Population Systems: Stochastic versus Deterministic · 2018-06-07 · Stochastic Modelling Well-known Models Stochastic vs Deterministic Summary Population Systems: Stochastic versus

Stochastic ModellingWell-known Models

Stochastic vs DeterministicSummary

Population Systems: Stochastic versusDeterministic

Xuerong Mao

Department of Mathematics and StatisticsUniversity of Strathclyde

Glasgow, G1 1XH

Xuerong Mao SDE Models



Outline

1 Stochastic Modelling

2 Well-known ModelsLinear SDE modelsNon-linear SDE models

3 Stochastic vs DeterministicLotka-Volterra model

Variance dependent on x(t)Variance independent of x(t)

Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace

4 Summary




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4 Summary




One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are

number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
























Now suppose that at time t the underlying quantity is x(t). Letus consider a small subsequent time interval dt , during whichx(t) changes to x(t) + dx(t). (We use the notation d · for thesmall change in any quantity over this time interval when weintend to consider it as an infinitesimal change.) By definition,the intrinsic growth rate at t is dx(t)/x(t). How might we modelthis rate?




If, given x(t) at time t , the rate of change is deterministic, sayR = R(x(t), t), then

dx(t)x(t)

= R(x(t), t)dt .

This gives the ordinary differential equation (ODE)

dx(t)dt

= R(x(t), t)x(t).




However the rate of change is in general not deterministic as itis often subjective to many factors and uncertainties e.g.system uncertainty, environmental disturbances. To model theuncertainty, we may decompose

dx(t)x(t)

= deterministic change + random change.




The deterministic change may be modeled by

Rdt = R(x(t), t)dt

where R = r(x(t), t) is the average rate of change given x(t) attime t . So

dx(t)x(t)

= R(x(t), t)dt + random change.

How may we model the random change?




In general, the random change is affected by many factorsindependently. By the well-known central limit theorem thischange can be represented by a normal distribution with meanzero and and variance V 2dt , namely

random change = N(0,V 2dt) = V N(0,dt),

where V = V (x(t), t) is the standard deviation of the rate ofchange given x(t) at time t , and N(0,dt) is a normal distributionwith mean zero and and variance dt . Hence

dx(t)x(t)

= R(x(t), t)dt + V (x(t), t)N(0,dt).




A convenient way to model N(0,dt) as a process is to use theBrownian motion B(t) (t ≥ 0) which has the followingproperties:

B(0) = 0,dB(t) = B(t + dt)− B(t) is independent of B(t),dB(t) follows N(0,dt).




The stochastic model can therefore be written as

dx(t)x(t)

= R(x(t), t)dt + V (x(t), t)dB(t),

or

dx(t) = R(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)

which is a stochastic differential equation (SDE).




Linear SDE modelsNon-linear SDE models

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Exponential growth model

If both R and V are constants, say

R(x(t), t) = µ, V (x(t), t) = σ,

then the SDE becomes

dx(t) = µx(t)dt + σx(t)dB(t).

This is a linear SDE. It is known as the geometric Brownianmotion in finance and the exponential growth model in thepopulation systems.





Mean-reverting process

IfR(x(t), t) =

α(µ− x(t))

x(t), V (x(t), t) = σ,


dx(t) = α(µ− x(t))dt + σx(t)dB(t).

This is known as the mean-reverting process.





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Logistic model

IfR(x(t), t) = b + ax(t), V (x(t), t) = σx(t),


dx(t) = x(t)(

[b + ax(t)]dt + σx(t)dB(t)).

This is the well-known Logistic model in population.





Square root process

IfR(x(t), t) = µ, V (x(t), t) =

σ√x(t)

,

then the SDE becomes the well-known square root process

dx(t) = µx(t)dt + σ√

x(t)dB(t).

This is used widely in engineering and finance.





Mean-reverting square root process

IfR(x(t), t) =

α(µ− x(t))

x(t), V (x(t), t) =

σ√x(t)

,


dx(t) = α(µ− x(t))dt + σ√

x(t)dB(t).

This is the mean-reverting square root process used widely infinance and population.





Theta process

IfR(x(t), t) = µ, V (x(t), t) = σ(x(t))θ−1,


dx(t) = µx(t)dt + σ(x(t))θdB(t),

which is known as the theta process.




Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace

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Consider the Lotka-Volterra model for a system with ninteracting species, namely

x(t) = diag(x1(t), · · · , xn(t))[b + Ax(t)], (3.1)

where

x = (x1, · · · , xn)T , b = (b1, · · · ,bn)T , A = (aij)n×n.

We need some condition on A, e.g., A + AT is negative-definite,for the system not to explode to infinity at a finite time (namely,no explosion).





We first consider the corresponding SDE model where thevariance dependent on the state, say

dx(t) = x(t)(

[b + Ax(t)]dt + σx(t)dB(t)), (3.2)

where σ = (σij)n×n and B(t) is a scalar Brownian motion.

How is this SDE different from its corresponding ODE?





Theorem 1

Assume that

σii > 0 for 1 ≤ i ≤ n whilst σij ≥ 0 for i 6= j .

Then for any system parameters b ∈ Rn and A ∈ Rn×n, and anygiven initial data x(0) = x0 ∈ Rn, there is a unique solution x(t)to equation (3.2) on t ≥ 0 and the solution will remain in Rn

+

with probability 1, namely x(t) ∈ Rn+ for all t ≥ 0 almost surely.





Significant Difference between ODE (3.1) and SDE(3.2)

ODE (3.1): The solution may explode to infinity at a finitetime if, e.g., A + AT is positive definite.SDE (3.2): With probability one, the solution will no longerexplode to infinity in a finite time, even in the case whenA + AT is positive definite, as long as σii > 0 for 1 ≤ i ≤ nwhilst σij ≥ 0 for i 6= j . Moreover, the stochastic populationsystem is persistent.





Significant Difference between ODE (3.1) and SDE(3.2)

ODE (3.1): The solution may explode to infinity at a finitetime if, e.g., A + AT is positive definite.SDE (3.2): With probability one, the solution will no longerexplode to infinity in a finite time, even in the case whenA + AT is positive definite, as long as σii > 0 for 1 ≤ i ≤ nwhilst σij ≥ 0 for i 6= j . Moreover, the stochastic populationsystem is persistent.





Example

dy(t)dt

= y(t)[1 + y(t)], t ≥ 0, x(0) = x0 > 0

has the solution

y(t) =1

−1 + e−t (1 + x0)/x0(0 ≤ t < T ) ,

which explodes to infinity at the finite time

T = log(

1 + x0

x0

).

However, the SDE

dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]

will never explode as long as σ 6= 0.Xuerong Mao SDE Models




0 2 4 6 8 10

01

23

45

t

x(t)

or

y(t)

x(t)

y(t)

0 2 4 6 8 100

.51

.01

.5

t

x(t)

or

y(t)

x(t)

y(t)

Left: σ = 3; right: σ = 6





Key Point:When a > 0 and ε = 0 the solution explodes at the finite timet = T ; whilst conversely, no matter how small ε > 0, thesolution will not explode in a finite time. In other words,

noise may suppress explosion.





Consider the SDE model where the noise is independent of thestate, namely

dx(t) = x(t)(

[b + Ax(t)]dt + σdB(t)), (3.3)

where σ = (σij) ∈ Rn×n and B(t) is an n-dimensional Brownianmotion.

How is this SDE different from others?





For this SDE model, we require, e.g. A + AT isnegative-definite, in order to have no explosion.If λ+max(Q) < 0, where Q = (qij)n×n is defined by

qij = bi + bj −n∑

k=1

σikσjk ,

then limt→∞ x(t) = 0 with probability 1, namely the noisemakes the population extinct.If both σ and −A are non-singular M-matrices while∑n

k=1 σ2ik < 2bi for 1 ≤ i ≤ n, the the population system is

persistent. In this case

limt→∞

Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,

where ξi = bi − 0.5∑n

k=1 σ2ik .







k=1

σikσjk ,




limt→∞

Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,


k=1 σ2ik .







k=1

σikσjk ,




limt→∞

Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,


k=1 σ2ik .





Example

The SDE

dx(t) = x(t)[(1− x(t))dt + σx(t)dw(t)]

will become extinct if σ >√

2 but will be persistent if σ <√

2.





0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

t

x(t)

0 2 4 6 8 100.5

1.0

1.5

2.0

2.5

t

x(t)

Left: σ = 2; right: σ = 0.5





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Consider the linear scalar ordinary differential equation

y(t) = a + by(t) on t ≥ 0 (3.4)

with initial value y(0) = y0 > 0, where a,b > 0. This equationhas its explicit solution

y(t) =(

y0 +ab

)ebt − a

b.

Hencelim

t→∞

1t

log(y(t)) = b,

that is, the solution tends to infinity exponentially.





On the other hand, considr its corresponding SDE

dx(t) = [a + bx(t)]dt + σx(t)dB(t) on t ≥ 0 , (3.5)

where σ > 0 and B(t) is a scalar Brownian motion. Given initialvalue x(0) = x0 > 0, this SDE has its explicit solution

x(t) = exp[(b−1

2σ2)t+σB(t)

](x0+a

∫ t

0exp

[(b−1

2σ2)s+σB(s)

]ds).

If σ2 > 2b then the solution of equation (3.5) obeys

lim supt→∞

log(x(t))

log t≤ σ2

σ2 − 2ba.s. (3.6)





This shows that for any ε > 0, there is a positive randomvariable Tε such that, with probability one,

x(t) ≤ tε+σ2/(σ2−2b) ∀t ≥ Tε.

In other words, the solution will grow at most polynomially withorder ε+ σ2/(σ2 − 2b). In particular, by increasing the noiseintensity σ, we can make the order as close to 1 as we like.Comparing this polynomial growth with the exponential growthof the solution (3.4), we see the important fact that the noisesuppresses the exponential growth.





Generally, consider a nonlinear system described by ann-dimensional ordinary differential equation

y(t) = f (y(t), t),

whose coefficient obeys

〈y , f (y , t)〉 ≤ α + β|y |2, ∀(y , t) ∈ Rn × R+.

Clearly, the solution of this equation may grow exponentially.





However, its stochastically linearly perturbed system

dx(t) = f (x(t), t)dt +m∑

i=1

Aix(t)dBi(t) (3.7)

may grow at most polynomially with probability one, where Bi(t)are m indepdenent Brownian motions and Ai ∈ Rn×n.





In fact, if there are two positive constants γ and δ such that

m∑i=1

|Aix |2 ≤ γ|x |2,m∑

i=1

|xT Aix |2 ≥ δ|x |4 (3.8)

for all x ∈ Rn, andδ > β + 1

2γ, (3.9)

then, for any initial value x(0) = x0 ∈ Rn, the solution of thestochastically controlled system (3.7) obeys

lim supt→∞

log(|x(t)|)log t

≤ δ

2δ − 2β − γa.s. (3.10)





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Consider the n-dimensional linear ordinary differential equation

dy(t)dt

= q + Qy(t), t ≥ 0 (3.11)

with initial value y(0) = y0 ∈ Rn, where q ∈ Rn and Q ∈ Rn×n.Assume that

λmax(Q + QT ) < 0. (3.12)

Equation (3.12) can be solved explicitly, which implies easilythat

lim supt→∞

|y(t)| ≤ |q|. (3.13)

That is, under condition (3.12), the solution of equation (3.11) isasymptotically bounded and the bound is independent of theinitial value.





Generally, consider an n-dimensional ordinary differentialequation

dy(t)dt

= f (y(t), t), t ≥ 0. (3.14)

Assume that f is sufficiently smooth and, in particular, it obeys

− 2〈x , f (x , t)〉 ≤ c1 + c2|x |2, ∀(x , t) ∈ Rn × R (3.15)

for some non-negative numbers c1 and c2. Our aim here isshow that the solution of its corresponding SDE

dx(t) = f (x(t), t)dt + g(x(t))dB(t) (3.16)

may grow exponentially with probability one.





First, let the dimension of the state space n be an even numberand choose the dimension of the Brownian motion m to be 2.Let g : Rn → Rn×2 by

g(x) = (ξ,Ax),

where ξ = (ξ1, · · · , ξn)T ∈ Rn and, of course, ξ 6= 0, while

A =

0 σ−σ 0

. . .0 σ−σ 0

with σ > 0. So the SDE (3.16) becomes





dx(t) = f (x(t), t)dt + ξdB1 + σ

x2(t)−x1(t)

...xn(t)−xn−1(t)

dB2(t). (3.17)





If we have

|ξ|2 > c1 and σ2 = c1 + c2 + |ξ|2 (3.18)

then the solution of equation (3.17) obeys

lim inft→∞

1t

log(|x(t)|) ≥|ξ|2 − c2

14|ξ|2

a.s. (3.19)

Alternatively, if we choose

|ξ|2 > c1 and σ2 = 3|ξ|2 + c2 − c1 (3.20)

thenlim inft→∞

1t

log(|x(t)|) ≥ 12(|ξ|2 − c1) a.s. (3.21)





We next consider the dimension of the state space n be an oddnumber but n ≥ 3. Choose the dimension of the Brownianmotion m to be 3 and design g : Rn → Rn×3 by

g(x) = (ξ,A2x ,A3x),

where ξ = (ξ1, · · · , ξn)T ∈ Rn and, of course, ξ 6= 0, while

A2 =

0 σ−σ 0

. . .0 σ−σ 0

0

, A3 =

00 σ−σ 0

. . .0 σ−σ 0

with σ > 0.





So the stochastically perturbed system (3.16) becomes

dx(t) = f (x(t), t)dt+ξdB1+σ

x2(t)−x1(t)

...xn−1(t)−xn−2(t)

0

dB2+σ

0x2(t)−x3(t)

...xn(t)−xn−1(t)

dB3.

(3.22)





If we choose ξ and σ as (3.18) then the solution of equation(3.22) obeys (3.19), while if we choose ξ and σ as (3.20) thenthe solution of equation (3.22) obeys (3.21).





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Following Bailey (The Mathematical Theory of InfectiousDiseases, Second Edition, Griffin, London, 1975), let X be theproportion of the human population that is infective and Y bethe proportion of infective (female) mosquitoes. Thedeterministic epidemic equations are then

dX/dt = abmY (t)[1− X (t)]− rX (t),dY/dt = aX (t)[1− Y (t)]− µY (t), (3.23)

where m is the number of female mosquitoes per human host,a the number of bites per unit time on man by a singlemosquito, b the proportion of infected bites on man thatproduce an infection, r the per capita rate of recovery inhumans and µ the per capita mortality rate for mosquitoes.





The deterministic steady state endemic levels, whendX/dt = dY/dt = 0, are

X∞ =a2bm − µra(abm + r)

and

Y∞ =a2bm − µrabm(a + µ)

Ifa2bm ≤ µr

the endemic level is zero, and initial outbreaks will collapse.





If a = b = m = r = 1 and µ = 0.9, then the equations are

dX/dt = Y (t)[1− X (t)]− X (t),dY/dt = X (t)[1− Y (t)]− 0.9Y (t). (3.24)

The endemic state (X∞ = 0.05,Y∞ = 0.0526) is a stable state.





Consider the corresponding SDE model

dX (t) = (Y (t)[1− X (t)]− X (t))dt + 0.4Y (t)dB1(t),dY (t) = (X (t)[1− Y (t)]− 0.9Y (t))dt + 0.4X (t)dB2(t).

The following simulation shows that the system divergesmarkedly from the low endemic level (initially (0.05,0.05)),generating repeated and increasing peaks of infection.





0 10 20 30 40 50

0.02

0.06

0.10

t

X(t)

0 10 20 30 40 50

0.02

0.06

0.10

t

Y(t)





On the other hand, consider another SDE model

dX (t) = (Y (t)[1− X (t)]− X (t))dt+0.4X (t)dB1(t) + 0.4Y (t)dB2(t),

dY (t) = (X (t)[1− Y (t)]− 0.9Y (t))dt+0.4X (t)dB1(t) + 0.4Y (t)dB2(t).

The following simulation shows that Individual outbreaks do notconverge on the endemic level, but fall to zero, in the presenceof sufficient symmetric stochasticity.





0 10 20 30 40 50

0.00

0.02

0.04

t

X(t)

0 10 20 30 40 50

0.00

0.02

0.04

0.06

t

Y(t)




Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.


















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Population Systems: Stochastic versus Deterministic · 2018-06-07 · Stochastic Modelling Well-known Models Stochastic vs Deterministic Summary Population Systems: Stochastic versus