Chapter 3: Linear & Non-Linear Interaction Modelsmcrane/CA659/CA659NewSlidesCh3_2Up.pdf · 2011. 3....

Introduction & BasicsGrowth and Decay

Linear & Non-Linear Interaction Models

IntroductionLinear ModelsNon-Linear Models

Chapter 3:


Dr. Martin Crane CA659Mathematical Methods/Computational Science




Introduction

Chapter develops the models above to examine models whichinvolve interacting species or quantities.

Models lead to simultaneous differential equations for coupledquantites due to the interaction.

Firstly, deal with Linear models which differ from thechemostat in that they can be uncoupled to reduce the pair offirst order O.D.E.s to a single second order one and hence besolved exactly.

As in previous chapter discuss steady states for this solutionfor the example of Diabetes Detection.

Move on to more realistic non-linear interaction models.

These don’t have an exact solution & demonstrate howstability concepts may be used with Phase-Plane models toshow model behaviour under various conditions.





Linear Models

Models of the form

dxdt = Ax + By + P

dydt = Cx + Dy + Q

(3.1)

for constant A, B, C , D are said to be linear (i.e. no productterms in x , y), first order, & if P, Q = 0, homogeneous.

Such models are also known as compartmental models asmodel quantities can be thought of as beingcompartmentalised & are hence useful for modelling mixingproblems.





Linear Models: Diabetes Detection

Example: Diabetes Detection

Glucose, an end product of carbohydrate digestion, isconverted into energy in the body’s cells.

Insulin, A hormone secreted by the pancreas, facilitatesglucose absorption by cells other than those of the brain &nervous system.A delicate balance is maintained between glucose (G ) &insulin (H) in the bloodstream:

if H is too low, too little glucose is absorbed (& is normallyexcreted);if H is too high too much glucose is absorbed by organs otherthan the brain.

end result in either case can be coma & even death.

Diabetes Mellitus patients need regular injections of insulinsupplements to compensate for lack of pancreatic insulin.





Linear Models: Diabetes Detection cont’d

Present a simple model for G/ H interaction & use this todiscuss a clinical test for detection of mild forms of diabetes.

Model must account for 4 features in glucose-insulinregulation:

1 G ↑⇒ liver absorbs more glucose, converting it & storing it asglycogen. G ↓ reverses this process,

2 H ↑⇒ more glucose absorbed from bloodstream through cellmembranes.

3 G ↑⇒ stimulates the pancreas to produce insulin at a fasterrate; G ↓ lowers insulin production rate.

4 Insulin is constantly being produced by the pancreas &degraded by the liver.






Model ignores biochemistry & treates whole process as acompartmental one: bloodstream is a single ‘box’ in which Gand H are instantaneously uniform.

Provided no recent digestion, glucose & insulin concentrationsshould be in equilibrium, interested in how system responds tochange in equilibrium.

Let G (t) & H(t) denote excess glucose & insulin conc’ns attime t.

⇒ G = H = 0 at equilibrium & +ive & -ive values refer todeviations from equilibrium.

If either G , H are given a non-zero value, body will try torestore the equilibrium.






Assuming rates of change of G , H depend only on values G ,H, may write:

dGdt = −αG − βH

dHdt = γG − δH

(3.2)

for some constants α, β, γ, δ.

To see why these constants must be all +ive, examine secondeqn in Eqn.(3.2) & condition 4 above.

If, initially, G = 0 & H > 0, ⇒ dH/dt < 0; this correspondsto the liver in action as per condition 4. Similar reasoningmay be applied for other constants.






Eqn.s(3.2) can be reorganised by expressing everything interms of a single variable:

G + (α + δ) G + (αδ + βγ)G = 0 (3.3)

which has an exact solution of form G (t) = c1eλ1t + c2e

λ2t

for constants c1, c2.

Eigenvalues λ1, λ2 can be found from substituting thissolution into Eqn.(3.3), from this, get:

λ2 + (α + δ)λ + (αδ + βγ) = 0 (3.4)

Insulin concentration H may be got from:

H = − 1

β

(G + αG

)(3.5)






To test for diabetes, a glucose tolerance test is administered.

In this, after fasting overnight, an injection of glucose is given.

Blood samples are then taken at subsequent times and theglucose concentration is measured to test the glucose-insulinregulatory system.

This should return to equilibrium faster for healthy patientsthan diabetic ones.

Mathematically, test ≡ G = g0 & H = 0 at t = 0, assumingthat the glucose concentration spikes while the insulinconcentration is instantaneously zero.






Initial conditions for Eqn.(3.3) can be found from inputtingthe above into Eqn.(3.5):

G = g0 and G = −αg0 at t = 0 (3.6)

Depending on the values of the constants α, β, γ, δ (discussedbelow) find a number of solution types for Eqn.(3.3) shown inTable 3.1.

Note: since all α, β, γ, δ are +ive, all the exponentials in thesolutions to Eqn.(3.3) will decay to zero with time (i.e. all areeither -ive or have -ive real part).






Solns of Eqn.(3.4) Solns of Eqn.(3.3)

λ1, λ2 real G = g0

((a + λ2) eλ1t − (a + λ1) eλ2t

)/ (λ2 − λ1)

one real, λ G = g0 (1− (α + λ)) eλt

λ = −υ ± iω G = g0

(cos ωt − α

ω sinωt)e−υt

Table 3.1: Diabetes Model Solutions






Results are backed up by exp’t; in 1961 Bolie extrapolateddata from dogs to humans. Using above model with exp’ldata determined that λ1 = −1.36 & λ2 = −2.34 and thefollowing equations held:

dGdt = −2.92G − 4.34H

dHdt = 0.208G − 0.78H

(3.7)

with the solutions

G (t) = g0

(−0.56eλ1t + 1.56eλ2t)

H(t) = 0.202g0

(−0.56eλ1t − eλ2t) (3.8)






Putting G = 0, see that G returns exponentially to zero inabout 1 hour (if our time units from the experimental resultsare hours) for a normal individual.

So, by measuring G , can see whether curve & time to returnto normality mirror these results.

Results for an initial Glucose concentrationg0 = 1[mol][litre]−1 are shown in Fig.3.1






0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Concentration

Time,t [hrs]

Figure 3.1: Insulin & Glucose Concentrations for g0 = 1 in Diabetes





Linear Models: Antibiotic Pharmacokinetics

Kinetics of drug absorption, distribution, & elimination isknown as pharmacokinetics.

A knowledge of pharmacokinetics is needed to administerdrugs optimally.

i.e. dose should be less than the lethal dose and greater thanthe effective dose for as long as is determined.

This range is known as the therapeutic range of the drug andmay (e.g. for chemotherapy drugs) be quite narrow

Hence knowing pharmacokinetics of some (especially novel)drugs and formulations is critical.

For this pharmacokinetic modelling use differential equations.





Linear Models: Antibiotic Pharmacokinetics cont’d

Tetracyclines (so-called because of four hydrocarbon rings)form quite an old, generally well-tolerated & broad-spectrumgroup of antibiotics.

Tetracyclines such as doxycycline used for prophylaxis againstBacillus anthracis (anthrax) & is effective against Yersiniapestis, the infectious agent of bubonic plague.

Also widely used for malaria treatment & prophylaxis, due tothe fact that it can be taken for long time periods without therisk of many side-effects (as for Mefloquine, for example).






For Tetracycline taken orally, represent respective amounts oftetracycline in G.I. tract & plasma at time t by x1(t) andx2(t).

Equations associated with this are:

dx1dt = −Ax1

dx2dt = +Ax1 − Bx2

(3.9)

i.e. amount of antibiotic in G.I. tract ↓ at rate defined by A &amount in plasma being excreted at rate defined by B.






Representing eqn.(3.9) as a matrix and:

K(A, B) =

( −A 0A B

)

x =

[x1

x2

]matrix representation of the tetracycline equations is:

x = Kx (3.10)

Assuming that A �= B, x1(0) = D, & x2(0) = 0, it can beshown that:

dx1dt = De−At

dx2dt = D A

B−A

[e−At − e−Bt

] (3.11)






Plasma conc’n (x2(t)) can be fitted as:

x2(t) = 2.65[e−0.15(t−0.41) − e−0.715(t−0.41)

](3.12)

Plasma concentration can be seen in Fig 3.2.

0 5 10 150

0.5

1

1.5

2

Concentration

Time, t [hrs]

Figure 3.2: Plasma Concentrations for Antibiotic





Non-Linear Models

Focus on more sophistiated interaction models betweensystems.

Distinguished from those so far in leading to non-linear, ratherthan linear, DEs; often not soluble exactly in analytical formso use Phase-Plane Analysis.

This is a method where a system of DEs is reduced to a singleDE, e.g.

dxdt = f (x , y)

dydt = g(x , y)

(3.13)

for some non-linear functions f , g , is written in terms of x , yonly.

Resulting curves plotted in (x , y) plane are known asphase-plane trajectories.





Non-Linear Models Cont’d

In this section we will consider three types of interactionmodels:

1 Mutually destructive interaction,2 Interaction beneficial to one or the other species,3 Mutually beneficial interaction (symbiosis)

Then proceed to the modelling of infectious diseases.





Non-Linear Models Cont’d: Guerrilla Combat

The Guerrilla Combat Model

Combat btw occupying & guerrilla force can be modelled(with some simplifying assumptions) as 2 coupled DEs.

Math models of combat used to understand what factors caninfluence battle outcomes e.g. how many occupying soldiers ittakes for victory.

Let number of ‘friendly’ soldiers be x & ‘enemy’ (i.e. guerrillaforces) be y at time t.

Also assume number of soldiers is so large as to permitapproximation by continuous variables.

In an isolated battle, major factor reducing each army isnumber of soldiers put out of action by opponents.

Finally assume neither army takes prisoners & that both useonly gunfire (reasonable assumptions in guerrilla combat).






If δx & δy denote changes in respective armies by gunfirefrom opponents then

δx = −RyPyyδt and δy = −RxPxxδt (3.14)

This, in mathematical form, says that number of soldiers hitin a small time δt is equal to product of

1 firing rate of each soldier (const. Rx ,Ry for respective armies),2 probability that a single shot hits target (Px ,Py for respective

armies, subscript denoting army firing),3 the number of soldiers firing.






Firing rates Rx , Ry are assumed constant, while probabilities,Px , Py depend on whether target is exposed/ hidden.

For ‘friendly’ troops, Py can be taken to be constant as eachsingle shot has equal probability of hitting.

For hidden targets, probability is ratio of area α exposed, tototal area occupied by enemy soldiers, hence Px = αy/A.

Inserting this into Eqn.(3.14) & letting δx , δy , δt → 0:

dxdt = −ay

dydt = −bxy

(3.15)

where a = RyPy and b = Rxα/A are positive constants.

Eqn(3.15) are Linear Lanchester Eqns for attritional combat.






Steady state analysis of eqns.(3.15) gives, at steady state(X , Y )

0 = −aY and 0 = −bXY (3.16)

i.e. Guerrilla army defeated with X unspecified.

However, phase-plane analysis shows that certain initialconditions lead to opposite outcome.

Taking Eqn.s(3.15) & dividing one by other get:

dy

dx=

b

ax (3.17)

This is first-order separable & may be integrated:

y =b

2ax2 + K (3.18)

where K = y0 − b2a (x0)

2 is a constant.






Thus, can see from Fig 3.3 (in (x , y) plane), solutions are afamily of parabolas as shown in for different values of K .

Note from this figure that, as x , y > 0 (+ive numbers ofsoldiers), x , y < 0 so trajectories point towards (0, 0).

Victory for ‘friendly’ army occurs when x > 0 and y = 0,corresponding to K < 0 & thus y0 < b

2a (x0)2 at t = 0.

Victory for guerrillas occurs when x = 0 and y > 0, henceK > 0 & so y0 > b

2a (x0)2.

Mutual annihilation occurs when

K = 0 & thusy0

(x0)2=

b

2a.

Generally guerrillas have advantage over friendly army cosb << a as ratio of areas α/A << 1, so two armies are evenlymatched if x0 is large and y0 is relatively small.






Figure 3.3: Guerrilla Combat Phase-Plane Plot





Non-Linear Models Cont’d: Predator-Prey

The Predator-Prey Model

Guerrilla Combat is a mutually destructive (or competitive)model (i.e. any interaction results in pop’n decrease).

More common for rate of pop’n increase of one species ↓ &other ↑, known as a predator-prey.

First models by Lotka (1925) & Volterra (1926) to explainoscillations in fish pop’ns in Mediterranean.

Lotka-Volterra model is based on a number of assumptions:1 Prey grow in an unlimited way when predators do not control

their numbers.2 Predators depend on presence of prey to survive.3 Predation rate depends on likelihood that a predator

encounters a victim.4 Predator pop’n growth rate ∝ rate of predation.






These can be expressed mathematically as

dxdt = ax − bxy

dydt = −cy + dxy

(3.19)

where x , y are prey & predator pop’n densities & a, b, c, d are+ive constants.

Note: item 1 corresponds to the ax term in Eqn.(3.19a), &item 2 to the −bxy term.

In Eqn.(3.19b), dxy term corresponds to item 3 & −cy termcorresponds to item 4.






Note: Eqn.s(3.19) depend on 4 parameters a, b, c, d ; bynon-dimens’n, subst’ing x = c

d u, y = abv , & t = 1

aτ , canrewrite Eqn.s(3.19):

dudτ = u(1− v)

dvdτ = γv(u − 1)

(3.20)

where γ = ca .

In the u, v phase plane, these give:

dv

du= γ

v(u − 1)

u(1− v)(3.21)

which has steady states at (u, v) = (0, 0) and (u, v) = (1, 1).






Can integrate this equation directly to give solution for thetrajectories:

γu + v − ln uγv = H (3.22)

for some H > Hmin = 1 + γ.

Hmin is minimum of H over all (u, v) & it occurs atu = v = 1.

For a given H > Hmin, trajectories given by Eqn.(3.22) areclosed & given by Fig. 3.4(a).






(a) Phase-Plane Plot (b) Time Series Plot

Figure 3.4: Predator Prey Plots for γ = 2






As noted above, system’s steady states are given by(u, v) = (0, 0) & (u, v) = (1, 1).

As in the chemostat model, can write Jacobian A(u, v) ofEqn.s(3.20) as

A(u, v) =

(1− v −uγv γ(u − 1)

)(3.23)

At 1st steady state, (u, v) = (0, 0) this reduces to diagonalmatrix with eigenvalues λ1 = 1 & λ2 = −γ.

Hence, solutions near (0, 0) take the form

u(t) =

(u(t)v(t)

)= c1x1e

λ1t + c2x2eλ2t .

where x1, x2 are e-vectors of A at (0, 0), as per Eqn.1.20






Since −γ < 0 < 1, say that (0, 0) is a saddle point as solution

u(t) =

(u(t)v(t)

)= c1x1e

t + c2x2e−γt . (3.24)

has eigenvalues of different signs & thus is shaped like asaddle at that point.

Note: this solution identifies the exp’l growth & decay of prey& predator respectively on the u, v axes referred to in items 1& 2 above.

As one of the eigenvalues is always greater than zero, rhs ofEqn.3.24 increases exp’ly with increasing t so (0, 0) isunstable.






At the equilibrium (u, v) = (1, 1), the Jacobian is

A(1, 1) =

(0 −1γ 0

)(3.25)

e-values of which are purely imaginary & given by ±i√

γ, thusthe point is a centre.Mathematically say that complex eigenvalues are only stable ifthey lie to the left of the imaginary axis.Since these eigenvalues lie on the axis, this makes the modelstructurally unstable mathematically.The reason for this can be seen qualitatively from Fig 3.4(a).As trajectories are separate but close to touching, any smallperturbation can have a very marked effect (not least on theamplitudes of the oscillation of (u, v), Fig 3.4(b) which can beseen to vary over time).






Although it is possible to flip between trajectories Fig 3.4(a)through small perturbations (i.e. inexact numerical solutions),trajectories are closed curves corresponding to different valuesof H in Eqn.(3.22).

Considering the areas round the centre (1, 1) as quadrants,can examine the slopes given by Eqn.s(3.20) & hence thedirections of the trajectories.

Considering the 1st quadrant to be that closest to the origin,can see from Eqn.s(3.20) that u > 0, v < 0, indicating thatprey ↑ & predators ↓.This indicates that phase-plane trajectories move in acounter-clockwise direction.






Although there have been many attempts to apply the LVmodel to real-world oscillatory phenomena (i.e. prey withlogistic growth etc.), have inevitably failed due to system’sstructural instability & have thus been of limited practical use.

However they permit important conclusions to be drawnregarding the qualitative behaviour of a system using purelymathematical methods and very simple assumptions about thesystem.

Most important of these being that systems like Eqn.s(3.19)give rise to the existence of periodic solutions.





Non-Linear Models Cont’d: Symbiosis

Symbiosis

3rd interaction regime will consider is symbiosis whereinteraction of benefit to all species. Model has form:

dxdt = μ1x

(1− x

K1+ c12

yK1

)dydt = μ2y

(1− y

K2+ c21

xK2

) (3.26)

where pop’n sizes x , y grow logistically in absence of otherwith different carrying capacities K1, K2 respectively.

The +ive parameters c12, c21 indicate +ive effect that eachspecies has on the other.






Setting u1 = xKi

, u2 = yK2

, & τ = μ1t, Eqn.(3.26) reduces to:

du1dτ = u1 (1− u1 + α12u2)du2dτ = ξu2 (1− u2 + α21u1)

(3.27)

where (non-dim’l) parameters given by ξ = μ2/μ1,α12 = c12K2/K1 & α21 = c21K1/K2.

The steady-states of Eqn.(3.27) are given by

(u1, u2) = (0, 0) ≡ (u, v) = (0, 0) or (1 + α12u2, 1 + α21u1)

Leads to 4 cases:

(0, 0) or (0, 1) or (1, 0) or

(1 + α12

1− α12α21,

1 + α21

1− α12α21

)

where final case is only relevant if α12α21 < 1.






Next, can find Jacobian matrix, at a steady-state (u1, u2):

A(u1, u2) =

(1− 2u1 + α12u2 α12u1

ξα21u2 ξ(1− 2u2 + α21u2)

)(3.28)

So at the steady states, get

for (0, 0) A =

(1 00 ξ

)

for (1, 0) A =

( −1 α12

0 ξ(1 + α21)

)

for (0, 1) A =

(1 + α12 0ξα21 −ξ

)






for

(1 + α12

1− α12α21,

1 + α21

1− α12α21

),

A =1

1− α12α21

( −(1 + α12) α12(1 + α12)ξα21(1 + α21) −ξ(1 + α21)

)

α’s +ive, ⇒ (0, 0) is unstable node, (1, 0) & (0, 1) saddles.4th steady-state has e-values given by zeros of

λ2 − trace(A)λ + det(A) = 0 (3.29)

Stability depends on (λ1, λ2) being -ive, only true if coeffs ofeqn(3.29) +ive thus need trace(A) < 0 - always true forα12α21 < 1 and

det(A) > 0 ≡ ξ(1 + α12)(1 + α21) [1 + α12α21]

1− α12α21> 0

which is again satisfied with α12α21 < 1.Dr. Martin Crane CA659Mathematical Methods/Computational Science





See this on phase-plane plots in Fig 3.5 (a),(b).

These were calculated for α12 = 0.5, α21 = 0.25 & ξ = 2 inFig 3.5(a) (i.e. α12α21 < 1) giving a stable steady-state (i.e.stable node) for the fourth case at

(127 , 10

7

).

Fig 3.5(b) shows a similar plot withα12 = 1.5, α21 = 1.25 & ξ = 2 (i.e. α12α21 > 1) givinginstabilities everywhere.






(a) (u1, u2) for α12α21 < 1 (b) (u1, u2) for α12α21 > 1

Figure 3.5: Symbiosis Phase-Plane Plots





Non-Linear Models Cont’d: Infectious Diseases

Infectious Diseases

Can be classified into 2 broad categories:1 those caused by viruses and bacteria (microparasitic diseases

such as smallpox measles),2 those due to worms (macroparasitic diseases such as malaria).

Main distinction btw them:1 former reproduce within the host & are transmitted directly

from one host to another,2 latter require some vector (i.e. must be insect-borne,

entertainment- borne etc.) for transmission.

whereas

We will focus purely on microparasitic diseases.






Cannot view virus & host organism as a predator-prey system,for a number of reasons:

1 virus pop’n in the individual host can vary greatly, thus modelwould not provide a definite answer to question of how manysuccumb to disease.

2 predator-prey models also presume random interaction betweenpredator and prey whereas μparasitic disease is typically spreadby close proximity or contact btw infected and healthyindividuals.

3 key question is how the disease spreads in the population,something difficult to model using the Predator-prey model.






To model diseases, pop’n divided into 3 groups:1 suscepibles (i.e. those not immune to disease),2 infectives (those who can infect non-immunes),3 removed (dead or in quarantine or immune).

Symbols S , I , R denote these resp groups at time t.Following assumptions are made:

1 rate of change of infectives ∝ number of contacts btw I & S(or each I infects a constant fraction β of S per unit time)

2 the number of I who become R is proportional to I

Equations are thus:

S = −βIS

I = βIS −νI

R = νI

(3.30)

β, ν are infection & removal rates, S = dSdt etc.






β governs speed of disease spread

ν governs rate infected hosts die/otherwise removed.

By adding 3 equations,it will be obvious that

dN

dt=

dS

dt+

dI

dt+

dR

dt= 0,

i.e. S + I + R = N for constant (initial) population N.

Rest of the initial conditions are

S(0) = S0 > 0, I (0) = I0 > 0, R(0) = 0.

Given these initial conditions from Eqn.(3.30b) get:

dI

dt

∣∣∣∣t=0

= I0 (βS0 − ν)

{< 0, if S0 < ν/β> 0, if S0 > ν/β

(3.31)






Ratio ρ = ν/β, known as relative removal rate.

Corresponding to different values of ρ, we have two distinctcases

1 From Eqn.(3.30a), dSdt < 0, so S(t) ≤ S0 and (if S0 < ρ) so

S(t) < ρ for all t. From Eqn.(3.30b), if S < ρ then dIdt < 0 &

as t → ∞, infection will die out.2 If S0 > ρ, then, by same reasoning dI

dt > 0 for all t such thatS(t) > ρ. This means that for some time interval t ∈ [0, t0),must have I (t) > I0. Say there is an epidemic situation in suchcases.






A plot of S , I and R is shown in Fig 3.6.

Population

Time, t

Figure 3.6: SIR Model for ρ = 100: Susceptable, Infected,Removed






To further analyse the model, consider the (S , I ) phase planein Fig. 3.7. Can express Eqn.s(3.30a,b) as

dI

dS=

I (βS − ν)

−βIS= −1 +

ρ

S, (3.32)

provided I �= 0. Can derive:

I (t) = I0 − S(t) + ρ lnS(t) + S0 − ρ lnS0 (3.33)

From Eqn.(3.33) & Fig. 3.7 that Imax reached when S = ρ.

Trajectories are shown for various values of (S0, I0) &, asabove with predator-prey, all move anti-clockwise.

A trajectory starts on line I + S = N (as R(0) = 0 &susceptible pop’n decreases with time.

If (S0, I0) is in epidemic region (right of ρ = 4 in Fig. 3.7), I ↑initially until S = ρ.






Figure 3.7: SIR Model Phase-Plane Plot for ρ = 4






How does disease eventually die out with SIR model?

From Eqn.s(3.30a,c):

dS

dR=−βIS

νI= −S

ρ, (3.34)

thus S = S0e−R/ρ. But, ∀t ≥ 0, R ≤ N this means

S(t) ≥ S0e−N/ρ, i.e. as t →∞, S > 0.

But, from Eqn.(3.30c) dRdt > 0 ∀t ≥ 0, so, if need to keep

R ≤ N, then need dRdt → 0, as t →∞.

From Eqn.(3.30c), this can only happen if I (∞) = 0.

Upshot of this is that lack of infectives wipes out disease &not lack of susceptibles.






More general model than Kermack-McKendrik or SIR (⇒more applicable), is one where only partial immunity isconferred.

This is known as SIRS & permits previously infected (i.e.removed) individuals to return to susceptible pop’n at a rateproportional to the number removed.

Mathematically SIRS can be expressed as:

dSdt = −βIS +γR

dIdt = βIS −νI

dRdt = νI −γR

(3.35)

Again the total population, S + I + R = N, is constant.






SIRS can be analysed using standard methods, steady statescan be found

S = 0 ⇒ βISγ = R

I = 0 ⇒ βIS = νI

R = 0 ⇒ νIγ = R

(3.36)

These yield 2 steady states:1 first is trivial

S1 = N, I1 = 0 R1 = 0,

i.e. all pop’n healthy but susceptible & disease eradicated;






A further steady state2 second found from inserting S = ν/β (from Eqn.(3.36b)) &

R = νI/γ (from Eqn.(3.36c)) into S + I + R = N to give:

S2 =ν

β, I2 =

γ

β

βN − ν

γ + ν, R2 =

ν

β

βN − ν

γ + ν(3.37)

(S2, I2, R2) is only meaningful if all values are +ive.

i.e. βν N > 1 (i.e. +ive numerators for I2, R2).

This threshold effect corresponds to the minimum populationnecessary for a disease to become endemic.






Have seen ρ = ν/β (relative removal rate) above;

Now define its reciprocal β/ν as follows:

as removal rate from infective class is ν (with units 1/time),average period of infectivity is obviously 1/ν.as β is fraction of contacts (between I and S) that result ininfections, then β × 1/ν gives pop’n fraction that comes intocontact with infective during infectious period.

Hence σ = βN/ν is defined as disease’s infectious contactnumber or intrinsic reproducive rate sometimes denoted R0.

So, from above, and usefully enough, the disease will becomeendemic in the population if σ > 1.






Can see this effect quite clearly in phase-plane. By usingR = N − S − I , can write Eqn.s(3.35a,b) as:

dSdt = −βIS + γ(N − S − I )dIdt = βIS − νI

(3.38)

In Fig 3.8(a) pop’n cannot sustain disease & it dies out;

In Fig 3.8(b), get steady-state

S2 =ν

β, I2 =

γ

β

βN − ν

γ + ν.

Jacobian corresponding to Eqn.(3.38) at (S2, I2) is

A(S2, I2) =

( −(β I2 + γ) −(βS2 + γ)−β I2 βS2 − ν

)(3.39)






(a) (S , I ) for σ < 1 (b) (S , I ) for σ > 1

Figure 3.8: SIRS Phase-Plane Plots






As seen above, conditions of stability are for the trace of theJacobian to be always -ive and its determinant to be always+ive.

It is left as an exercise to show that the stability of thesteady-states (S2, I2), (S2, I2) is assured when the thresholdcondition σ > 1 is satisfied.






The SIS Model

A special case of SIR where infection does not confer any longlasting immunity.

Such infections (e.g. tuberculosis, meningitis, & infectionsleading to the common cold) do not have a recovered state &individuals become susceptible again after infection. Theequations are thus:

dSdt = −βIS +νI

dIdt = βIS −νI

(3.40)






From this can see that for a total population, N, it holds that

dN

dt=

dS

dt+

dI

dt= 0

i.e. S + I = N for constant (initial) population N.

Expressing I in terms of S in eqn.(3.40), can be seen that:

dI

dt= (βN − ν)I − βI 2

This is a form of the logistic growth equation withr = βN − ν and K = N − ν

β so that we have two cases:

1 for βν N > 1, limt→∞ I (t) = βN−ν

β & disease will spread,

2 for βν N ≤ 1, limt→∞ I (t) = 0 & disease will die out.






A plot of the former SIS model case is shown in Fig 3.9.

Population

Time, t

Figure 3.9: SIS Model for ρ = 100: Susceptable, Infected






Eradication & Control for the Models above

For SIR model, for eqn(3.37) S0 ≈ N, the total pop’n.

Hence, from eqn(3.37) & eqns(??,3.37) for the SIS & SIRSmodels, resp that infectious contact number or intrinsicreproductive rate,

σ or R0 = βN/ν

is highly important.

It is fraction of population that comes into contact with aninfective individual during the period of infectiousness.

≡ mean number of secondary cases one infected case willcause in a pop’n with no immunity & without interventions tocontrol the infection.

Useful because it helps determine if an infectious disease willspread thro a pop’n.






See from βN/ν that can reduce infectious disease spread by:1 ↑ ν, removal rate of infectives. Seen in the UK

Foot-and-Mouth epidemic by slaughtering those infectedcattle.

2 β ↓, infectious contact rate btw susceptibles and infectives.Disinfection & movement controls in Foot-and-Mouth ↓ β.

3 Decrease the effective number of N which has the effect of↓ S . Again for the Foot-and-Mouth example, slaughteringpotential contacts surrounding infected farms was employed.

Vaccination of susceptibles for the epidemic, which becameincreasingly politically controversial.






Immunizing an entire population from a disease is impracticaldue to matters of cost & the logistics of administering thevaccine to (potentially) hundreds of thousands.

Costs:

direct costs of producing & administering the dosage andindirect costs of providing information to the public andmaking sure that everyone that is vaccinated has been.

Thus would like to be able to provide safety from disease atthe lowest possible cost.






Actually only have to immunize a fraction of population inorder to give the entire population herd immunity.

Specifically need to reduce the effective value of N so thatthat the disease disappears of its own accord.

In terms of SIR model need to move enough people such that(from eqn(3.31)), βS0 − ν < 0 so that the rate of increase ofinfectives dI

dt is negative.

i.e. need to lower epidemic threshold below one.

Or to write it more directly the fraction of people that needsto be immunized is such that S0 < ν/β i.e. 1− ν/β percentof the susceptible pool need to be immunized.






This makes intuitive sense since if ν is small that means ittakes longer to recover from infection & an infective personhas more time to infect people.

Thus as ν ↓, 1− ν/β ↑; need to inoculate a larger fraction ofthe population.

As β ↑, each infected person contacts more people in a givenperiod and 1− ν/β ↑.Thus again need to inoculate a larger fraction of population.






R0 & 1− 1/R0 shown in % Table 3.2 for common diseases.

Disease Transmission R0 1− 1R0

%

Measles Airborne 12 to 18 92 to 94.5Pertussis Airborne droplet 12 to 17 92 to 94Diphtheria Saliva 6 to 7 84Smallpox Social contact 5 to 7 80 to 85Polio Fecal-oral route 5 to 7 80 to 85Rubella Airborne droplet 5 to 7 80 to 85HIV/AIDS Sexual contact 2 to 5 50 to 80SARS Airborne droplet 2 to 5 50 to 80Influenza (1918) Airborne droplet 2 to 3 50 to 80Cholera Fecal-oral route 2.9 65.5

Table 3.2: Values for R0 for Several Common Infectious Diseases





Non-Linear Models Cont’d: The Chemostat Revisited

The Chemostat Revisited

Returning to chemostat above, can look at phase-plane plot.Derived the equations:

dn

dτ= f (n, c) = α1

(nc

1 + c

)− n (3.41)

and

dc

dτ= g(n, c) = −

(nc

1 + c

)− c + α2 (3.42)

containing (dimensionless) parameters:

α1 =VKmax

Fand α2 =

C0

Kn






Eqns(3.41, 3.42) also contain dimensionless time, bacterialpopulation & nutrient concentrations respectively:

τ =tF

V, n =

NαVKmax

FKn, c =

C

Kn

The phase-plane plot for nVc is shown in Fig 3.10. It will beseen from the figure that when α1 = 3, α2 = 1 there are twoequilibrium points:

(n1, c1) =

(α1

(α2 − 1

α1 − 1

),

1

α1 − 1

)= (

3

2,1

2)

and(n2, c2) = (0, α2) = (0, 1)

as predicted.






Figure 3.10: Chemostat Phase-Plane Plot, α1 = 3, α2 = 1





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Chapter 3: Linear & Non-Linear Interaction Modelsmcrane/CA659/CA659NewSlidesCh3_2Up.pdf · 2011. 3....

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