ B.7

4 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 4 of 8 and [ log I + β ] ϕ + σ B = by.5 and he propery of he Brown moion. If R <,from.7, we have sup log I β + γ + ɛ + σ = + γ + ɛ + σ R <, as required. Theorem. Assume > σ σ σ 3. Le S, I, R be he soluion of sysem. wih any iniial value S, I, R R 3 +. If R <,hen Sr dr =, I=, R= Proof We have R <,le κ = + γ + ɛ + σ R <. I follows from he resul of Theorem. ha, for arbirary small consan < ξ < min{/, κ/},hereexisaconsan T = Tωandase ξ such ha P ξ ξ and for T, ω ξ, log I κ,andso I e κ/. Therefore sup I, which ogeher wih he posiiviy of he soluion implies I=.8 Therefore, from.6,.8, and he hird equaions of., i is easy o obain Ss ds =, R= The proof of his resul is compleed. In he previous, we show he exincion of he disease. Now, we invesigae condiions for he prevalence of he disease. Theorem.3 Assume > σ σ σ 3. Le S, I, R be he soluion of sysem. wih any iniial value S, I, R R 3 +. If R >,hen Ss ds = R, Is ds = + γ + ɛ + σ R, β + γ + ɛ Rs ds = γ + γ + ɛ + σ R β + γ + ɛ

5 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 5 of 8 Proof If R >, hen from he las equaliy of.7 and Lemmas 5., 5. in [7], we have Is ds = β + γ + ɛ + σ β+γ +ɛ = + γ + ɛ + σ R.9 + γ + ɛ Togeher wih.6, hisyields Ss ds = + γ + ɛ + σ R = β R Moreover, from he hird equaion of sysem., we obain R R = γ Is ds Rs ds + σ 3 Rs db 3 s, which ogeher wih.3and.9implies Rs ds = γ + γ + ɛ + σ R β + γ + ɛ Remark. Alhough here is no endemic equilibrium of sysem., Theorem.3 ells us he soluion of i ends o a poin in ime on average whose value is less han he value of S, I, R [3], and we consider i is sable in ime on average. Furhermore, from he resuls of Theorem. and Theorem.3, we see ha he value of R mainly deermines he exincion and persisence of he disease. As deerminisic epidemic models, R can be called he basic reproducion of he sochasic epidemic sysem.. 3 Simulaions In his secion, using he mehod menioned in [4], we give simulaions o suppor our resuls. The discreized equaion is S k+ = S k + S k βs k I k + σ ɛ,k Sk, I k+ = I k +[βs k I k + γ + ɛi k ] + σ ɛ,k Ik, R k+ = R k +γi k R k + σ 3 ɛ,k Rk. Le S, I, R =, 5, 3, =,β =, =,γ =4,ɛ =, =..Inhissiuaion, R = 4 >, hen he soluion of he corresponding deerminisic sysem will end 3 o S, I, R = 5, 5 3, and he disease will prevail see he black real lines in Figures 3 and. Choosing differen values of σ, σ, σ 3,wegewocases. Case Assume σ =.5,σ =4,σ 3 =.,hen > σ σ σ 3 and R = 3 <.As expeced, he infeced populaion I will end o very quickly and he disease will die ou. The suscepible populaion S willendo = in ime average see he second picure in Figure. Case Le σ =.5,σ =.5,σ 3 =.suchha > σ σ σ 3 and R = 6 9 >. As Theorem.3 said, he disease will prevail and he soluion is flucuaing around he endemic equilibrium S, I, R of he corresponding deerminisic sysem see he upper hree picures in Figure. Furhermore, we show ha he suscepible, infeced and re-

6 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 6 of 8 Figure Simulaion of he pah S, Ss ds, IandR for he sochasic sysem.andhe corresponding deerminisic sysem wih R >and R <. Figure Simulaion of he pah S, I, R, and Ss ds, Is ds, Rs ds for he sochasic sysem. and he corresponding deerminisic sysem wih R >and R >. covered populaion will end o a posiive value in ime on average, respecively, in deail, Ss ds = 9 6, Is ds = 3 4, Rs ds = 3 6 The oher hree picures in Figure illusrae his.

7 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 7 of 8 4 Discussions In his paper, by qualiaive analysis we have sudied he global behavior of an SIR model wih sochasic perurbaion. As he deerminisic epidemic models, we give a basic reproducion number R of sysem., which is also less han he value of he basic reproducion number in he corresponding deerminisic sysem. From he discussion, we can hink whie noise can make he value of he basic reproducion number lower, and so i is beneficial for conrolling he disease. Infac,forasochasicSIRSepidemicmodel, ds=[ S βsi+ρr] d + σ S db, di=[βsi + γ + ɛi] d + σ I db, dr=[γi R ρr] d + σ 3 R db 3, 4. we can obain he following properies hrough a similar analysis. If > σ σ σ 3 and R <,hen log I = + γ + ɛ + σ R < Sr dr =, I=, R= If > σ σ σ 3 and R >,hen Ss ds = R, Is ds = + ρ + γ + ɛ + σ β[ + ρ + ɛ+γ ] R Rs ds = γ + γ + ɛ + σ β[ + ρ + ɛ+γ ] R, Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors conribuions CJ conribued o Secions,, 3,and4. Daqing Jiang conribued o Secions and 4. Auhor deails School of Mahemaics and Saisics, Changshu Insiue of Technology, Changshu, Jiangsu 55, P.R. China. School of Mahemaics and Saisics, Norheas Normal Universiy, Changchun, Jilin 34, P.R. China. 3 College of Science, China Universiy of Peroleum Eas China, Qingdao, 6658, P.R. China. 4 Nonlinear Analysis and Applied Mahemaics NAAM-Research Group, Deparmen of Mahemaics, King Abdulaziz Universiy, Jeddah, Saudi Arabia. Acknowledgemens The work was suppored by he Naional Naural Science Foundaion of China No. 643, he Naional Science Foundaion for Pos-docoral Scieniss of China No. 6M5943 and sponsored by Qing Lan Projec of Jiangsu Province 6. Received: 9 July 6 Acceped: 6 December 6 References. Anderson, RM, May, RM: Populaion biology of infecious diseases, par I. Naure 8, Berea, E, Takeuchi, Y: Convergence resuls in SIR epidemic model wih varying populaion sizes. Nonlinear Anal. 8,

8 Jiand JiangAdvances in Difference Equaions 7 7:3 Page 8 of 8 3. Berea, E, Hara, T, Ma, W, Takeuchi, Y: Global asympoic sabiliy of an SIR epidemic model wih disribued ime delay. Nonlinear Anal. 47, Franceschei, A, Pugliese, A: Threshold behaviour of a SIR epidemic model wih age srucure and immigraion. J. Mah. Biol. 57, Jana, S, Haldar, P, Kar, TK: Mahemaical analysis of an epidemic model wih isolaion and opimal conrols. In. J. Compu. Mah Zaman, G, Kang, YH, Jung, IH: Sabiliy analysis and opimal vaccinaion of an SIR epidemic model. Biosysems 93, Ji, CY, Jiang, DQ: Threshold behaviour of a sochasic SIR model. Appl. Mah. Model. 38, Lin, YG, Jiang, DQ, Xia, PY: Long-ime behavior of a sochasic SIR model. Appl. Mah. Compu. 36, Zhao, DL: Sudy on he hreshold of a sochasic SIR epidemic model and is exensions. Commun. Nonlinear Sci. Numer. Simul. 38, Zhou, YL, Zhang, WG: Threshold of a sochasic SIR epidemic model wih Lévy jumps. Physica A 446, Ji, CY, Jiang, DQ, Yang, QS, Shi, NZ: Dynamics of a muligroup SIR epidemic model wih sochasic perurbaion. Auomaica 48, -3. Zhao, YN, Jiang, DQ: The hreshold of a sochasic SIS epidemic model wih vaccinaion. Appl. Mah. Compu. 43, Hehcoe, HW: Qualiaive analyses of communicable disease models. Mah. Biosci. 7, Higham, DJ: An algorihmic inroducion o numerical simulaion of sochasic differenial equaions. SIAM Rev. 43,