Stochastic dynamics of feline immunodeficiency virus within cat populations

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Journal of the Franklin Institute 353 (2016) 4191–4212 www.elsevier.com/locate/jfranklin

Stochastic dynamics of feline immunodeficiency virus within cat populations Jingli Lia, Meijing Shanb, Malay Banerjeec, Weiming Wanga,d,n a

b

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, PR China Institute of Information Science and Technology, East China University of Political Science and Law, Shanghai 201620, PR China c Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India d School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China Received 26 November 2015; received in revised form 29 June 2016; accepted 5 August 2016 Available online 11 August 2016

Abstract In this paper, we investigate the basic features of a simple susceptible-infected (SI) epidemic model of Feline immunodeficiency virus (FIV) within cat populations in presence of multiplicative noise terms to understand the effects of environmental driving forces on the disease dynamics. The value of this study lies in two aspects. Mathematically, we propose three threshold parameters, Rhs , R1 and R2 to utilize in identifying the stochastic extinction and persistence. In the case of stochastic persistence, we prove that there is a stationary distribution. Based on the statistical data for rural cat populations Barisey-la-Côte in France, we perform some numerical simulations to verify/extend our analytical results. Epidemiologically, we find that: (1) Large environment fluctuations can suppress the outbreak of FIV; (2) The distributions are governed by Rhs ; (3) White noise perturbations of the birth rate for infectious cats (i.e., the vertical transmission) can induce the susceptible-free dynamics. & 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction In 1987, Pedersen and coworkers first described the Feline Immunodeficiency Virus (FIV; formerly feline T-lymphotropic lentivirus) [1], they realized its effectiveness towards the modelling n Corresponding author at: College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, PR China. E-mail addresses: [email protected] (J. Li), [email protected] (M. Shan), [email protected] (M. Banerjee), [email protected] (W. Wang).

http://dx.doi.org/10.1016/j.jfranklin.2016.08.004 0016-0032/& 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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of human immunodeficiency virus (HIV) related disease and its dynamic behavior. FIV was isolated from cats, having immune deficiency syndrome that is similar to human AIDS in various respect [2]. Infection of pathogen-free kittens with FIV resulted in lymphadenopathy. FIV is morphologically similar to HIV-1, and its reverse transcriptase (RT, the retrovirus-encoded RNAdependent DNA polymerase) has a Mg2þ requirement that is similar to that of the HIV-1 RT [3]. And FIV is a member of the lentivirus subfamily and is distinct from characterized retroviruses, which also includes HIV and Simian Immunodeficiency Viruses (SIV) [4]. The expectation that FIV is a leading animal model for HIV studies, indeed, the similarities between these two viruses are so many and, studying FIV will not only help us to understand the pathogenesis of HIV and provide clues for the development of interventive strategies for its control but also will benefit the natural host species [5]. However, FIV is mainly transmitted through bites during aggressive contacts between cats, in contrast to HIV [6,7]. Because domestic cat populations live in a variety of ecological setup, they show a wide range of variability in their spatial and social structures which can be compared to other populations of FIV-infected non-domestic fields that are more difficult to study within natural habitats. The wide prevalence of FIV infection in natural populations of cats provides an opportunity to analyse the consequences of population structures on the circulation of the virus related diseases [8]. The mathematical models have been revealed as a powerful tool to understand the dynamic spread of FIV within cat populations [7–13]. In order to describe the spread of FIV through the cat populations, Courchamp et al. [8] proposed a susceptible-infected (SI) model for the transmission of FIV within domestic cats with logistic growth, the standard incidence rate with no vertical transmission, and found that FIV has a low impact on the population in that the total number of cats at equilibrium point. O'Nell et al. [9] claimed that vertical transmission of FIV may be a useful modelling approach to evaluate the intervention strategies for HIV transmission from mother to child. Hilker et al. [7] considered the influence of the Allee effect (a phenomenon in biology characterized by a correlation between population size or density and the mean individual fitness of a population or species [14]) on the host population towards the disease spread. Ducrot et al. [13] investigate the global stability of the endemic steady state and the existence of travelling wave solutions connecting the endemic to the disease-free state of an SI type model to FIV with vertical transmission (or call mother-to-child transmission, the transmission of a bacteria, viruses, or in rare cases, parasites transmitted directly from the mother to an embryo, fetus, or baby during pregnancy or childbirth) and a density dependent incidence rate. In the real world, biological populations exist inevitably under the influence of noisy random environmental condition which affect the environmental parameters and as a result affect the population dynamics significantly. It is well known that in several instances, environmental variations have a significant influence on the development and propagation of an epidemic [15,16]. For human disease related epidemics, the nature of epidemic growth and spread is inherently random due to the unpredictability in person-to-person contacts [17] and population is subject to a continuous spectrum of disturbances [18,19]. Hence the variability and randomness of the environment is fed through the state of the epidemic [20]. And in epidemic dynamics, stochastic differential equation (SDE) models could be more appropriate way of modeling epidemics under various circumstances [21–31]. However, despite the potential importance of parametric noise, it has received relatively less attention in the research on FIV. And there comes a question: How does environment fluctuations affect the dynamics of FIV within cat populations? The main focus of this paper is to investigate how environment fluctuations introduced to linear growth rates affect disease's dynamics through studying the global dynamics of a general SI-type FIV model for both deterministic and the corresponding stochastic version. The rest of

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this article is organized as follows: In Section 2, we derive a general SI FIV deterministic model (or without noise) and its corresponding stochastic version (or with noise), and illustrate the main results. In Section 3, we give the proofs of the main results in details. In Section 4, we provide some numerical simulations to support our findings. In the last section, we provide a brief discussion and the summary of the main results. 2. Model derivations and main results Throughout this paper, unless otherwise specified, let ðΩ; fF t gt Z 0 ; PÞ be a complete probability space with a filtration fF t gt Z 0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets. 2.1. Model derivations Following the insightful work of [7,8,13], in this subsection, we establish our model which deals with the spread of FIV through the cat population under assumptions that include the possibility of multiple modes of transmission, that is, horizontal and vertical. First, we give the basic model assumptions as follows: (i) Assume that the total cat population N(t) is divided into two groups, susceptible (uninfected) cats S(t) and infected cats I(t), i.e., NðtÞ ¼ SðtÞ þ IðtÞ; (ii) With regard to the demographic parameters, we assumed that the density dependence acts primarily on mortality, hence the natural birth rate for susceptible cats and infectious cats can be assumed as constants b and bI, respectively. The ratio 0 r ρ≔ bbI r 1 describes the reduced reproductive ability of infected cats: ρ ¼ 0 means that infected cats lose their reproducing ability, that is, in this case, there is no vertical transmission; while ρ ¼ 1 indicates that they experience no reduction in reproductive fitness. And 0oρo1 means that the FIV is assumed to be transmitted to offspring and implies the vertical transmission. Hence, in this case, newborns of the infected are in the infectious class. (iii) Assume that the mortality rate of the cat population N is linearly related to N and has the bm is a positive real number, here form m þ kN, where m is the natural death rate and k ¼ K r≔b  m is the intrinsic growth rate of cat populations N in the absence of resource limits, K is the carrying capacity of the population N. (iv) The infected cats suffer an additional FIV-related mortality α40, which shall be referred to as virulence. (v) Assume that the rate of horizontal transmission (the transmission of a bacterial, fungal, or viral infection between members of the same species that are not in a parent–child relationship) is proportional to the density of susceptible cats S times the density of infected cats I, with proportionality constant β, a “mass-action” assumption (or called bilinear incidence). Obviously, if β ¼ 0, there is no horizontal transmission. Such a two-component epidemiological system modelling the spread of the FIV is as follows: 8 dS > > < ¼ bS ðm þ kN ÞS  βSI; dt ð1Þ ; dI > > : ¼ βSI  αI þ bI I  ðm þ kN ÞI; dt with initial conditions Sð0Þ ¼ S0 40, Ið0Þ ¼ I 0 40.

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In the absence of FIV, from model (1), we can get: 0 1 dN N C B ¼ bN  ðm þ kN ÞN ¼ rN @1  A; b  m dt k this incorporates the carrying capacity K ¼ b k m. To incorporate the effect of environmental fluctuations, here we formulate the stochastic model by introducing two multiplicative noise terms into the growth equations of both the susceptible and infected populations. Following the approach of Mao et al. [21], in this paper, we assume that during the spreading of a disease, the birth rate for susceptible and infectious b, bI will fluctuate randomly around some average value due to continuous fluctuation in the environmental conditions. And we introduce randomness into the deterministic model (1) by perturbing b by b þ s1 ζðtÞ and bI by bI þ s2 ζðtÞ to obtain the following stochastic differential equations: 8 dS > > < ¼ ðb þ s1 ζ ðt ÞÞS  ðm þ kN ÞS βSI; dt ð2Þ ; dI > > : ¼ βSI  αI þ ðbI þ s2 ζ ðt ÞÞI  ðm þ kN ÞI; dt where ζðtÞ is a Gaussian white noise and characterized by: 〈ζðtÞ〉 ¼ 0;

〈ζðtÞζðt 0 Þ〉 ¼ δðt  t 0 Þ;

here 〈  〉 denotes ensemble average and δðÞ is the Dirac-δ function. s1 and s2 denote the intensity of environmental forcing. Now we can rewrite model (2) into the form of stochastic differential equations as follows: ( dSðtÞ ¼ ½bS ðm þ kNÞS  βSIdt þ s1 SdB1 ðtÞ; ; ð3Þ dIðtÞ ¼ ½βSI þ bI I  ðm þ kNÞI  αIdt þ s2 IdB2 ðtÞ: where B1 ðtÞ and B2 ðtÞ are two independent standard one-dimensional Wiener processes defined   over the complete probability space Ω; F ; fF t gt Z 0 ; P , the relations between the white noise terms and Wiener process are defined by dBi ðtÞ ¼ si ζðtÞdtði ¼ 1; 2Þ where the differentials stand for pathwise derivative. 2.2. Main results One of our main goals in the paper is to derive the FIV dynamics of the deterministic model (1). Following Driessche and Watmough [32], set the basic demographic reproduction number βK T dd ; K ¼ bI  m  α: ð4Þ 0 ¼ b þ α  bI Theorem 2.1. 2 (i) If T dd 0 o0 or r≔b  mo0, then for any given initial value ðS0 ; I 0 ÞA Rþ , the whole cat population goes to extinction. That is, ðSðtÞ; IðtÞÞ tend to ð0; 0Þ as t-1. (ii) If 0oT dd 0 o1, then the disease-free steady state DFE ¼ ðK; 0Þ is globally asymptotically stable with respect to initial value ðS0 ; I 0 Þ AR2þ . (iii) If T dd 0 41,

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(iii-1) if K 40,   (a) when β Kk 4b þ α  bI , then the susceptible-free steady state SFE ¼ 0; Kk is globally asymptotically stable with respect to initial value ðS0 ; I 0 ÞA R2þ ; (b) when β K ob þ α  bI , then there exists a unique endemic state, En k   dd kðbþα  bI Þ  βK ðT 0  1Þðbþα  bI Þk ; β2 β2 2 ðS0 ; I 0 ÞA Rþ .

¼

, which is global asymptotic stability for initial data

  ðT dd  1Þðbþα  bI Þk (iii-2) if K r 0, then there exists a unique endemic state, E n ¼ kðbþα β2bI Þ  βK ; 0 , 2 β which is global asymptotic stability for initial data ðS0 ; I 0 ÞA R2þ . The proof of this theorem is similar to that in [13] and hence is omitted. The other goal of this paper is to study the FIV dynamics of the stochastic model (2). We define the horizontal transmission reproduction number [32,29] for the SDE model (3) as: Rhs ≔

2βðb  mÞ þ 2kbI  ks22 : 2kðb þ αÞ

ð5Þ

Furthermore, for simplicity to show the theoretical results, we define two threshold parameters as: R1 ≔b  m 

s21 2

ð6Þ

and R2 ≔bI  m  α 

s22 s2 ¼K 2: 2 2

ð7Þ

Theorem 2.2. (i) If R1 40, when β4k and Rhs o1, or β r k and R2 o0 hold, then for any given initial value ðS0 ; I 0 ÞA R2þ , the solution of the SDE model (3) obeys lim sup t-1

lim inf t-1

  log I log I R2 r ðb þ αÞ Rhs  1 o0 a:s: or lim sup r o0 a:s .. t t k t-1

1 t

Z

t

Sds Z

0

ð8Þ

R1 40 a:s ., k

namely, only the infected population I(t) goes to extinction almost surely. (ii) if R1 o0, (ii-1) (a) when β Z k and R2 40, or (b) βok and Rhs 41 hold, then for any given initial value ðS0 ; I 0 ÞA R2þ , the solution of the SDE model (3) obeys lim sup t-1

log S r R1 o0 a:s: t

and lim inf t-1

1 t

Z

t

Ids4 0

R2 40 a:s ., k

namely, only the susceptible S(t) goes to extinction almost surely.

ð9Þ

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(ii-2) when β4k and Rhs o1, or β r k and R2 o0 hold, then for any given initial value ðS0 ; I 0 ÞA R2þ , the susceptible S(t) and infectious I(t) of the SDE model (3) go to extinction almost surely. Theorem 2.3. (i) Assume β4k, if Rhs 41, R1 40 and R2 40 hold, then the solution ðSðtÞ; IðtÞÞ of the SDE model (3) has the following properties: Z 1 t R2 I ðsÞdsZ 40 a:s ., lim inf t-1 t 0 k Z 1 t ðb þ αÞðRhs  1Þ a:s ., ð10Þ lim sup I ðsÞds r k t-1 t 0 and 1 0olim sup t-1 t

Z

t

SðsÞds r

0

R1 k

a:s ..

ð11Þ

That is, the solutions of model (3) with respect to initial value ðS0 ; I 0 Þ A R2þ weakly persist in mean. (ii) If all assumptions in (i) hold, and assume that T dd 41, If K r 0 or K 40 and βK n n n n 2 1 n 2 1 n 2 k ob þ α bI , and M 1 ¼ 2 S s1 þ 2 I s2 okðS þ I Þ , where ðS ; I Þ is the endemic equilibrium of model (1), then there exists a stationary distribution μðÞ for the SDE model (3).

3. Proofs of main results The following definitions are commonly used and we list them below. Definition 3.1 ([33]). The population x(t) is said to go to extinction if limt-1 xðtÞ ¼ 0: Definition 3.2 ([33]). The population x(t) is said to be weakly persistent in mean if Z 1 t xðsÞds40: lim sup t-1 t 0 Definition 3.3 ([33]). The population x(t) is said to be strongly persistent in mean if Z 1 t lim inf xðsÞds40: t-1 t 0 Lemma 3.4 ([33]). Let xðtÞ A C½Ω  ½0; 1Þ; ð0; 1Þ. (i) If there exist positive constants λ, γ such that Z t log xðtÞr λt  γ xðsÞds þ FðtÞ; a:s: 0

for all t Z 0, where F A C½Ω  ½0; 1Þ; R and limt-1 FðtÞ t ¼ 0, then Z t 1 λ xðsÞdsr a:s .. lim sup t γ t-1 0

ð12Þ

J. Li et al. / Journal of the Franklin Institute 353 (2016) 4191–4212

(ii) If there exist positive constants λ, γ such that Z t xðsÞds þ FðtÞ; a:s: log xðtÞ Z λt  γ

4197

ð13Þ

0

for all t Z 0, where F A C½Ω  ½0; 1Þ; R and limt-1 FðtÞ t ¼ 0, then Z t 1 λ a:s .. xðsÞds Z lim inf t-1 t 0 γ 3.1. Existence and uniqueness of the global positive solution Theorem 3.5. For any initial condition ðS0 ; I 0 ÞA R2þ , there is a unique solution ðSðtÞ; IðtÞÞ of the SDE model (3) for all t Z 0 and the solution will remain in R2þ with probability one, namely, ðSðtÞ; IðtÞÞ A R2þ for all t Z 0 almost surely. Proof. Consider the diffusion process, corresponding to the SDE system (3), as follows:   1 2 u ν ν du ¼ b  m  k ðe þ e Þ βe  s1 dt þ s1 dB1 ðt Þ; 2   1 2 u u ν dν ¼ βe þ bI  m  kðe þ e Þ α  s2 dt þ s2 dB2 ðt Þ; ð14Þ 2 subjected to the initial condition uð0Þ ¼ log S0 ; νð0Þ ¼ log I 0 . The functions involved with drift part of above stochastic differential system satisfy the linear growth condition and they are locally Lipschitz. Hence there is a unique local solution ðuðtÞ; νðtÞÞ, for t A ½0; τe Þ where τe is any finite positive real number. Clearly, the unique positive local solution of model (3) starting from an interior point of the first quadrant R2þ is given by SðtÞ ¼ euðtÞ ; IðtÞ ¼ eνðtÞ . To show that this solution is global in R2þ , we need to show that τe ¼ 1 a:s:. We choose a sufficiently large non-negative number r0 such that both of S0 and I0 lie within the interval h i 1 r0

; r 0 . For each integer r Z r 0 , we can define the stopping time

   1 1 ; r or I ðt Þ= 2 ;r ; τr ¼ inf t A ½0; τe Þ : Sðt Þ2 = r r

where inf∅ ¼ 1 (as usual ∅ denotes the empty set). Clearly, τr is increasing as r-1. Set τ1 ¼ limt-1 τr , then τ1 r τe a:s:. In the following, we need to show that τe ¼ 1 a:s:. If this statement is violated, there exists two constants T40 and ϵA ð0; 1Þ such that Pfτ1 r Tg4ϵ:

ð15Þ

Hence we can find an integer r 1 Z r 0 such that Pfτr r Tg Z ϵ: for all r Z r 1 . Define a C2-function V : R2þ -Rþ by VðS; IÞ ¼ ðS þ 1  log SÞ þ ðI þ 1  log IÞ:

ð16Þ

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Considering that ðz þ 1 log zÞ Z 0 for all z40, the function VðÞ is positive definite for all ðS; IÞ A R2þ . Calculating the differential of V along the solution trajectories of the system (3) by using Itô's formula, we get    1 1 dV ¼ ðbS  ðm þ kðS þ I ÞÞS βSI Þ 1 þ ðβSI þ bI I  ðm þ kðS þ I ÞÞI  αI Þ 1 S I

 s21 þ s22 þ dt þ s1 ðS 1ÞdB1 þ s2 ðI  1ÞdB2 : 2

The positivity of S(t) and I(t) implies that   s21 þ s22 dV r bS  b þ 2m þ 2kðS þ I Þ þ βI þ bI I  bI þ α þ dt 2 þs1 ðS  1ÞdB1 þ s2 ðI  1ÞdB2 : r

  s2 þ s22 þ ð2k þ bÞS þ ðβ þ 2k þ bI ÞI dt þ s1 ðS 1ÞdB1 þ s2 ðI  1ÞdB2 : 2m þ α þ 1 2

ð17Þ We define two positive constants c1 ¼ 2m þ α þ

s21 þ s22 ; 2



c2 ¼ max 4k þ 2b; 4k þ 2β þ 2bI :

For z Z 0, using the fact z r 2ðz þ 1  log zÞ  ð4 2log 2Þ; then we get ðb þ 2kÞS þ ðβ þ bI þ 2kÞI r ð2b þ 4kÞðS þ 1 log SÞ þ ð4k þ 2β þ 2bI ÞðI þ 1  log IÞ r c2 V:

ð18Þ Considering Eqs. (17) and (18), we can write dV r ðc1 þ c2 VÞ dt þ s1 ðS  1Þ dB1 þ s2 ðI  1Þ dB2 : Choose c3 ¼ maxfc1 ; c2 g, then dV r c3 ð1 þ VÞ dt þ s1 ðS 1Þ dB1 þ s2 ðI  1Þ dB2 : Therefore, for t 1 r T, by doing the integration on both sides of inequality from 0 to τr 4 t 1 , we get Z τr 4 t 1 Z τr 4 t 1 Z τr 4 t 1 Z τr 4 t 1 dV r c3 ð1 þ VÞdt þ s1 ðS  1ÞdB1 þ s2 ðI  1ÞdB2 ; 0

0

0

0

where τr 4 t 1 ¼ minfτr ; t 1 g. Using the property of Itô's integral formula [16], we get from the above inequality Z τr 4 t1 VðSðτr 4 t 1 Þ; Iðτr 4t 1 ÞÞ r VðS0 ; I 0 Þ þ c3 ð1 þ VÞdt: 0

Taking expectation of both sides of the above inequality and applying Fubin's theorem

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[16,34], we can get

Z

4199

τr 4 t 1

EVðSðτr 4 t 1 Þ; Iðτr 4 t 1 ÞÞ r VðS0 ; I 0 Þ þ c3 E

ð1 þ VÞdt Z τr 4 t 1 r VðS0 ; I 0 Þ þ c3 t 1 þ c3 E Vdt 0 Z τr 4 t 1 VðSðτr 4 t 1 Þ; Iðτr 4 t 1 ÞÞdt r VðS0 ; I 0 Þ þ c3 T þ c3 E Z τ0r 4 t1 EVðSðτr 4 t 1 Þ; Iðτr 4 t 1 ÞÞdt: ¼ VðS0 ; I 0 Þ þ c3 T þ c3 0

0

Using Gronwall's inequality [23], we get from the inequality above EVðSðτr 4 t 1 Þ; Iðτr 4 t 1 ÞÞ r c4 ;

ð19Þ

where c4 ¼ ðVðS0 ; I 0 Þ þ c3 TÞec4 T : Set Ωr ¼ fτr r Tg for r Z r 1 . So by Eq. (16), we have PðΩr ÞZ ϵ. Note that for every ω A Ωr , there is at least one of Sðτr ; ωÞ, Iðτr ; ωÞ which is equal to either r or 1r and hence  1 1 þ 1  log V ðSðτr Þ; I ðτr ÞÞZ ðr þ 1 log r Þ4 : r r Therefore from Eqs. (15) and (19), it follows that      1 1 þ 1 log c4 Z E 1Ωr ðωÞ V ðSðτr Þ; I ðτr ÞÞ Z ϵ ðr þ 1 log r Þ4 ; r r where 1Ωr stands for the indicator function of Ωr. Letting r-1, we get 14c4 ¼ 1 which leads us to a contradiction. So we must have τ1 ¼ 1. This completes the proof. □ 3.2. Proof of Theorem 2.2 In this section, we give the proof of Theorem 2.2, namely, the stochastic extinction. First of all, we give the proof of the extinction of infectious cats I(t), i.e, Theorem 2.2(i). 3.2.1. Stochastic extinction of the infectious cats I(t) Proof. For simplicity, here we only prove the case of (i-2) follows similarly. R (i-1), and the proof 1 t bm If β4k, it is easy to verify that lim sup ð S ð t Þ þ I ð t Þ Þdsr t-1 t 0 k implies for any 8 ε40, Rt then there is a TðωÞ40 such that 1t 0 ðSðt Þ þ I ðt ÞÞds r b k m þ ε for t Z TðωÞ. Then Z 1 t b m þ ε: ð20Þ Sðt Þdsr t 0 k By the Itô's formula, we have  s2 dlog I ðt Þ ¼ βS þ bI  m  kðS þ I Þ  α  2 dt þ s2 dB2 ðt Þ: 2 Integrating it from 0 to t and then dividing both sides by t on the both sides, we have Rt Rt Rt  k ðS þ IÞds s2 dB2 ðsÞ log I log I 0 β 0 Sds s2 þ bI  m  α  2  0 þ 0 ; ¼ þ t t t t t 2

ð21Þ

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Rt Rt  Sds k Ids s2 þ bI  m  α  2  0 þ ϕ1 ð t Þ r ðβ  k Þ 0 t t 2  b m s2 þ ε þ bI  m  α  2 þ ϕ1 ðt Þ; r ðβ  k Þ 2 R kt

ð22Þ

s2 dB2

where ϕ1 ðt Þ ¼ logt I 0 þ 0 t , and limt-1 ϕ1 ðtÞ ¼ 0 a:s: according to the strong law of large numbers for martingales [23]. According to Lemma 3.4 (i) and ε arbitrary, we can get   log I βðb mÞ s2 r þ bI b α 2 ¼ ðb þ αÞ Rhs 1 o0 a:s .. lim sup ð23Þ t k 2 t-1 so, there exists some constants λ40 such that lim sup t-1

  log I βðb mÞ s2 r þ bI b α 2 ¼ ðb þ αÞ Rhs 1 r λo0 a:s .. t k 2

Hence, lim supt-1 logt I r  λ implies that for any ε1 40, there is a Tðω1 Þ40 such that r  λ þ ε1 for all t Z Tðω1 Þ. Hence we have

log I t

IðtÞr e  λtþε1 :

ð24Þ

By the Itô's formula, we have  s2 dlog S ¼ b  m  kðS þ I Þ  βI  1 dt þ s1 dB1 : 2 Integrating it from 0 to t and dividing by t on the both sides, then from Eq. (24), we have Rt  Z Z s1 dB1 ðsÞ log S log S0 s21 k t β t ; ¼ þ bm ðS þ I Þds  Ids þ 0  t t t 0 t 0 t 2 R t  λtþε Rt  Z 1 ds k t s1 dB1 ðsÞ log S0 s21 0e Z  ; þ bm Sds þ 0  ðk þ β Þ t t t 0 t 2 Z s2 k t Zbm 1  Sds þ ϕ2 ðt Þ; ð25Þ t 0 2 where ϕ2 ð t Þ ¼

log S0  ðk þ β Þ t

Rt 0

e  λtþε1 ds þ t

Rt 0

s1 dB1 ðsÞ : t

As we can choose ε1 small enough, according to the strong law of large numbers for martingales [23], we get limt-1 ϕ2 ðtÞ ¼ 0 a:s:. From Lemma 3.4 (ii), we have Z 1 t R1 Sds Z 40: lim inf t-1 t 0 k This completes the proof of (i).

□

Remark 3.6. Theorem 2.2 talks about the case that only infected compartment I(t) goes to extinction almost surely. Therefore a natural question arises: in this situation, what will be the limiting magnitude of the susceptible compartment S(t)?

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In fact, when Pflimt-1 IðtÞ ¼ 0g ¼ 1, then from the SDE (3), we get the following limiting equation: dS ¼ ½ðb  mÞS  kS2 dt þ s1 SdB1 : The distribution of S(t) converges to a stationary distribution which has a density function [15]  2ðb  mÞ=s21  1 2k 2ðb  mÞ  2  2kS s2 2 2 S s1 e s1 ; PðSÞ ¼ 1 2ðb mÞ Γ 1 s21 R1 for s21 o2ðb  mÞ, where ΓðμÞ ¼ 0 t μ  1 e  t dt, hence the mean of density P(S) is b  m s21  : k 2k It is easy to verify that,  b m s21 bm  ; lim EðSÞ ¼ lim ¼ s1 -0 s1 -0 k k 2k E ð SÞ ¼

which means that the mean of S(t) will tend to K ¼ b k m. That is, the dynamics of the disease-free situation for the stochastic model is same as the result provided in Theorem 2.1(iii)(a) for the deterministic model (1). 3.2.2. Stochastic extinction of the susceptible cats S(t) We now focus on the proof of Theorem 2.2(ii-1). Proof. Here also we will prove the case (ii-1)(a) only, the proof of (b) can be carried out in similar fashion. If β Z k, from Eq. (22), we get Rt  Z Z s2 dB2 ðsÞ log I log I 0 β t s22 k t ¼ þ Sds þ bI  m  α  ðS þ I Þds þ 0  t t t 0 t 0 t 2 Z t 2 s k Z bI  m  α  2  Ids þ ϕ1 ðt Þ; ð26Þ t 0 2 Rt s2 dB2 where ϕ1 ðt Þ ¼ logt I 0 þ 0 t , and limt-1 ϕ1 ðtÞ ¼ 0 a:s:. From Lemma 3.4 (ii), we get Z s2 bI  m  α  22 1 t R2 ¼ 40 a:s .. ð27Þ Ids Z lim inf t-1 t 0 k k On the other hand, considering (25) and (27) together, we get Rt  Z Z s1 dB1 ðsÞ log S log S0 s21 k t β t ; ¼ þ b m  ðS þ I Þds Ids þ 0  t t t 0 t 0 t 2 s2 r b m  1 þ ϕ3 ðt Þ; R2t s1 dB1 ðsÞ log S0 , and limt-1 ϕ3 ðtÞ ¼ 0 a:s:, we have where ϕ3 ðt Þ ¼ t þ 0 t lim sup t-1

log S s2 r b m  1 ¼ R1 o0 t 2

The proof of the first part is completed.

a:s .. □

ð28Þ

ð29Þ

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Remark 3.7. Theorem 2.2(ii-1) shows the susceptible-free dynamics of the SDE model (3). If limt-1 SðtÞ ¼ 0 a:s:, we can get the limiting equation dI ¼ ½ðbI  m  αÞI  kI 2 dt þ s2 IdB2 : And the distribution of the process I(t) converges to a stationary distribution which has a density [15]  2ðbI  m  αÞ=s22  1 2k s2 2 2 I 2ðbI  m  αÞ=s2  2 e  2ks2 I ; PðI Þ ¼  2 2ðbI  m  αÞ Γ 1 s22 R1 for s22 o2ðbI  m  αÞ, where ΓðμÞ ¼ 0 t μ  1 e  t dt, hence the mean of density P(I) is E ðI Þ ¼

bI  m  α s22  : k 2k

It is easy to verify the limiting dynamics as follows:  bI  m  α s22 bI  m  α lim E ðI Þ ¼ lim  : ¼ s2 -0 s2 -0 k k 2k Here also the susceptible-free dynamics within the stochastic setup is analogous to the deterministic dynamics, the relevant result is provided at Theorem 2.1(iii)(b). Remark 3.8. The proof of Theorem 2.2(ii-2) (i.e., the property of stochastic extinction of the cat population N(t)) is similar to that in Theorem 2.2(i) or (ii-1) and hence omitted. 3.3. Proof of Theorem 2.3 Now, we give the proof of Thereom 2.3, namely, the persistence of the infectious I(t) and the susceptible S(t) (or called endemic) and the existence of the stationary distribution. Firstly, we prove Theorem 2.3(i). 3.3.1. Persistence of the infectious cats and the susceptible cats Proof. Assume β4k, Set V ¼ log I, by Itô's formula for Eq. (3), we have Rt  Z Z s2 dB2 ðsÞ log I log I 0 β t s2 k t ¼ þ Sds þ bI  m  α  2  ðS þ I Þds þ 0 t t t 0 t 0 t 2 Rt  Z t Z t 2 s2 dB2 ðsÞ log I 0 ðβ  kÞ s k ¼ þ Sds þ bI  m  α  2  Ids þ 0 t t t 0 t 2 0  Z s22 k t Z bI  m  α  Ids þ ϕ1 ðt Þ; ð30Þ  t 0 2 Rt s2 dB2 where ϕ1 ðt Þ ¼ logt I 0 þ 0 t and limt-1 ϕ1 ðtÞ ¼ 0 a:s:. Then from Lemma 3.4 (ii), we get Z t 1 bI  m  α  s22 =2 R2 ¼ 40 a:s .. I ðsÞdsZ lim inf t-1 t 0 k k

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In addition,

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Rt  Z s2 dB2 ðsÞ s2 k t Sds þ bI  m  α  2  ðS þ I Þds þ 0 t t 2 0 0 Rt Z t Z t 2 s2 dB2 ðsÞ log I 0 ðβ  kÞ s k þ r Sds þ bI  m  α  2  Ids þ 0 t t t t 2 0 Rt 0 Z t 2 s2 dB2 ðsÞ log I 0 bm s2 k þ ε þ bI  m  α   r þ ðβ  k Þ Ids þ 0 t k t 0 t 2 Z t 2 βðb  mÞ s k þ bI  b α 2  r Ids þ ϕ4 ðt Þ; ð31Þ k t 0 2 Rt s2 dB2 ðsÞ , and when ε is small enough, according to the strong where ϕ4 ðt Þ ¼ logt I 0 þ ðβ  k Þε þ 0 t law of large numbers for martingales [23], we have log I log I 0 β ¼ þ t t t

lim ϕ4 ðtÞ ¼ 0

Z

t

a:s

t-1

..

From Lemma 3.4 (i), we have βðb  mÞ Z þ bI  b α s22 =2 ðb þ αÞðRh  1Þ 1 t k s lim sup ¼ I ðsÞdsr k k t-1 t 0

a:s ..

And according to Eq. (25), we can get Rt  Z Z s1 dB1 ðsÞ log S log S0 s21 k t β t ¼ þ b m  ðS þ I Þds Ids þ 0  t t t 0 t 0 t 2 Rt Z t 2 s dB ðsÞ log S0 s k 1 1 þ bm 1  r Sds þ 0 t t 0 t 2 Z s2 k t r b m  1  Sds þ ϕ2 ðt Þ; t 0 R2t where ϕ2 ðt Þ ¼ can get 1 lim sup t-1 t

log S0 t

Z

t 0

þ

0

s1 dB1 ðsÞ t

SðsÞdsr

ð32Þ

ð33Þ

and limt-1 ϕ2 ðtÞ ¼ 0 a:s:. Then according to Lemma 3.4 (i), we s21 2 ¼ R1 a:s .. k k

b m 

Besides, taking the limit superior in Eq. (25), we can get Rt  Z Z s1 dB1 ðsÞ log S=S0 s21 k t ðk þ βÞ t : ¼ b m  Sds  Ids þ 0  t t 0 R t t 2 0 t

ð34Þ

s1 dB1 ðsÞ

0 In fact, lim supt-1 log S=S r 0, limt-1 0 t ¼ 0 a:s:, hence t Z t Z t k ðk þ βÞ Sds þ Ids t 0 t 0 Rt Z Z s1 dB1 ðsÞ log S k t ðk þ βÞ t þ Z lim sup Sds þ Ids  lim inf 0 t-1 t t t 0 t t-1 0 s21 Zbm ¼ R1 40: 2

ð35Þ

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Thus

Z 1 t Sds40 a:s .. t-1 t

0 Rt  Sðs;ω1 Þds And for 8 ω1 A lim supt-1 0 t ¼ 0 , it follows from Eq. (35) that Rt Iðs; ω1 Þds 40: lim sup 0 t t-1 lim sup

On the other hand, taking the limit superior in Eq. (22) and then using lim supt-1 we can get Z log Iðt; ω1 Þ s2 1 t r bI  m  α  2  klim inf I ðsÞds r 0: lim sup t-1 t 0 t 2 t-1 Consider Iðs; ω1 ÞZ 0, then Rt Iðs; ω1 Þds ¼ 0: lim sup 0 t t-1 This is a contradiction. Therefore, we must have lim supt-1 1t The proof is completed. □

Rt 0

Rt 0

SðsÞds t

¼ 0,

ð36Þ

SðsÞds40 a:s:.

Theorem 3.9. Assume β r k, If Rhs 41, R1 40 and R2 40 hold, then the solutions of model (3) with respect to initial value ðS0 ; I 0 Þ A R2þ are weakly persistent in mean. The proof of Theorem 3.9 is similar to that in Theorem 2.3(i) and hence is omitted. 3.3.2. Existence of stationary distribution For the sake of understanding of the stochastic endemic dynamics of the SDE model (3), we now focus on the existence of stationary distribution. We firstly state a useful lemma from Ref. [35] which will be useful to obtain the main result. Let X(t) be homogeneous Markov process defined over the l-dimensional Euclidean space, denoted by El and is described by the following system of stochastic differential equations: dXðtÞ ¼ AðXÞdt þ

l X

f r ðXÞdBr ðtÞ:

ð37Þ

r¼1

The diffusion matrix is defined by [35]: AðxÞ ¼ ðaij ðxÞÞ;

aij ðxÞ ¼

l X r¼1

f ir f jr ðxÞ:

We assume there exists a bounded domain U  El with regular boundary Γ and satisfies the following properties: (H1) In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero. (H2) If xA E l ⧹U, the mean time τ at which a path emerging from x reaches the set U is finite, and supx A Q Ex τo1 for every compact subset Q  E l .

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Lemma 3.10 ([35]). If the assumptions (H1) and (H2) hold, then the Markov process X(t) has a stationary distribution μðÞ. Let gðÞ be a integrable function with respect to the measure μ, then

 Z Z 1 T Px lim gðX ðt ÞÞdt ¼ gðxÞμðdxÞ ¼ 1 T-1 T 0 El for all x A El . Remark 3.11. For model (3), the diffusion matrix is A ¼ diagðs21 S2 ; s22 I 2 Þ: Proof of Theorem 2.3(ii). Proof. Since T dd 41, K r 0 or K 40 and equilibrium point of model (1). Hence b m ¼ kðSn þ I n Þ þ βI n ;

βK k

ob þ α  bI , then there exists a positive

bI  m  α ¼ kðSn þ I n Þ βSn :

ð38Þ

Define a positive definite function V : R2þ -Rþ as follows:     VðS; IÞ ¼ S Sn  Sn log ðS=Sn Þ þ I  I n  I n log ðI=I n Þ  V 1 þ V 2 : Applying Itô's formula and using Eq. (38), we can get       1 n 2 n dV 1 ¼ S  S ðb  m  kðS þ IÞ βI Þ þ S s1 dt þ s1 S  Sn dB1 2    1     ¼ S  Sn kðSn þ I n Þ þ βI n  kðS þ IÞ βI þ Sn s21 dt þ s1 S  Sn dB1 2         1 n n n n 2 ¼ S  S kðS  SÞ  ðk þ βÞðI  I Þ þ S s1 dt þ s1 S  Sn dB1 2         1 n 2 2 n n n ¼  kðS  S Þ  ðk þ βÞ S  S I  I þ S s1 dt þ s1 S Sn dB1 ; 2 and

    1 I  I n ðβS þ bI  m  α  kðS þ IÞÞ þ I n s22 dt þ s2 I  I n dB2 2         1 n 2 n n n n ¼ I  I βS βS þ kðS þ I Þ  kðS þ IÞ þ I s2 dt þ s2 I  I n dB2 2      1 n 2   n 2 n n ¼  kðI  I Þ þ ðβ  kÞ S S I  I þ I s2 dt þ s2 I  I n dB2 : 2

dV 2 ¼

 

Therefore dV ¼ dV 1 þ dV 2 ¼ LVdt þ s1 ðS  Sn ÞdB1 þ s2 ðI  I n ÞdB2 ; where

ð39Þ

      LV ¼  kðS  Sn Þ2  ðk þ βÞ S  Sn I  I n  kðI  I n Þ2 þ ðβ  kÞ S Sn I  I n 1 1 þ Sn s21 þ I n s22 : 2 2

ð40Þ

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1 Set M 1 ¼ Sn s21 þ 12 I n s22 , then 2 LV ¼  kðS Sn Þ2  2kðS  Sn ÞðI  I n Þ kðI  I n Þ2 þ M 1 r  kðS Sn Þ2 þ 2kjS Sn JI  I n j  kðI  I n Þ2 þ M 1  2 ¼  k jS Sn j þ jI  I n j þ M 1 :

ð41Þ

Now if M1 satisfies the following condition: M 1 okðSn þ I n Þ2 ; then the ellipse  2 k jS  Sn j þ jI  I n j ¼ M 1 lies entirely in R2þ . We can take U to be a neighborhood of the ellipse with U DE 2 ¼ R2þ , where U is the compact closure of U. so for x AU⧹E 2 , LVo0, which implies conditions (ii) in Lemma 3.10. Besides, there is M ¼ minfs21 S2 þ s22 I 2 ; ðS; IÞA U g40 such that 2 X i;j ¼ 1

aij ξi ξj ¼ s21 S2 ξ2i þ s22 I 2 ξ2j Z Mjξj2 ;

for all ðS; IÞA U , ξA R2þ , which shows that conditions (i) of Lemma 3.10 is also satisfied. Therefore we can conclude that the SDE model (3) has a stationary distribution μðÞ.

□

4. Numerical simulations and dynamics comparisons In this section, we provide some numerical simulation results to compare disease dynamics resulting from the deterministic model (1) (in the case of s1 ¼ s2 ¼ 0 in model (3)) with its stochastic counterpart (3) for same set of parameter values. Here we use the Milstein's method [36] to simulate the stochastic model (3). The numerical scheme for the Milstein's method applied to the stochastic model (3) under considerations is given by, 8  pffiffiffiffiffi s21  2      > > < Sjþ1 ¼ Sj þ bSj  m þ k Sj þ I j Sk  βSj I j Δt þ s1 ϵ1j Δt þ Δt ϵ1j  1 ; 2  p ffiffiffiffiffi       s2  > > : I jþ1 ¼ I j þ βSj I j þ bI I  m þ k Sj þ I j I j  αI Δt þ s2 ϵ2j Δt þ 2 Δt ϵ22j  1 : 2 where ϵ1j and ϵ2j are two independent Gaussian random variables Nð0; 1Þ for j ¼ 1; 2; …; n. The parameter values are chosen as follows (see, Table 1 for details). In rural areas, preliminary observations led us to a value of 2.4 for fecundity b and the mortality rate m, per cat, at 0.6 year. The virulence α ¼ 0:2 year  1 is kept constant. The carrying capacity of the habitat is K ¼ 46 [8,37]. Consider the vertical transmission, the transmission coefficient has been estimated to be approximately 3:0=Nð0Þ year  1. And the initial values Nð0Þ can be adopted the data for

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rural cat populations (Barisey-la-Côte) in France (see, Table 1 in [8]): ðSð0Þ; Ið0ÞÞ ¼ ð30; 3ÞA R2þ :

ð42Þ

3  0:09091: 33 Under these parameters above, we can easily calculate that if bI Z 1:34, model (1) has no positive equilibrium. Hence, we can conclude that 0obI o1:34. As an example, we adopt bI ¼ 1.0 in the numerical simulations. For this choice of the parameters, we can calculate for the deterministic model (1), T dd 0 ¼ 2:6137, K ¼ bI  m  α ¼ 0:240 and β Kk ¼ 0:4647ob þ α  bI ¼ 1:6. Hence, according to Theorem 2.1 (iii), for model (1), all the trajectories eventually approach towards the endemic equilibrium point E n ¼ ð5:3756; 12:2244Þ which is globally asymptotically stable. In addition, the susceptible-free equilibrium SFE ¼ ð0; 5:1111Þ and the disease-free equilibrium DFE ¼ ð46; 0Þ are saddle points, ð0; 0Þ is a nodal point (see, Fig. 1). Next we focus on the role of noise strength on the resulting dynamics for the SDE model (3). For simplicity, we fix s1 ¼ 0:1, and consider three different values of s2: 0.1, 0.5 and 2.3. The corresponding values of Rhs are 1.9911, 1.9449 and 0.9757, respectively. By Theorem 2.2 we can know that the FIV is persistent in the first two cases whilethe infectious I(t) is extinction in the last case. The numerical simulations shown in Figs. 2 clearly support these results. Fig. 2(Left) shows the evolution of a single path of I(t) for system (3) and its corresponding deterministic model with three different values of s2: 0.1, 0.5 and 2.3. One can see that, in the case of FIV persistence, i.e., s2 ¼ 0:1 (see Fig. 2(a)) and 0.5 (see Fig. 2(b)), after some initial transients the population density fluctuates around the deterministic steady state values I n ¼ 12:2244. The irregularity of stochastic variation and range of fluctuation increases with the increase in the magnitude of and s2. In Fig. 2(c), one can see that large noise s2 ¼ 2:3 induce the extinction of I(t). And in Fig. 2(Right), we show the probability densities of the values of the path I(t) at time t¼ 100 for system (3) based on 10 000 stochastic simulations, and the numerical method for them can be found in [29]. One can find that, in the persistent cases s2 ¼ 0:1 (see Fig. 2(a)) and s2 ¼ 0:5 (see Fig. 2(b)), I(t) distributes around the mean values I n ¼ 12:2244, and the solution to the SDE model (3) for higher s2 (e.g., s2 ¼ 0:5 that the amplitude of fluctuation is remarkable and the distribution of the solution is skewed, while for lower s2 (e.g., s2 ¼ 0:1, the amplitude of fluctuation is slight and the oscillations are more symmetrically distributed. More precisely, when s2 ¼ 0:1, the distribution appears closer to a normal distribution (see Fig. 2(a)),

Hence, Nð0Þ ¼ Sð0Þ þ Ið0Þ ¼ 33 and β ¼

Table 1 Parameter values in numerical simulations for model (3). Parameters

Value range

Reference

b: The birth rate for susceptible cats bI : The birth rate for infectious cats m: The natural death rate K: The carrying capacity of the habitat α: The FIV-related mortality β: The transmission coefficient

2.4 year  1 1.0 year  1 0.6 year  1 46 km  2 0.2 year  1 3:0 year  1 Nð0Þ

[8,7,37] Estimated [8,7,37] [8] [8,7] [8,7]

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E*

I

12 10 8 6

SFE

4 2 0

DFE

(0,0) 0

10

20

S

30

40

50

Fig. 1. The phase portrait of model (1). The parameters are taken as β ¼ 0:09091; b ¼ 2:4; k ¼ 0:03913; m ¼ 0:6; α ¼ 0:2; bI ¼ 1. The endemic equilibrium point En ¼ ð5:3756; 12:2244Þ is globally asymptotically stable. In this case, the susceptible-free equilibrium SFE ¼ ð0; 5:1111Þ and the disease-free equilibrium DFE ¼ ð46; 0Þ are all saddle points, and the extinction state ð0; 0Þ is a nodal point.

but as s2 increases to 0.5, the distribution is positively skewed (see Fig. 2(b)). Obviously, in these two persistent cases, the SDE model (3) has a stationary distribution. However, in the extinct case s2 ¼ 2:22, the mass of the distribution of I(t) is concentrated on the small neighborhood of zero (see Fig. 2(c)). Similar to that in [30], Fig. 2 indicates that the small environmental perturbations can generate the irregular cycling phenomena of recurrent diseases, while the large ones will eradicate diseases. This means the small perturbations of the white noise can sustain the irregular recurrence of FIV in cat populations between two pandemics, and larger ones may be beneficial, leading to the extinction of FIV.

5. Concluding remarks In this paper, we have considered the basic features of a simple SI-type epidemic model with varying size in presence of multiplicative noise terms to understand the effects of environmental driving forces on the disease dynamics. The value of this study lies in two aspects: Mathematically, for the SDE model (3), we have introduced three threshold quantities, the horizontal transmission reproduction number Rhs ≔

2βðb  mÞþ2kbI  ks22 , 2kðbþαÞ

and R1 ≔b  m  s21 =2,

R2 ≔bI  m  α  s22 =2. These three threshold parameters are utilized in identifying the stochastic extinction and persistence (c.f., Theorem 2.2) for the stochastic model (3). In the case of stochastic persistence, we have proved the existence of a stationary distribution (c.f., Theorem 2.3). For the sake of better understanding the stochastic disease dynamics of the SDE model (3), we summarize the theoretical results in the case of β ¼ k in the R1 R2 -plane and presented them in Fig. 3. Epidemiologically, we partially provide the effects of the environment fluctuations on the FIV spreading to the SDE model (3).

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Fig. 2. (Left) The evolution of a single path of I(t) for system (3) and its corresponding deterministic model. (Right) Probability densities of the values of the path I(t) for system (3) based on 10 000 stochastic simulations.

1. Large environment fluctuations can suppress the outbreak of disease: For the deterministic model (1), Theorem 2.1 shows that model (1) admits a unique endemic equilibrium En which h dd is globally asymptotically stable if T dd 0 41. Notice that Rs oT 0 , thus it is possible that h dd Rs o1oT 0 . This is the case when the deterministic model (1) is at endemic situation (see, Fig. 1) while the infectious I(t) is extinct in the stochastic model (3) with probability one (see Theorem 2.2 and Fig. 2(c)). This implies that large environment fluctuations can suppress the outbreak of FIV.

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Fig. 3. Schematic stochastic dynamics of extinction and endemic of the SDE model (3) in the case of β ¼ k in the ðR1 ; R2 Þplane.

2. The distributions governed by Rhs : When the noise intensities si ði ¼ 1; 2Þ are small enough and satisfy the conditions Rhs 41, R1 40 and R2 40, from Theorem 2.3 and Fig. 3, one can see that the environmental noise can drive the solutions of model (3) to oscillate significantly around the endemic steady state (see, Fig. 2), and slight increase of the amplitude of fluctuation leads to the oscillations more symmetrically distributed. In addition, as suggested by Theorems 2.3 and numerical simulation results presented in Fig. 2, the SDE model (3) has an endemic stationary distribution which leads to the stochastic persistence of the FIV disease. And Theorem 2.2 says that under the influence of strong noise intensity, the infectious I(t) will extinct exponentially almost surely; and the susceptible cats S(t) will extinct exponentially almost surely. That is, the extinctions of I(t) or S(t) are induced by the environmental noise, and hence in these cases, we cannot ignore the effect of the noise. 3. The effect of the vertical transmission: In the case of S(t) going to extinct exponentially almost surely, from Theorem 2.2, one can see that 1 lim inf t-1 t

Z 0

t

bI  m  α  Ids4 k

s22 2

40 a:s ..

And from Remark 3.7, we know the distribution of the process I(t) converges to a stationary distribution which has a density P(I) and the mean of density P(I) is  bI  m  α s22 bI  m  α  : lim E ðI Þ ¼ lim ¼ s2 -0 s2 -0 k k 2k That is, the susceptible-free situation occurring in this case, which is similar to that in Theorem 2.1 for the deterministic model (1). Also, we can conclude that the susceptible-free situation is induced by the vertical transmission (i.e., bI 40).

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