Chapter 3: Linear & Non-Linear Interaction Models

Transcription

1 & Basics Chapter 3: & Basics Chapter develops the models above to examine models which involve interacting species or quantities. Models lead to simultaneous differential equations for coupled quantites due to the interaction. Firstly, deal with Linear models which differ from the chemostat in that they can be uncoupled to reduce the pair of first order O.D.E.s to a single second order one and hence be solved exactly. As in previous chapter discuss steady states for this solution for the example of Diabetes Detection. Move on to more realistic non-linear interaction models. These don t have an exact solution & demonstrate how stability concepts may be used with Phase-Plane models to show model behaviour under various conditions.

2 & Basics Models of the form dx = Ax + By + P dy = Cx + Dy + Q (3.1) for constant A, B, C, D are said to be linear (i.e. no product terms in x, y), first order, &ifp, Q =0,homogeneous. Such models are also known as compartmental models as model quantities can be thought of as being compartmentalised & are hence useful for modelling mixing problems. & Basics : Diabetes Detection Example: Diabetes Detection Glucose, an end product of carbohydrate digestion, is converted into energy in the body s cells. Insulin, A hormone secreted by the pancreas, facilitates glucose 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 normally excreted); if H is too high too much glucose is absorbed by organs other than the brain. end result in either case can be coma & even death. Diabetes Mellitus patients need regular injections of insulin supplements to compensate for lack of pancreatic insulin.

3 & Basics : Diabetes Detection cont d Present a simple model for G/ H interaction & use this to discuss a clinical test for detection of mild forms of diabetes. Model must account for 4 features in glucose-insulin regulation: 1 G liver absorbs more glucose, converting it & storing it as glycogen. G reverses this process, 2 H more glucose absorbed from bloodstream through cell membranes. 3 G stimulates the pancreas to produce insulin at a faster rate; G lowers insulin production rate. 4 Insulin is constantly being produced by the pancreas & degraded by the liver. & Basics : Diabetes Detection cont d Model ignores biochemistry & treates whole process as a compartmental one: bloodstream is a single box in which G and H are instantaneously uniform. Provided no recent digestion, glucose & insulin concentrations should be in equilibrium, interested in how system responds to change in equilibrium. Let G(t) & H(t) denoteexcess glucose & insulin conc ns at time t. G = H = 0 at equilibrium & +ive & -ive values refer to deviations from equilibrium. If either G, H are given a non-zero value, body will try to restore the equilibrium.

4 & Basics : Diabetes Detection cont d Assuming rates of change of G, H depend only on values G, H, maywrite: dg = αg βh for some constants α, β, γ, δ. dh = γg δh (3.2) To see why these constants must be all +ive, examine second eqn in Eqn.(3.2) & condition 4 above. If, initially, G =0&H > 0, dh/ < 0; this corresponds to the liver in action as per condition 4. Similar reasoning may be applied for other constants. & Basics : Diabetes Detection cont d Eqn.s(3.2) can be reorganised by expressing everything in terms of a single variable: G +(α + δ) Ġ +(αδ + βγ) G = 0 (3.3) which has an exact solution of form G(t) =c 1 e λ 1t + c 2 e λ 2t for constants c 1, c 2. Eigenvalues λ 1,λ 2 can be found from substituting this solution into Eqn.(3.3), from this, get: λ 2 +(α + δ) λ +(αδ + βγ) = 0 (3.4) Insulin concentration H may be got from: H = 1 ) (Ġ + αg β (3.5)

5 & Basics : Diabetes Detection cont d 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 the glucose concentration is measured to test the glucose-insulin regulatory system. This should return to equilibrium faster for healthy patients than diabetic ones. Mathematically, test G = g 0 & H =0att = 0, assuming that the glucose concentration spikes while the insulin concentration is instantaneously zero. & Basics : Diabetes Detection cont d Initial conditions for Eqn.(3.3) can be found from inputting the above into Eqn.(3.5): G = g 0 and Ġ = αg 0 at t =0 (3.6) Depending on the values of the constants α, β, γ, δ (discussed below) find a number of solution types for Eqn.(3.3) shown in Table 3.1. Note: since all α, β, γ, δ are +ive, all the exponentials in the solutions to Eqn.(3.3) will decay to zero with time (i.e. all are either -ive or have -ive real part).

6 & Basics : Diabetes Detection cont d Solns of Eqn.(3.4) Solns of ( Eqn.(3.3) λ 1,λ 2 real G = g 0 (a + λ2 ) e λ1t (a + λ 1 ) e λ 2t ) / (λ 2 λ 1 ) one real, λ G = g 0 (1 ( (α + λ)) e λt λ = υ ± iω G = g 0 cos ωt α ω sin ωt) e υt Table 3.1: Diabetes Model Solutions & Basics : Diabetes Detection cont d Results are backed up by exp t; in 1961 Bolie extrapolated data from dogs to humans. Using above model with exp l data determined that λ 1 = 1.36 & λ 2 = 2.34 and the following equations held: dg = 2.92G 4.34H dh = 0.208G 0.78H (3.7) with the solutions G(t) = g 0 ( 0.56e λ 1 t +1.56e λ 2t ) H(t) = 0.202g 0 ( 0.56e λ 1 t e λ 2t ) (3.8)

7 & Basics : Diabetes Detection cont d Putting G = 0,seethatG returns exponentially to zero in about 1 hour (if our time units from the experimental results are hours) for a normal individual. So, by measuring G, can see whether curve & time to return to normality mirror these results. Results for an initial Glucose concentration g 0 = 1[mol][litre] 1 are shown in Fig.3.1 & Basics : Diabetes Detection cont d Concentration Time,t [hrs] Figure 3.1: Insulin & Glucose Concentrations for g 0 = 1 in Diabetes

8 & Basics : Antibiotic Pharmacokinetics Kinetics of drug absorption, distribution, & elimination is known as pharmacokinetics. A knowledge of pharmacokinetics is needed to administer drugs optimally. i.e. dose should be less than the lethal dose and greater than the effective dose for as long as is determined. This range is known as the therapeutic range of the drug and may (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. & Basics : Antibiotic Pharmacokinetics cont d Tetracyclines (so-called because of four hydrocarbon rings) form quite an old, generally well-tolerated & broad-spectrum group of antibiotics. Tetracyclines such as doxycycline used for prophylaxis against Bacillus anthracis (anthrax) & is effective against Yersinia pestis, the infectious agent of bubonic plague. Also widely used for malaria treatment & prophylaxis, due to the fact that it can be taken for long time periods without the risk of many side-effects (as for Mefloquine, for example).

9 & Basics : Antibiotic Pharmacokinetics cont d For Tetracycline taken orally, represent respective amounts of tetracycline in G.I. tract & plasma at time t by x 1 (t) and x 2 (t). Equations associated with this are: dx 1 = Ax 1 dx 2 = +Ax 1 Bx 2 (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. & Basics : Antibiotic Pharmacokinetics cont d Representing eqn.(3.9) as a matrix and: ( A 0 K(A, B) = A B [ ] x1 x = matrix representation of the tetracycline equations is: x 2 ) ẋ = Kx (3.10) Assuming that A B, x 1 (0) = D, &x 2 (0) = 0, it can be shown that: dx 1 = De At dx 2 = D A B A [ e At e Bt] (3.11)

10 & Basics : Antibiotic Pharmacokinetics cont d Plasma conc n (x 2 (t)) can be fitted as: [ x 2 (t) = 2.65 e 0.15(t 0.41) e 0.715(t 0.41)] (3.12) Plasma concentration can be seen in Fig Concentration Time, t [hrs] Figure 3.2: Plasma Concentrations for Antibiotic & Basics Focus on more sophistiated interaction models between systems. Distinguished from those so far in leading to non-linear, rather than linear, DEs; often not soluble exactly in analytical form so use Phase-Plane Analysis. This is a method where a system of DEs is reduced to a single DE, e.g. dx = f (x, y) (3.13) dy = g(x, y) for some non-linear functions f, g, is written in terms of x, y only. Resulting curves plotted in (x, y) plane are known as phase-plane trajectories.

11 & Basics Cont d In this section we will consider three types of interaction models: 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. & Basics 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 can influence battle outcomes e.g. how many occupying soldiers it takes for victory. Let number of friendly soldiers be x & enemy (i.e. guerrilla forces) be y at time t. Also assume number of soldiers is so large as to permit approximation by continuous variables. In an isolated battle, major factor reducing each army is number of soldiers put out of action by opponents. Finally assume neither army takes prisoners & that both use only gunfire (reasonable assumptions in guerrilla combat).

12 & Basics Cont d: Guerrilla Combat If δx & δy denote changes in respective armies by gunfire from opponents then δx = R y P y yδt and δy = R x P x xδt (3.14) This, in mathematical form, says that number of soldiers hit in a small time δt is equal to product of 1 firing rate of each soldier (const. R x, R y for respective armies), 2 probability that a single shot hits target (P x, P y for respective armies, subscript denoting army firing), 3 the number of soldiers firing. & Basics Cont d: Guerrilla Combat Firing rates R x, R y are assumed constant, while probabilities, P x, P y depend on whether target is exposed/ hidden. For friendly troops, P y can be taken to be constant as each single shot has equal probability of hitting. For hidden targets, probability is ratio of area α exposed, to total area occupied by enemy soldiers, hence P x = αy/a. Inserting this into Eqn.(3.14) & letting δx,δy,δt 0: dx = ay dy = bxy (3.15) where a = R y P y and b = R x α/a are positive constants. Eqn(3.15) are Linear Lanchester Eqns for attritional combat.

13 & Basics Cont d: Guerrilla 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 initial conditions lead to opposite outcome. Taking Eqn.s(3.15) & dividing one by other get: dy dx = b a x (3.17) This is first-order separable & may be integrated: where K = y 0 b 2a (x 0) 2 is a constant. y = b 2a x 2 + K (3.18) & Basics Cont d: Guerrilla Combat Thus, can see from Fig 3.3 (in (x, y) plane), solutions are a family of parabolas as shown in for different values of K. Note from this figure that, as x, y > 0 (+ive numbers of soldiers), ẋ, ẏ < 0 so trajectories point towards (0, 0). Victory for friendly army occurs when x > 0andy =0, corresponding to K < 0 & thus y 0 < b 2a (x 0) 2 at t =0. Victory for guerrillas occurs when x =0andy > 0, hence K > 0&soy 0 > b 2a (x 0) 2. Mutual annihilation occurs when y 0 K = 0 & thus (x 0 ) 2 = b 2a. Generally guerrillas have advantage over friendly army cos b << a as ratio of areas α/a << 1, so two armies are evenly matched if x 0 is large and y 0 is relatively small.

14 & Basics Cont d: Guerrilla Combat Figure 3.3: Guerrilla Combat Phase-Plane Plot & Basics 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, knownasapredator-prey. First models by Lotka (1925) & Volterra (1926) to explain oscillations 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.

15 & Basics Cont d: Predator-Prey These can be expressed mathematically as dx = ax bxy dy = 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), & item2tothe bxy term. In Eqn.(3.19b), dxy term corresponds to item 3 & cy term corresponds to item 4. & Basics Cont d: Predator-Prey Note: Eqn.s(3.19) depend on 4 parameters a, b, c, d; by non-dimens n, subst ing x = c d u, y = a b v,&t = 1 a τ,can rewrite Eqn.s(3.19): du dτ = u(1 v) dv dτ = γv(u 1) (3.20) where γ = c a. In the u, v phase plane, these give: dv du = γ v(u 1) u(1 v) (3.21) which has steady states at (ū, v) =(0, 0) and (ū, v) =(1, 1).

16 & Basics Cont d: Predator-Prey Can integrate this equation directly to give solution for the trajectories: γu + v ln u γ v = H (3.22) for some H > H min =1+γ. H min is minimum of H over all (u, v) & it occurs at u = v =1. For a given H > H min, trajectories given by Eqn.(3.22) are closed & given by Fig. 3.4(a). & Basics Cont d: Predator-Prey (a) Phase-Plane Plot (b) Time Series Plot Figure 3.4: Predator Prey Plots for γ = 2

17 & Basics Cont d: Predator-Prey As noted above, system s steady states are given by (ū, v) =(0, 0) & (ū, v) =(1, 1). As in the chemostat model, can write Jacobian A(u, v) of Eqn.s(3.20) as ( ) 1 v u A(u, v) = (3.23) γv γ(u 1) At 1st steady state, (ū, v) =(0, 0) this reduces to diagonal matrix with eigenvalues λ 1 =1&λ 2 = γ. Hence, solutions near (0, 0) take the form ( ) u(t) u(t) = = c v(t) 1 x 1 e λ1t + c 2 x 2 e λ2t. where x 1, x 2 are e-vectors of A at (0, 0), as per Eqn.1.20 & Basics Cont d: Predator-Prey Since γ <0 < 1, say that (0, 0) is a saddle point as solution ( ) u(t) u(t) = = c v(t) 1 x 1 e t + c 2 x 2 e γt. (3.24) has eigenvalues of different signs & thus is shaped like a saddle 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 &2above. As one of the eigenvalues is always greater than zero, rhs of Eqn.3.24 increases exp ly with increasing t so (0, 0) is unstable.

18 & Basics Cont d: Predator-Prey At the equilibrium (ū, v) =(1, 1), the Jacobian is ( ) 0 1 A(1, 1) = (3.25) γ 0 e-values of which are purely imaginary & given by ±i γ, thus the point is a centre. Mathematically say that complex eigenvalues are only stable if they lie to the left of the imaginary axis. Since these eigenvalues lie on the axis, this makes the model structurally unstable mathematically. The reason for this can be seen qualitatively from Fig 3.4(a). As trajectories are separate but close to touching, any small perturbation can have a very marked effect (not least on the amplitudes of the oscillation of (u, v), Fig 3.4(b) which can be seen to vary over time). & Basics Cont d: Predator-Prey 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 values of 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 the directions 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 that prey & predators. This indicates that phase-plane trajectories move in a counter-clockwise direction.

19 & Basics Cont d: Predator-Prey Although there have been many attempts to apply the LV model to real-world oscillatory phenomena (i.e. prey with logistic growth etc.), have inevitably failed due to system s structural instability & have thus been of limited practical use. However they permit important conclusions to be drawn regarding the qualitative behaviour of a system using purely mathematical methods and very simple assumptions about the system. Most important of these being that systems like Eqn.s(3.19) give rise to the existence of periodic solutions. & Basics Cont d: Symbiosis Symbiosis 3rd interaction regime will consider is symbiosis where interaction of benefit to all species. Model has form: ( ) dx = μ 1 x 1 x y K 1 + c 12 K 1 dy = μ 2 y ( 1 y K 2 + c 21 x K 2 ) (3.26) where pop n sizes x, y grow logistically in absence of other with different carrying capacities K 1, K 2 respectively. The +ive parameters c 12, c 21 indicate +ive effect that each species has on the other.

20 & Basics Cont d: Symbiosis Setting u 1 = x K i, u 2 = y K 2,&τ = μ 1 t, Eqn.(3.26) reduces to: du 1 dτ = u 1 (1 u 1 + α 12 u 2 ) du 2 dτ = ξu 2 (1 u 2 + α 21 u 1 ) where (non-dim l) parameters given by ξ = μ 2 /μ 1, α 12 = c 12 K 2 /K 1 & α 21 = c 21 K 1 /K 2. The steady-states of Eqn.(3.27) are given by (3.27) ( u 1, u 2 )=(0, 0) (u, v) =(0, 0) or (1 + α 12 u 2, 1+α 21 u 1 ) Leads to 4 cases: (0, 0) or (0, 1) or (1, 0) or ( 1+α12 1 α 12 α 21, where final case is only relevant if α 12 α 21 < 1. 1+α 21 1 α 12 α 21 ) & Basics Cont d: Symbiosis Next, can find Jacobian matrix, at a steady-state (ū 1, ū 2 ): ( ) 1 2ū1 + α A(ū 1, ū 2 )= 12 ū 2 α 12 ū 1 ξα 21 ū 2 ξ(1 2ū 2 + α 21 ū 2 ) (3.28) So at the steady states, get for (0, 0) A = ( ξ ( ) 1 α12 for (1, 0) A = 0 ξ(1 + α 21 ) ( ) 1+α12 0 for (0, 1) A = ξ ξα 21 )

21 & Basics Cont d: Symbiosis A = ( ) 1+α12 1+α 21 for,, 1 α 12 α 21 1 α 12 α 21 ( 1 (1 + α12 ) α 12 (1 + α 12 ) 1 α 12 α 21 ξα 21 (1 + α 21 ) ξ(1 + α 21 ) ) α s +ive, (0, 0) is unstable node, (1, 0) & (0, 1) saddles. 4th steady-state has e-values given by zeros of λ 2 trace(a)λ +det(a) = 0 (3.29) Stability depends on R(λ 1,λ 2 ) being -ive, only true if coeffs of eqn(3.29) +ive thus need trace(a) < 0 - always true for α 12 α 21 < 1and det(a) > 0 ξ(1 + α 12)(1 + α 21 )[1+α 12 α 21 ] > 0 1 α 12 α 21 which is again satisfied with α 12 CA659Mathematical α 21 < 1. Methods/Computational Science & Basics Cont d: Symbiosis See this on phase-plane plots in Fig 3.5 (a),(b). These were calculated for α 12 =0.5, α 21 =0.25 & ξ =2in Fig 3.5(a) (i.e. α 12 α 21 < 1) giving a stable steady-state (i.e. stable node) for the fourth case at ( 12 7, 10 ) 7. Fig 3.5(b) shows a similar plot with α 12 =1.5, α 21 =1.25 & ξ =2(i.e. α 12 α 21 > 1) giving instabilities everywhere.

22 & Basics Cont d: Symbiosis (a) (u 1, u 2 )forα 12 α 21 < 1 (b) (u 1, u 2 )forα 12 α 21 > 1 Figure 3.5: Symbiosis Phase-Plane Plots & Basics 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.

23 & Basics Cont d: Infectious 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 model would not provide a definite answer to question of how many succumb to disease. 2 predator-prey models also presume random interaction between predator and prey whereas μparasitic disease is typically spread by close proximity or contact btw infected and healthy individuals. 3 key question is how the disease spreads in the population, something difficult to model using the Predator-prey model. & Basics Cont d: Infectious Diseases 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: Ṡ = βis İ = βis νi Ṙ = νi (3.30) β,ν are infection & removal rates, Ṡ = ds etc.

24 & Basics Cont d: Infectious Diseases β governs speed of disease spread ν governs rate infected hosts die/otherwise removed. By adding 3 equations,it will be obvious that dn = ds + di + dr =0, i.e. S + I + R = N for constant (initial) population N. Rest of the initial conditions are S(0) = S 0 > 0, I (0) = I 0 > 0, R(0) = 0. Given these initial conditions from Eqn.(3.30b) get: { di < 0, if S = I 0 (βs 0 ν) 0 <ν/β t=0 > 0, if S 0 >ν/β (3.31) & Basics Cont d: Infectious Diseases Ratio ρ = ν/β, knownasrelative removal rate. Corresponding to different values of ρ, wehavetwodistinct cases 1 From Eqn.(3.30a), ds < 0, so S(t) S 0 and (if S 0 <ρ)so S(t) <ρfor all t. From Eqn.(3.30b), if S <ρthen di < 0& as t, infection will die out. 2 If S 0 >ρ, then, by same reasoning di > 0forallt such that S(t) >ρ. This means that for some time interval t [0, t 0 ), must have I (t) > I 0.Saythereisanepidemic situation in such cases.

25 & Basics Cont d: Infectious Diseases AplotofS, I and R is shown in Fig 3.6. Population Time, t Figure 3.6: Removed SIR Model for ρ = 100: Susceptable, Infected, & Basics Cont d: Infectious Diseases To further analyse the model, consider the (S, I ) phase plane in Fig Can express Eqn.s(3.30a,b) as di ds = I (βs ν) βis provided I 0. Can derive: = 1+ ρ S, (3.32) I (t) = I 0 S(t)+ρ ln S(t)+S 0 ρ ln S 0 (3.33) From Eqn.(3.33) & Fig. 3.7 that Imax reached when S = ρ. Trajectories are shown for various values of (S 0, I 0 )&,as above 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 (S 0, I 0 ) is in epidemic region (right of ρ = 4 in Fig. 3.7), I initially until S = ρ.

26 & Basics Cont d: Infectious Diseases Figure 3.7: SIR Model Phase-Plane Plot for ρ = 4 & Basics Cont d: Infectious Diseases 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 = S 0 e R/ρ. But, t 0, R N this means S(t) S 0 e N/ρ,i.e.ast, S > 0. But, from Eqn.(3.30c) dr > 0 t 0, so, if need to keep R N, then need dr 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.

27 & Basics Cont d: Infectious Diseases More general model than Kermack-McKendrik or SIR ( more applicable), is one where only partial immunity is conferred. This is known as SIRS & permits previously infected (i.e. removed) individuals to return to susceptible pop n at a rate proportional to the number removed. Mathematically SIRS can be expressed as: ds = βis +γr di = βis νi (3.35) dr = νi γr Again the total population, S + I + R = N, isconstant. & Basics Cont d: Infectious Diseases SIRS can be analysed using standard methods, steady states can be found Ṡ = 0 βis γ = R İ = 0 βis = νi Ṙ = 0 νi γ = R (3.36) These yield 2 steady states: 1 first is trivial S 1 = N, Ī 1 =0 R1 =0, i.e. all pop n healthy but susceptible & disease eradicated;

28 & Basics Cont d: Infectious Diseases A further steady state 2 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: S 2 = ν β, Ī 2 = γ β βn ν γ + ν, R2 = ν β βn ν γ + ν (3.37) ( S 2, Ī2, R 2 ) is only meaningful if all values are +ive. i.e. β ν N > 1 (i.e. +ive numerators for Ī2, R 2 ). This threshold effect corresponds to the minimum population necessary for a disease to become endemic. & Basics Cont d: Infectious Diseases 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 in infections, then β 1/ν gives pop n fraction that comes into contact with infective during infectious period. Hence σ = βn/ν is defined as disease s infectious contact number or intrinsic reproducive rate sometimes denoted R 0. So, from above, and usefully enough, the disease will become endemic in the population if σ>1.

29 & Basics Cont d: Infectious Diseases Can see this effect quite clearly in phase-plane. By using R = N S I, can write Eqn.s(3.35a,b) as: ds = βis + γ(n S I ) di = βis νi In Fig 3.8(a) pop n cannot sustain disease & it dies out; In Fig 3.8(b), get steady-state S 2 = ν β, Ī 2 = γ β βn ν γ + ν. (3.38) Jacobian corresponding to Eqn.(3.38) at ( S 2, Ī2) is ( ) (βī2 + γ) (β S A( S 2, Ī 2 ) = 2 + γ) βī 2 β S 2 ν (3.39) & Basics Cont d: Infectious Diseases (a) (S, I )forσ<1 (b) (S, I )forσ>1 Figure 3.8: SIRS Phase-Plane Plots

30 & Basics Cont d: Infectious Diseases As seen above, conditions of stability are for the trace of the Jacobian to be always -ive and its determinant to be always +ive. It is left as an exercise to show that the stability of the steady-states ( S 2, Ī2), ( S 2, Ī2) is assured when the threshold condition σ>1issatisfied. & Basics Cont d: Infectious Diseases The SIS Model A special case of SIR where infection does not confer any long lasting immunity. Such infections (e.g. tuberculosis, meningitis, & infections leading to the common cold) do not have a recovered state & individuals become susceptible again after infection. The equations are thus: ds = βis +νi di = βis νi (3.40)

31 & Basics Cont d: Infectious Diseases From this can see that for a total population, N, itholdsthat dn = ds + di =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 = (βn ν)i βi 2 This is a form of the logistic growth equation with r = βn ν and K = N ν β so that we have two cases: 1 for β ν N > 1, lim t I (t) = βn ν β & disease will spread, 2 for β ν N 1, lim t I (t) = 0 & disease will die out. & Basics Cont d: Infectious Diseases 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

32 & Basics Cont d: Infectious Diseases Eradication & Control for the Models above For SIR model, for eqn(3.37) S 0 N, the total pop n. Hence, from eqn(3.37) & eqns(??,3.37) for the SIS & SIRS models, resp that infectious contact number or intrinsic reproductive rate, σ or R 0 = βn/ν is highly important. It is fraction of population that comes into contact with an infective individual during the period of infectiousness. mean number of secondary cases one infected case will cause in a pop n with no immunity & without interventions to control the infection. Useful because it helps determine if an infectious disease will spread thro a pop n. & Basics Cont d: Infectious Diseases 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 infected cattle. 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, slaughtering potential contacts surrounding infected farms was employed. Vaccination of susceptibles for the epidemic, which became increasingly politically controversial.

33 & Basics Cont d: Infectious Diseases Immunizing an entire population from a disease is impractical due to matters of cost & the logistics of administering the vaccine to (potentially) hundreds of thousands. Costs: direct costs of producing & administering the dosage and indirect costs of providing information to the public and making sure that everyone that is vaccinated has been. Thus would like to be able to provide safety from disease at the lowest possible cost. & Basics Cont d: Infectious Diseases Actually only have to immunize a fraction of population in order to give the entire population herd immunity. Specifically need to reduce the effective value of N so that that the disease disappears of its own accord. In terms of SIR model need to move enough people such that (from eqn(3.31)), βs 0 ν<0sothat the rate of increase of infectives di is negative. i.e. need to lower epidemic threshold below one. Or to write it more directly the fraction of people that needs to be immunized is such that S 0 <ν/βi.e. 1 ν/β percent of the susceptible pool need to be immunized.

34 & Basics Cont d: Infectious Diseases This makes intuitive sense since if ν is small that means it takes longer to recover from infection & an infective person has more time to infect people. Thus as ν, 1 ν/β ; need to inoculate a larger fraction of the population. As β, each infected person contacts more people in a given period and 1 ν/β. Thus again need to inoculate a larger fraction of population. & Basics Cont d: Infectious Diseases R 0 &1 1/R 0 shown in % Table 3.2 for common diseases. Disease Transmission R R 0 % Measles Airborne 12 to to 94.5 Pertussis Airborne droplet 12 to to 94 Diphtheria Saliva 6to7 84 Smallpox Social contact 5to7 80 to 85 Polio Fecal-oral route 5to7 80 to 85 Rubella Airborne droplet 5to7 80 to 85 HIV/AIDS Sexual contact 2to5 50 to 80 SARS Airborne droplet 2to5 50 to 80 Influenza (1918) Airborne droplet 2to3 50 to 80 Cholera Fecal-oral route Table 3.2: Values for R 0 for Several Common Infectious Diseases

35 & Basics Cont d: The Chemostat Revisited The Chemostat Revisited Returning to chemostat above, can look at phase-plane plot. Derived the equations: ( ) dn nc dτ = f (n, c) = α 1 n (3.41) 1+c and dc dτ ( ) nc = g(n, c) = c + α 2 (3.42) 1+c containing (dimensionless) parameters: α 1 = VK max F and α 2 = C 0 K n & Basics Cont d: The Chemostat Revisited Eqns(3.41, 3.42) also contain dimensionless time, bacterial population & nutrient concentrations respectively: τ = tf V, n = NαVK max FK n, c = C K n The phase-plane plot for nvc is shown in Fig It will be seen from the figure that when α 1 =3,α 2 =1therearetwo equilibrium points: and ( n 1, c 1 ) = as predicted. (α 1 ( α 2 1 α 1 1 ), ) 1 α 1 1 ( n 2, c 2 ) = (0,α 2 ) = (0, 1) = ( 3 2, 1 2 )

36 & Basics Cont d: The Chemostat Revisited Figure 3.10: Chemostat Phase-Plane Plot, α 1 =3,α 2 =1 blocs & Basics title of the bloc bloc text title of the bloc bloc text title of the bloc bloc text