DSpace at VNU: Classification of Asymptotic Behavior in a Stochastic SIR Model

Classification of Asymptotic Behavior in a Stochastic SIR Model∗N.. Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential

Classification of Asymptotic Behavior in a Stochastic SIR Model∗

N T Dieu†, D H Nguyen‡, N H Du§, and G Yin‡

Abstract Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of

stochastic differential equations with a degenerate diffusion, this paper provides sufficient conditions that are very close to the necessary ones for the permanence In addition, this paper develops ergodicity of the underlying system It is proved that the transition probabilities converge in total variation norm to the invariant measure Our result gives a precise characterization of the support

of the invariant measure Rates of convergence are also ascertained It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree.

Key words SIR model, extinction, permanence, stationary distribution, ergodicity

AMS subject classifications 34C12, 60H10, 92D25

DOI 10.1137/15M1043315

1 Introduction Since epidemic models were first introduced by Kermack andMcKendrick in [15, 16], the study of mathematical models has flourished Much attentionhas been devoted to analyzing, predicting the spread of, and designing controls of infectiousdiseases in host populations; see [1, 2, 4, 6, 9, 18, 19, 15, 16, 24, 26, 29] and the refer-ences therein One of the classic epidemic models is the SIR (susceptible-infected-removed)model which is suitable for modeling some diseases with permanent immunity such as rubella,whooping cough, measles, and smallpox In the SIR model, a homogeneous host population

is subdivided into the following three epidemiologically distinct types of individuals:

• (S) The susceptible class: those individuals who are capable of contracting the diseaseand becoming infected

• (I) The infected class: those individuals who are capable of transmitting the disease

to others

• (R) The removed class: infected individuals who are deceased, or have recovered andare either permanently immune or isolated

∗ Received by the editors October 9, 2015; accepted for publication (in revised form) by L Billings March 7, 2016; published electronically May 26, 2016.

http://www.siam.org/journals/siads/15-2/M104331.html

† Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam ( dieunguyen2008@gmail com ) The research of this author was supported in part by the Foundation for Science and Technology Development

of Vietnam’s Ministry of Education and Training under grant B2015-27-15.

‡ Department of Mathematics, Wayne State University, Detroit, MI 48202 ( dangnh.maths@gmail.com ,

DMS-1207667.

§ Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam ( dunh@vnu.edu.vn ) The research of this author was supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 101.03-2014.58.

1062

If we denote by S(t), I(t), and R(t) the number of individuals at time t in classes (S), (I),and (R), respectively, the spread of infection can be formulated by the following deterministicsystem of differential equations:

where α is the per capita birth rate of the population, µ is the per capita disease-free deathrate, ρ is the excess per capita death rate of the infective class, β is the effective per capitacontact rate, and γ is the per capita recovery rate of the infected individuals For the abovedeterministic model (1.1), if λd = βαµ − (µ + ρ + γ) ≤ 0, then the population tends to thedisease-free equilibrium (αµ, 0, 0); if λd> 0, the population approaches an endemic equilibrium.Thus, using the critical threshold value λd, the asymptotic behavior of the system has beencompletely classified In [29], similar results were given for a general epidemic model withreaction-diffusion in terms of basic reproduction numbers

It is well recognized that random effect is often not avoidable, and a population is alwayssubject to random disturbances Thus, it is important to investigate stochastic epidemicmodels A resurgent effort has been devoted to finding the corresponding classification bymeans of threshold levels Jiang et al [14] investigated the asymptotic behavior of globalpositive solution for the nondegenerate stochastic SIR model

where B1(t), B2(t), and B3(t) are mutually independent Brownian motions, and σ1, σ2, and

σ3 are the intensities of the white noise The model is more difficult to deal with comparedwith the deterministic counterpart Moreover, in reality, the classes (S), (I), and (R) areusually subject to the same random factors such as temperature, humidity, pollution, andother extrinsic influences As a result, it is more plausible to assume that the random noisesperturbing the three classes are from the same source If we assume that the Brownian motions

B1(t), B2(t), and B3(t) are the same, we obtain the model

which has been considered in [20] An important question is whether the transition to either adisease-free state or the disease state will become permanent In [20], the authors attempted

to answer this question for (1.3) for the cases σ1 > 0 and σ2 > 0 By using Lyapunov-typefunctions, they provided some sufficient conditions for extinction or permanence, as well asergodicity, for the solution of system (1.3) Using these same methods, the authors of [13,30]studied extinction and permanence in some different stochastic SIR models

In contrast to (1.2), (1.3) is more difficult to deal with due to the degeneracy of thediffusion Moreover, although one may assume the existence of an appropriate Lyapunovfunction, it is fairly difficult to find an effective Lyapunov function in practice Because of thesedifficulties, there has been no decisive classification for stochastic SIR models that is similar

to the deterministic case Our main goal in this paper is to provide such a classification Weshall derive a sufficient and almost necessary condition for permanence (as well as ergodicity)and extinction of the disease for the stochastic SIR model (1.3) by using a value λ, which issimilar to λd in the deterministic model Note that such results are obtained for a stochasticsusceptible-infective-susceptible (SIS) model in [10] However, the model studied there can bereduced to a one-dimensional equation that is much easier to investigate The method used in[10] cannot treat the stochastic SIR model (1.3) Estimation for the convergence rate is alsonot given in [10] Therefore, a more general method needs to be introduced The new methodnot only can remove most assumptions in [20] but also can treat cases σ1 > 0 and σ2 < 0,which have not been taken into consideration in [20] Note that cases σ1 > 0 and σ2 < 0indicate that the random factors have opposite effects on healthy and infected individuals.For instance, patients with tuberculosis or some other pulmonary disease do not endure well

in cool and humid weather, while healthy people may be fine in such weather In addition,individuals with a disease usually have weaker resistance to other diseases Our new method

is also suitable for dealing with other stochastic variants of (1.1) such as models introduced

in [5,13,30]

The rest of the paper is arranged as follows Section 2 derives a threshold that is used

to classify the extinction and permanence of the disease To establish the desired result,

by considering the dynamics on the boundary we obtain a threshold λ that enables us todetermine the asymptotic behavior of the solution In particular, it is shown that if λ < 0,the disease will decay at an exponential rate In case λ > 0, the solution converges to astationary distribution in total variation This means that the disease is permanent Therate of convergence is proved to be bounded above by any polynomial decay The ergodicity

of the solution process is also proved Finally, section 3 is devoted to some discussion andcomparison to existing results in the literature Some numerical examples are provided toillustrate our results

2 Threshold between extinction and permanence Let (Ω, F , {Ft}t≥0, P) be a completeprobability space with the filtration {Ft}t≥0satisfying the usual condition, i.e., it is increasingand right continuous while F0 contains all P-null sets Let B(t) be an Ft-adapted, Brownianmotion Because the dynamics of the recovered class has no effect on the disease transmissiondynamics, we consider only the following system:

(2.1)

(dS(t) = [α − βS(t)I(t) − µS(t)]dt + σ1S(t)dB(t),dI(t) = [βS(t)I(t) − (µ + ρ + γ)I(t)]dt + σ2I(t)dB(t)

Assume that σ1, σ2 6= 0 By the symmetry of Brownian motion, without loss of generality

we suppose throughout this paper that σ1 > 0 Using standard arguments, it can be easilyshown that for any positive initial value (u, v) ∈ R2,◦+ := {(u0, v0) : u0> 0, v0> 0}, there exists

a unique global solution (Su,v(t), Iu,v(t)), t ≥ 0, that remains in R2,◦+ with probability 1 (see,e.g., [14]), where the subscripts u and v denote the dependence on the initial data (u, v) To

obtain further properties of the solution, we first consider the equation on the boundary,

Let bSu(t) be the solution to (2.2) with initial value u It follows from the comparison theorem[11, Thm 1.1, p 437] that Su,v(t) ≤ bSu(t) ∀t ≥ 0 a.s By solving the Fokker–Planck equation,the process bSu(t) has a unique stationary distribution with density

Z t 0

2



Remark 2.1 Because λ is key, we explain its definition and use

(i) To determine whether or not Iu,v(t) converges to 0, we consider the Lyapunov exponent

of Iu,v(t) when Iu,v(t) is small for a sufficiently long time Hence, we look at the followingequation derived from Itˆo’s formula:

Z T 0

2



dt

When T is large, the first and second terms on the right-hand side are small Intuitively, if

Iu,v(t) is small for t ∈ [0, T ], Su,v(t) is close to bSu(t) Using the ergodicity (2.4), we arrive atthe following approximation:

1T

Z T 0

(ii) It is interesting to note that λ does not depend on σ1 This is because the long-termaverage of bS(t), which is R0∞xf∗(x)dx, does not depend on σ1

Based on the above idea, we now consider two cases λ < 0 and λ > 0 separately Then weprovide rigorous proofs of the desired results

2.1 Case 1 λ < 0.

Theorem 2.1 If λ < 0, then for any initial value (S(0), I(0)) = (u, v) ∈ R2,◦+ we havelim supt→∞ln Iu,v(t)t ≤ λ a.s., and the distribution of Su,v(t) converges weakly to the uniqueinvariant probability measure µ∗ with the density f∗

Proof Let bIv(t) be the solution to

d bI(t) = bI(t) − (µ + ρ + γ) + β bSu(t)
dt + σ2I(t)dB(t),b I(0) = v,bwhere bS(t) is the solution to (2.2) By the comparison theorem, Iu,v(t) ≤ bIv(t) a.s given thatb

S(0) = S(0) = u, I(0) = bI(0) = v In view of Itˆo’s formula and the ergodicity of bSu(t),

Z t 0

2

+ β bSu(τ )

2



= λ < 0 a.s

That is, Iu,v(t) converges almost surely to 0 at an exponential rate

For any ε > 0, it follows from (2.6) that there exists t0 > 0 such that P(Ωε) > 1 − ε, where

Ωε:=

n

Iu,v(t) ≤ exp

nλt2

Clearly, we can choose t0 satisfying −2βλ exp{λt02 } < ε Let bSu(t), t ≥ t0, be the solution

to (2.2) given that bS(t0) = S(t0) We have from the comparison theorem that P{Su,v(t) ≤b

Su(t) ∀t ≥ t0} = 1 In view of Itˆo’s formula, for almost all ω ∈ Ωε we have

0 ≤ ln bSu(t) − ln Su,v(t) = α

Z t t0

Iu,v(τ )dτ

≤ β

Z t t0

expnλτ2

o

dτ = −2β

λ

expnλt02

distri-|g(x) − g(y)| ≤ |x − y| and distri-|g(x)| < 1 ∀x, y ∈ R, we have

Since the diffusion given by (2.2) is nondegenerate, it is well known that the distribution ofb

Su(t) weakly converges to µ∗ as t → ∞ (see, e.g., [11]) Thus,

lim sup

t→∞ |Eg(ln Su,v(t)) − g| ≤ 3ε

Since ε is taken arbitrarily, we obtain the desired conclusion The proof is complete

2.2 Case 2 λ > 0 Let P (t, (u, v), ·) be the transition probability of (Su,v(t), Iu,v(t)).Since the diffusion is degenerate, to obtain properties of P (t, (u, v), ·) we check its hypo-ellipticity First, we rewrite (2.1) in Stratonovich’s form,

(2.10)

(dS(t) = [α − c1S(t) − βS(t)I(t)]dt + σ1S(t) ◦ dB(t),dI(t) = [−c2I(t) + βS(t)I(t)]dt + σ2I(t) ◦ dB(t),



σ1x

σ2y



To proceed, we first recall the notion of Lie bracket If Φ(x, y) = (Φ1, Φ2)> and Ψ(x, y) =(Ψ1, Ψ2)> are vector fields on R2, then the Lie bracket [Φ, Ψ] is a vector field given by

Denote by L(x, y) the Lie algebra generated by A(x, y), B(x, y) and by L0(x, y) the ideal inL(x, y) generated by B We have the following lemma

Lemma 2.1 For σ1 > 0, σ2 6= 0, the H¨ormander condition holds for the diffusion (2.10)

To be more precise, we have dim L0(x, y) = 2 at every (x, y) ∈ R2,◦+ or, equivalently, the set

of vectors B, [A, B], [A, [A, B]], [B, [A, B]], spans R2 at every (x, y) ∈ R2,◦+ As a result, thetransition probability P (t, (u, v), ·) has smooth density p(t, u, v, u0, v0)

Proof This lemma has been proved in [20] for the case σ2 > 0 Assume that r = −σ2σ1 > 0.

−ry

,

D :=[A, C](x, y) =α − rβxy

−βxy

,

E :=[C, D](x, y) =−α + r2βxy

−βxy

,

F :=[C, E](x, y) =α − r3βxy

−βxy



Observe that det(D, F ) = 0 only if r2 = 1 or r = 1 (since r > 0) When r = 1, solvingdet(D, E) = 0 obtains βxy = α which implies

det(C, D) =

6= 0

As a result, B, D, E, F span R2 ∀ (x, y) ∈ R2,◦+ The lemma is proved

In order to describe the support of the invariant measure π∗ (if it exists) and to prove theergodicity of (2.1), we need to investigate the following control system on R2,◦+ :

by O+(u, v) the reachable set from (u, v) ∈ R2,◦+ , that is, the set of (u0, v0) ∈ R2,◦+ such thatthere exist a t ≥ 0 and a control φ(·) satisfying uφ(t, u, v) = u0, vφ(t, u, v) = v0 We now recallsome concepts introduced in [17] Let X be a subset of R2,◦+ satisfying the property that forany w1, w2 ∈ X, w2 ∈ O+(w1) Then there is a unique maximal set Y ⊃ X such that thisproperty still holds for Y Such a Y is called a control set A control set W is said to beinvariant if O+(w) ⊂ W ∀ w ∈ W

Putting r := −σ2σ1 and zφ(t) = urφ(t)vφ(t), we have an equivalent system

g(u, z) = −c1u + α − βzu1−rand

h(u, z) = u−rz − (c1r + c2)ur+ βu1+r+ αrur−1− βrz

Lemma 2.2 For the control system (2.12), the following claims hold.

1 For any u0, u1, z0 > 0, and ε > 0, there exist a control φ and T > 0 such that

uφ(T, u0, z0) = u1, |zφ(T, u0, z0) − z0| < ε

2 For any 0 < z0< z1, there are a u0> 0, a control φ, and T > 0 such that zφ(T, u0, z0) =

z1 and that uφ(t, u0, z0) = u0 ∀ 0 ≤ t ≤ T

3 Let d∗ = infu>0{−(c1r + c2)ur+ βu1+r+ αrur−1}

(a) If d∗ ≤ 0, then for any z0 > z1 there are u0 > 0, a control φ, and T > 0 such that

zφ(T, u0, z0) = z1 and that uφ(t, u0, z0) = u0 ∀ 0 ≤ t ≤ T

(b) Suppose that d∗ > 0 and z0 > c∗ := dβr∗ If c∗ < z1 < z0, there are u0> 0 and a control

φ and T > 0 such that zφ(T, u0, z0) = z1 and that uφ(t, u0, z0) = u0 ∀ 0 ≤ t ≤ T However,there are no control φ and T > 0 such that zφ(T, u0, z0) < c∗ for any u0 > 0

Proof Suppose u0 < u1 and ε > 0 are as in claim 1 Let ρ1= sup{|g(u, z)|, |h(u, z)| : u0 ≤

u ≤ u1, |z − z0| ≤ ε} We choose φ(t) ≡ ρ2with (σ1ρ2u0ρ1 − 1)ε ≥ u1− u0 It is easy to check thatwith this control, there is 0 ≤ T ≤ ρ1ε such that uφ(T, u0, z0) = u1, |zφ(T, u0, z0) − z0| < ε If

u0 > u1, we can construct φ(t) similarly Then claim 1 is proved

By choosing u0 to be sufficiently large, there is a ρ3 > 0 such that h(u0, z) > ρ3 ∀z0 ≤

z ≤ z1 This property, combined with (2.12), implies the existence of a feedback control φand T > 0 satisfying that zφ(T, u0, z0) = z1 and that uφ(t, u0, z0) = u0 ∀ 0 ≤ t ≤ T

We now prove claim 3 If r < 0, then

lim

u→0 − (c1r + c2)ur+ βu1+r+ αrur−1 = −∞,and if r > 1, then

lim

u→0 − (c1r + c2)ur+ βu1+r+ αrur−1 = 0

As a result, d∗ ≤ 0 if r /∈ (0, 1], which implies that for any z0 > z1 we choose u0 such that

zφ(T, u0, z0) = z1 and uφ(t, u0, z0) = u0 ∀ 0 ≤ t ≤ T

If r ∈ (0, 1], there exists u0 such that −(c1r + c2)ur0 + βu1+r0 + αrur−10 = d∗ If d∗ ≤ 0,then for any z0 > z1 > 0 we have supz∈[z1,z0]h(u0, z) ≤ u−r0 supz∈[z1,z0]{−βrz2} < 0, whichimplies the desired claim

Consider the remaining case when r ∈ (0, 1] and d∗ > 0 First, assume c∗ < z1 < z0 Let

u0 satisfy −(c1r + c2)ur0+ βu1+r0 + αrur−10 = d∗= βrc∗ Hence,

Thus, there are a feedback control φ and T > 0 satisfying zφ(T, u0, z0) = z1and uφ(t, u0, z0) =

u0 ∀0 ≤ t ≤ T The final assertion follows from the fact that h(u, c∗) ≥ 0 ∀ u > 0

Note that our main goal is not only to prove the permanence of the disease when λ > 0 butalso to estimate the convergence rate of the transition probability to an invariant probabilitymeasure In order to do this, we construct a function V : R2,◦+ → [1, ∞) satisfying

EV Su,v(t∗), Iu,v(t∗)
≤ V (u, v) − κ1Vγ(u, v) + κ21{(u,v)∈K}

for some petite set K and some γ ∈ (0, 1), κ1, κ2 > 0, t∗ > 1, and then apply [12, Thm 3.6].Recall that a set K is said to be petite with respect to the Markov chain (Su,v(nt∗), Iu,v(nt∗)),

n ∈ N, if there exist a measure ψ with ψ(R2,◦+ ) > 0 and a probability distribution ν(·)concentrated on N such that

Because the proofs are quite technical, we explain briefly the main ideas and steps toobtain the convergence rate In the literature (see [7,12, 22,23,28] and references therein),the key to examining convergence rate of ergodic Markov processes (or chains) is to estimatehow fast the processes (or chains) enter a petite set (usually a compact set) and practicalcriteria are often given in terms of Lyapunov functions Basically, our proofs adopt thatidea First, we estimate how fast the solution (Su,v(t), Iu,v(t)) enters a compact subset K :=[κ−1, κ]×[κ−1, κ] for suitable κ > 1 To estimate how fast the solution (Su,v(t), Iu,v(t)) starting

in D1 := {(u0, v0) ∈ R2,◦+ : u0 > κ or u0 < κ−1 or v0> κ} enters K, we use Lemma2.3in which

a Lyapunov function U (u, v) is utilized We estimate the moment (Su,v(t), Iu,v(t)) hits K for(u, v) ∈ D2 := {(u0, v0) ∈ R2,◦+ : v0 < κ−1} in Propositions 2.1 and 2.2 in which the function[ln v]− (which is − ln v if v < 1) is considered based on the idea in Remark 2.1 Lemmas2.4 and 2.5 are auxiliary results needed for Propositions 2.1 and 2.2 Finally, we show thatthe Markov chain Su,v(nt∗), Iu,v(nt∗)
, n ∈ N, is irreducible and aperiodic and that everycompact set is petite in Lemma 2.6with help of Lemmas2.1 and 2.2

Lemma 2.3 For any 0 < p∗< min{2µσ2,2(µ+ρ+γ)σ2 }, let U (u, v) = (u + v)1+p∗+ u−p∗2 Thereexist positive constants K1, K2 such that

(2.13) eK1tE(U (Su,v(t), Iu,v(t))) ≤ U (u, v) + K2(e



u2+

By Young’s inequality, we have

LU (u, v) + K1U (u, v) ≤ κ1(u + v)p∗− κ2(u + v)p∗+1+ κ3(u + v)3p∗4 +1

+−κ4u−1+ κ3u−2 + κ5u−p∗2 ,where κ1 = (1 + p∗)α, κ2 = p∗K1, κ3 = βp2∗, κ4 = p∗2α, and κ5 = p2∗(2+p∗)σ 2

4 + µ + K1 It iseasy to derive from this estimate that

{LU (u, v) + K1U (u, v)} < ∞

As a result,

For n ∈ N, define the stopping time

ηn= inf{t ≥ 0 : U (Su,v(t), Iu,v(t)) ≥ n}

Then Itˆo’s formula and (2.16) yield that

≤ U (u, v) + E

Z t∧ηn 0

eK1τ



LU (Su,v(τ ), Iu,v(τ )) + K1U (Su,v(τ ), Iu,v(τ ))

dτ

≤ U (u, v) + K2(e

K1

By letting n → ∞, we obtain from Fatou’s lemma that

(2.17) EeK1t(U (Su,v(t), Iu,v(t))) ≤ U (u, v) + K2(e

The lemma is proved

Lemma 2.4 There are positive constants K3, K4 such that, for any u ≥ 0, v > 0, t ≥ 1,and A ∈ F ,

(2.18) E [ln Iu,v(t)]2−1A
≤ [ln v]2

−P(A) + K3

pP(A)t[ln v]−+ K4t2pP(A),where [ln x]−= max{0, − ln x}

Proof We have

− ln Iu,v(t) = − ln Iu,v(0) − β

Z t 0

Su,v(τ )dτ + c2t − σ2B(t)

≤ − ln v + c2t + |σ2B(t)|,where c2 = µ + ρ + γ + σ22 Therefore,

[ln Iu,v(t)]− ≤ [ln v]−+ c2t + |σ2B(t)|

This implies that

[ln Iu,v(t)]2−1A≤ [ln v]2−1A+ c22t2+ σ22B2(t)
1A+ 2c2t[ln v]−1A

+ 2|σ2B(t)|1A[ln v]−+ 2c1t|σ2B(t)|1A

By using the H¨older inequality, we obtain for t ≥ 1 that

E|B(t)|1A≤pEB2(t)P(A) ≤ptP(A) ≤ tpP(A)and

EB2(t)1A≤pEB4(t)P(A) ≤p6t2P(A) ≤ 3t

pP(A)

Taking expectation on both sides and using the two above estimates as well as the fact thatP(A) <pP(A) and

√

t ≤ t for t ≥ 1, we haveE[ln Iu,v(t)]2−1A≤ [ln v]2

−P(A) + K3tpP(A)[ln v]−+ K4t2pP(A)for some positive constants K3, K4

We now choose ε ∈ (0, 1) satisfying



−βH − 2c22σ2 2

By using the Hăolder inequality, we obtain for t that

E|B(t)|1A< /small>≤pEB2(t)P (A) ... ≤p6t2P (A) ≤ 3t

pP (A)

Taking expectation on both sides and using the two above estimates as well as the fact thatP (A) <pP (A) and

√

t