Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response

Transcription

1 Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA IMA Workshop in Memory of George R. Sell June 016

2 Outline 1. Stochastic SIR model with the functional response. Positive invariant region 3. Stationary distribution 4. Pathwise and moment bounds 5. Moment Lyapunov exponent 6. Persistence and extinction

3 1. Stochastic Model Equations ( βxy dx = λ δx 1 + ax + by ( βxy dy = 1 + ax + by qy dz = (ky γz)dt + σ 3 z db 3 (t), ) dt + σ 1 x db 1 (t), ) dt + σ y db (t), (1) where the susceptible cells with density x(t) are generated at a constant rate λ, die at a density-proportional rate δx, and become infective cells y(t) at a rate βxy/(1 + ax + by) called Beddington-DeAngelis functional response. The density of recovered cells is z(t).

4 Beddington-DeAngelis Response Classical infectious and epidemic disease models use bilinear incidence or interaction terms such as βxy. The nonlinear incidence rate with this Holling type II response βxy/(1 + ax + by) was introduced separately by Beddington (1975) and DeAngelis (1975) in ecology. It was also used in the study of the interaction dynamics of HIV-1 virus and uninfected CD4 + T cells, cf. Huang et al. (009).

5 Deterministic Dynamics and Stability If the comprehensive reproduction ratio of the virus, R 0 = kβλ δqγ + aqγλ 1, the disease-free equilibrium E 0 = (λ/δ, 0, 0) is asymptotically stable. Disease Extinction. If R 0 > 1, the endemic equilibrium E 1 = (x 0, y 0, z 0 ) in R 3 + is globally asymptotically stable Persistence, where cf. Huang-Ma-Takeuch (009) qγ + kbλ x 0 = kβ aqγ + bkδ, kβλ aλqγ δqγ y 0 = q(kβ aqγ + bkδ), k(kβλ aλqγ δqγ) z 0 = qγ(kβ aqγ + bkδ).

6 Related Models and Researches Quite a few research work on stochastic population dynamics, in particular Lotka-Volterra type equations for predator-prey models whose growth/death rates perturbed by noise and adding various functional responses: bilibear response, ratio-dependent response, Leslie-Gower response, and Beddington-DeAngelis response on many topics: Khasminskii and Klebaner (001), Rudnicki (003), Rudnicki-Pichor (007), Deng et al (008), Ji-Jiang-Li (011), Mao (011), Ton-Yagi (011), Li-Shuang (013), Zou-Fan-Wang (013), etc. Diffusive predator-prey Model with Bedington-DeAngelis response: Chen-Wang (005).

7 Findings by Shown Results With stochastic perturbation, there will be no positive equilibrium points. Instead one can try to prove that (open problem) there exists a stationary distribution on the positive invariant region R n +, which can be viewed as a counterpart in the stochastic dynamics arena. If intensity of noise is relatively small, hopefully (not always) one can show that stochastic asymptotic dynamics imitates the corresponding deterministic dynamics in some way, almost surely, in moment, or in time average. If noise is sufficiently large, something dramatic and/or unexpected in dynamics may occur: extinction of all species, suppress or express exponential growth or explosion. Large random environment, weather, or epidemic diseases effect seems to be the decisive responsible for the extinction of some species in the nature.

8 Preliminaries Assume that B i (t) are independent standard Brownian motion defined on the canonical probability space (Ω, F, P). R n + = {x R n : x i > 0 for all 1 i n}. Itô formula: For V C,1 (R n [0, ), R + ), dv (X (t), t) = LV (X, t) dt + V x (X, t)g(x, t) db, for a SDE dx = f (X, t) dt + g(x, t) db, and LV (X (t), t) = V t (X, t) + V x (X, t)f (X, t) + 1 Trace [ g T (X, t)v xx (X, t)g(x, t) ].

9 . Global Existence of Positive Solutions Theorem Under the condition q k, for any positive initial data (x 0, y 0, z 0 ), there is a unique positive solution (x(t), y(t), z(t)) of the model SDE (1) such that the solution will remain in R 3 + for all t 0 with probability one. It shows that R 3 + is a positive invariant region for the model system. The solution X (t) = (x(t), y(t), z(t)) is a time-homogeneous Markov process. Let the maximal existence interval of a unique local pathwise positive solution be τ e. We show τ e = a.s. V (x, y, z) = (x 1 log x) + (y 1 log y) + (z 1 log z).

10 Sketch of Proof dv Kdt + σ 1 (x 1) db 1 + σ (y 1) db + σ 3 (z 1) db 3, K = 1 (σ 1 + σ + σ 3) + λ + δ + q + γ + β b. It suffices to prove τ = lim m τ m = a.s. { ( ) } 3 1 τ m = inf t [0, τ e ) : (x(t), y(t), z(t)) m, m. Otherwise, if τ <, then P{τ T } > ε. It leads to [ ε (m 1 log m) > V (x 0, y 0, z 0 ) + KT ( 1 m 1 + log m )].

11 3. Stationary Distribution Definition Let P X0,t( ) be the probability measure induced by a stochastic process {X (t)} t 0 in R n + (or R n ) over (Ω, F, P) with X (0) = X 0, P X0,t(S) = P{ω Ω : X (t, ω) S}, S B(R n +), If there is a probability measure µ( ) on the measurable space (R n +, B(R n +)) such that P X0,t( ) µ( ) as t, in distribution for any X 0 R n +, then we say that X (t) has a stationary distribution µ( ).

12 Khasminskii Assumption Assumption: There is a bounded open set U R n + with regular boundary and the following properties: 1. In a neighborhood of U, the smallest eigenvalue of the diffusion matrix g(x) is uniformly bounded away from zero.. For any x R n +\U, the mean time τ at which a path from x reaches the set U satisfies sup x K E x [τ] <, for every compact set K R n +.

13 Khasminskii Lemma Lemma Under the Khasminskii Assumption, the Markov process X (t) has a stationary distribution µ( ), µ(r n \ R n +) = 0. For any bounded continuous function Q(ξ), it holds that E[Q(X (t, ξ)] dµ(ξ) = Q(ξ) dµ(ξ), t 0. R n + For any integrable function F (X ) with respect to µ, the R n + ergodic property holds, { 1 T P lim F (X (t, ξ)) dt = T T 0 R n + F (ξ) dµ(ξ) } = 1.

14 The Existence of Stationary Distribution Theorem If there is a positive equilibrium point (x 0, y 0, z 0 ) R 3 + for the corresponding deterministic system of ODE, then there exists a stationary distribution with respect to the Markov process generated by the positive solutions (x(t), y(t), z(t)) of the system of SDE (1). Proof. Construct another V -function ) 1 + by 0 V (x, y, z) = (x x 0 x 0 log xx0 1 + ax 0 + by 0 +y y 0 y 0 log y + q ) (z z 0 z 0 log zz0. y 0 k

15 Sketch of Proof By the equilibrium conditiom we have where and LV (x(t), y(t), z(t)) = Q(x(t), x(t), z(t)) + σ σ = (1 + by 0)x 0 σ1 (1 + ax 0 + by 0 ) + y 0σ + qz 0σ3 > 0, k Q(x, y, z) = δ(1 + by 0) x(1 + ax 0 + by 0 ) (x x 0) qy 0 b(1 + ax)(y y 0 ) z 0 (1 + ax + by)(1 + ax + by 0 ) + qy 0R(x, y, z) 0, R(x, y, z) = 4 (1 + ax + by 0)x 0 (1 + ax 0 + by 0 )x (1 + ax 0 + by 0 )xy 0 z (1 + ax + by)x 0 yz 0 yz 0 y 0 z (1 + ax + by) (1 + ax + by 0 ) 0.

16 Continued Q(x, y, z) = 0, iff (x, y, z) = (x 0, y 0, z 0 ). Q(x, y, z) > 0, if otherwise in R 3 +. Q(x, y, z) as (x, y, z). The diffusion matrix g(x, y, z) = diag (σ 1x, σ y, σ 3z ) has the smallest eigenvalue uniformly bounded from zero, on a nbhd of any compact set in R 3 +\U. The two conditions in Khasminskii Assumption are satisfied for the bounded open set U = {(x, y, z) : Q(x, y, z) < σ} R 3 +. By Khasminskii Lemma, the system of SDE (1) has a unique stationary distribution µ( ) on R 3 +.

17 4. Moment and Pathwise Estimates Lemma For p > 1, the solution of the Bernoulli equation [ ) ] dv dt = p v(t) (δ σ (p 1) + λv 1p (t), v(0) = v 0, is given by where v(t) = [ v 1 p 0 e θpt + λ θ p ( 1 e θ pt )] p, θ p = δ σ (p 1).

18 Uniform bounds of the p-th Moments Theorem Suppose the following condition is satisfied, δ σ 1 (p 1) > 0, q σ (p 1) > 0, γ σ 3 (p 1) > 0. Then for p > 1 and all positive solution of the stochastic viral equations (1) with (x 0, y 0, z 0 ) R 3 +, lim sup t lim sup t lim sup t E[x p (t)] L 1 (p), E[y p (t)] L (p), E[z p (t)] L 3 (p). Remark. For p = 1, bounds are given thru linear equations.

19 Uniform Moment Bounds ( ) p λ L 1 (p) = δ σ 1 (p 1) ( ) p 4βbλ L (p) = b(δ σ1 (p 1))(q σ (p 1)) ( 8kβλ L 3 (p) = b(δ σ1 (p 1))(q σ (p 1))(γ σ 3 (p 1)) where δ, q, γ are death rates, λ, β, k are production or growth rates, and σ 1, σ, σ 3 are noise intensities. ) p

20 Sketch of Proof For positive solutions, d(x p ) px p 1 ( λ δx + σ 1 ) (p 1)x dt + px p σ 1 db 1 (t). de[x p (t)] dt ( ) p δ σ 1 (p 1) E[x p ] + pλ E[x p ] 1 1 p. Use Lemma for Bernoulli equations and comparison theorem. For the y-equation, the nonlinear term of functional response βxy 1 + ax + by βx b.

21 Pathwise Upper/Lower Bounds Theorem Every positive solution (x(t), y(t), z(t)) of the system (1) with (x 0, y 0, z 0 ) R 3 + satisfies Φ l (t) x(t) Φ u (t), Ψ l (t) y(t) Ψ u (t), Γ l (t) z(t) Γ u (t), t 0, a.s. where { ( } t δ+ β b Φ u (t) = λ exp + σ 1 )(t s)+σ 1 (B 1 (t) B 1 (s)) ds 0 +x 0 exp { ( δ+ β b + σ 1 )t+σ 1 B 1 (t) }

22 Continued Ψ u (t) = β b Γ u (t) = k t 0 + y 0 e t 0 + z 0 e ( ) Φ u (s)e q+ σ (t s)+σ (B (t) B (s)) ds ( ) q+ σ t+σ B (t), ( ) Ψ u (s)e γ+ σ 3 (t s)+σ 3 (B 3 (t) B 3 (s)) ds ( ) γ+ σ 3 t+σ 3 B 3 (t).

23 Absorbing Property in Time Average Theorem Under the same condition that the noise intensities are relatively small, δ σ 1 (p 1) > 0, q σ (p 1) > 0, γ σ 3 (p 1) > 0, all the positive solutions has the property a.s. 1 t lim x p (s) ds = ξ p dµ(ξ, η, ζ) L 1 (p), t t 0 R+ 3 1 t lim y p (s) ds = η p dµ(ξ, η, ζ) L (p), t t 0 R+ 3 1 t lim z p (s) ds = ζ p dµ(ξ, η, ζ) L 3 (p). t t 0 R 3 +

24 5. Moment Lyapunov Exponent Definition The p-th moment Lyapunov exponent of a pathwise solution X (t, X 0 ) of the SDE (1) is defined by Λ(p) = lim sup t 1 t log E X (t, X 0) p, p 1. Theorem Under the same assumption as above, for p 1, Λ(p) = lim sup t 1 t log E X (t, X 0) p 0, R 3 + for any positive solution X (t, X 0 ) = (x(t, x 0 ), y(t, y 0 ), z(t, z 0 )) of the stochastic viral equations (1) with X 0 R 3 +.

25 Sketch of Proof The p-th moment of the geometric Brownian motion for p > 1 is given by ds = δs(t) dt + σs(t) db E S(t) p = S(0) p exp [ ( ) ] (p 1) p σ δ t. Directly estimate: in view of log(1 + x) x for x > 0, lim sup t lim sup t + lim sup t 1 t log (E x(t, x 0) 1 ) lim sup t t log (E Φ u(t) ) ( 1 t log 1 + x ) 0 (δ σ1) exp{(σ λ 1 δ)t} 1 t log λ δ σ 1 = 0. when p =. Then bootstrap for the y and z components.

26 6. Persistence and Extinction For a deterministic or stochastic model in population dynamics for ecology or viral and epidemic dynamics for disease control, the two most important questions are the analysis and prediction of persistence and extinction. Definition A component x(t) of the system is said to be persistent in mean, if lim inf E[x(t)] > 0, t or persistent a.s. if lim inf t x(t) > 0. Other types of persistence and extinction can be defined.

27 Useful Lemma for Stochastic Predator-Prey Eqns Lemma Consider a one-dimensional SDE dx = X (t)(a bx (t)) dt + σx (t) db. If a > σ, then for any solution X (t) with X 0 > 0, one has 1 lim log X (t) = 0, a.s. t t and 1 t lim t t X (s) ds = a σ, a.s. 0 b Thus the solutions are persistent in time mean. But this does not work for the stochastic SIR model with the Beddington-DeAngelis functional response.

28 x(t) Φ u (t) Lemma lim 1 t log x(t) = 0 +x 0 exp λ δ + β + σ 1 b [ ( exp [σ 1 ( δ + β b + σ 1 B 1 (t) min B 1(s) 0 s t ) ] t + σ 1 B 1 (t) )] The running max of BM, B 1 (t) min 0 s t B 1 (s) = B 1 (t) in distribution, and log(1 + x) < x lead to lim sup t + lim sup t 1 t log Φ u(t) lim sup 1 t log ( 1 [σ 1 B 1 (t) min t t B 1(s) 0 s t [ 1 + x ] 0(δ + β/b + σ1/) exp (Ratio) λ )] = 0.

29 The Exponential Transformation Let x(t) = e u(t), y(t) = u v(t), z(t) = e w(t). The SDE of this stochastic viral model is converted to ( ( ) du = δ + σ 1 + λe u βe v 1 + a e u + be ( ( ) v dv = q + σ βe u a e u + be ( ( )) v dw = ke v w γ + σ 3 dt + σ 3 db 3. Here we see dv ) dt + σ db, ( ( )) β a q + σ dt + σ db. ) dt + σ 1 db 1,

30 Extinction of the Infectives Proposition If β/a < q + σ /, then lim t y(t) = 0. The infective cells tends to extinction. By the stochastic comparison theorem, cf. Ikeda and Watanabe (1981), p. 35, ( ( )) β v(t) v 0 + a q + σ t + σ B (t) ( ( )) β = v 0 + a q + σ σ B (t) t 1 + ( ( )), β q + σ a t by SLLN for Brownian motion. Then lim y(t) = lim e v(t) = 0.

31 Persistence of the Susceptible From the u-equation, ( ( ) du = δ + σ 1 + λe u βe v 1 + a e u + be ( ( v λe u δ + σ 1 + β )) dt + σ 1 db 1, b ) dt + σ 1 db 1, and dũ = (λe u θ) dt + σ 1 db 1, θ = δ + σ 1/ + β/b, has a unique stationary distribution whose probability density is a solution of the stationary Fokker-Planck equation The solution is 1 d p σ 1 dξ d dξ [( λe ξ θ ) p] = 0. p(ξ) = C exp where C is the normalizing const. { θ ξ + λ } e ξ, σ1 σ1

32 Persistence of the Susceptible Accordingly, cf. Rudnicki (003), Skorokhod (1987), u(t) ũ(t) which converges in distribution to a stationary solution ũ whoe probability density is p(ξ), as t. Proposition The susceptible cells will be persistent in mean and = C lim inf t 0 E[x(t)] e ξ p(ξ) dξ 0 {( exp 1 θ ) ξ + λ } e ξ dξ σ1 σ1 Cσ 1 σ 1 θ = Cσ 1 (δ + β/b) > 0.

33 Persistence of the Recovered Proposition Under the condition β/a < q + σ /, The recovered cells will be persistent in mean. From the w-equation ( ( )) dw = ke v w γ + σ 3 dt + σ 3 db 3 with y(t) = e v(t) 0 under the condition, the proof is similar. Many questions still unanswered: stability, bifurcation, stochastic pattern formation for SPDE models, etc.

34 Conclusion The study shows that under the condition q k the positive cone R 3 + is invariant with probability one for the pathwise solutions of this model of stochastic viral dynamics. Construction of Lyapunov functions shows that there is a unique stationary distribution whose probability density function is a steady state of the Fokker-Planck equation. Moments and pathwise upper/lower bounds and some asymptotic estimates are obtained, which make possible to discuss the persistence and extinction of the virus diseases. All these can be used for better reaveal of features of models of stochastic SIR equations with different functional responses and for better understanding of medical/epidemic incident events.

35 THANKS