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Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Rumour Spread The spread of rumours can be modelled using a modified SIR model. Here, as before, the population is divided into 3 groups: 1 2 3 Susceptibles (i.e. those who haven’t heard the rumour), Influencers (those who spread the rumour to others), Repressers (those who do not spread the rumour). Again, symbols S, I, R denote these resp groups at time t. This time, the following assumptions are made: 1 2 3 When an Influncer meets a Susceptable, a certain fraction β of Susceptables are influenced and change to being Influncers, When one Influncer meets another, a certain fraction ν of Influncers get bored with the rumour and become Repressers, When an Influncer meets a Represser, a certain fraction ν of Influncers are bored and become Repressers. 207 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Rumour Spread (cont’d) Equations are thus: Ṡ İ Ṙ = = = −βIS +βIS −νI 2 −νIR +νI 2 +νIR (4.35) β is the influence rate and governs speed of rumour spread ν is the boredom rate and governs rate of rumour quashing. Again, S + I + R = N for constant population N, as before. Rest of the initial conditions are S(0) = S0 > 0, I(0) = I0 > 0, R(0) = 0. Given these initial conditions from Eqn.(4.35b) get: dI = I0 ((β + ν)S0 − ν) dt t=0 < 0, > 0, if S0 < ν/(β + ν) if S0 > ν/(β + ν) (4.36)208 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Rumour Spread Ratio ρ′ = ν/(β + ν), can be seen as relative boredom rate. As with model for infectious disease spread, for different ρ′ , we have two cases: 1 2 ′ From Eqn.(4.30a), dS dt < 0, so S(t) ≤ S0 and (if S0 < ρ ) so dI ′ ′ S(t) < ρ for all t. From Eqn.(4.30b), if S < ρ then dt < 0 & as t → ∞, rumour will be quashed. dI If S0 > ρ′ , then, by same reasoning dt > 0 for all t such that ′ S(t) > ρ . This means that for some time interval t ∈ [0, t0 ), must have I(t) > I0 . Say there is an chain reaction situation in such cases. 209 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Rumour Spread Again, as with SIR model of infection, consider the (S, I) phase plane in Fig. 4.8. Figure 4.8 : SIR Rumour Spread for ρ′ = 4/5 210 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Rumour Spread Compared to Fig. 4.7, the trajectories are much steeper. This is because, comparing Eqn (4.36) dI = I0 ((β + ν)S0 − ν) dt t=0 with the SIR model of Infectious Diseases Eqn(4.31): dI = I0 (βS0 − ν) dt t=0 there is now a factor of β + ν instead of β. This has the effect of causing ’epidemic’ behaviour even for small rate parameters. 211 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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 dt = dI dt = dR dt = +γR −βIS βIS −νI (4.37) νI −γR Again the total population, S + I + R = N, is constant. 212 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases SIRS can be analysed using standard methods, steady states can be found Ṡ = 0 İ = 0 Ṙ = 0 ⇒ ⇒ ⇒ βIS γ βIS νI γ = = = R νI R (4.38) These yield 2 steady states: 1 first is trivial S̄1 = N, Ī1 = 0 R̄1 = 0, i.e. all pop’n healthy but susceptible & disease eradicated; 213 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases A further steady state 2 second found from inserting S = ν/β (from Eqn.(4.38b)) & R = νI/γ (from Eqn.(4.38c)) into S + I + R = N to give: S̄2 = ν , β Ī2 = γ βN − ν , β γ+ν R̄2 = ν βN − ν β γ+ν (4.39) (S̄2 , Ī2 , R̄2 ) is only meaningful if all values are +ive. i.e. β νN > 1 (i.e. +ive numerators for Ī2 , R̄2 ). This threshold effect corresponds to the minimum population necessary for a disease to become endemic. 214 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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 R0 . So, from above, and usefully enough, the disease will become endemic in the population if σ > 1. 215 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases Can see this effect quite clearly in phase-plane. By using R = N − S − I, can write Eqn.s(4.37a,b) as: dS dt dI dt = = −βIS + γ(N − S − I) βIS − νI (4.40) In Fig 4.9(a) pop’n cannot sustain disease & it dies out; In Fig 4.9(b), get steady-state S̄2 = ν , β Ī2 = γ βN − ν . β γ+ν Jacobian corresponding to Eqn.(4.40) at (S̄2 , Ī2 ) is A(S̄2 , Ī2 ) = ! −(β Ī2 + γ) −(β S̄2 + γ) −β Ī2 β S̄2 − ν " (4.41) 216 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases (a) (S, I) for σ < 1 (b) (S, I) for σ > 1 Figure 4.9 : SIRS Phase-Plane Plots 217 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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 (S̄2 , Ī2 ), (S̄2 , Ī2 ) is assured when the threshold condition σ > 1 is satisfied. 218 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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 dt = −βIS +νI dI dt = βIS −νI (4.42) 219 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases From this can see that for a total population, N, it holds that dN dS dI = + =0 dt dt dt i.e. S + I = N for constant (initial) population N. Expressing I in terms of S in eqn.(4.42), can be seen that: dI = (βN − ν)I − βI 2 dt This is a form of the logistic growth equation with r = βN − ν and K = N − βν so that we have two cases: 1 for 2 for β νN β νN > 1, limt→∞ I(t) = βN−ν β & disease will spread, ≤ 1, limt→∞ I(t) = 0 & disease will die out. 220 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases A plot of the former SIS model case is shown in Fig 4.10. Population Time, t Figure 4.10 : SIS Model for ρ = 100: Susceptable, Infected 221 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases Eradication & Control for the Models above For SIR model, for eqn(4.39) S0 ≈ N, the total pop’n. Hence, from eqn(4.39) & eqns(4.42,4.39) for the SIS & SIRS models, resp that infectious contact number or intrinsic reproductive rate, σ or R0 = 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. 222 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases See from βN/ν that can reduce infectious disease spread by: 1 2 3 ↑ ν, removal rate of infectives. Seen in the UK Foot-and-Mouth epidemic by slaughtering those infected cattle. β ↓, infectious contact rate btw susceptibles and infectives. Disinfection & movement controls in Foot-and-Mouth ↓ β. 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. 223 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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. 224 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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(4.31)), βS0 − ν < 0 so that the rate of increase of dI is negative. infectives dt 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 S0 < ν/β i.e. 1 − ν/β percent of the susceptible pool need to be immunized. 225 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models 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. 226 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: Infectious Diseases R0 & 1 − 1/R0 shown in % Table 4.2 for common diseases. Disease Measles Pertussis Diphtheria Smallpox Polio Rubella HIV/AIDS SARS Influenza (1918) Cholera Transmission Airborne Airborne droplet Saliva Social contact Fecal-oral route Airborne droplet Sexual contact Airborne droplet Airborne droplet Fecal-oral route R0 12 to 18 12 to 17 6 to 7 5 to 7 5 to 7 5 to 7 2 to 5 2 to 5 2 to 3 2.9 1 − R10 % 92 to 94.5 92 to 94 84 80 to 85 80 to 85 80 to 85 50 to 80 50 to 80 50 to 80 65.5 Table 4.2 : Values for R0 for Several Common Infectious Diseases 227 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models 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 nc = f (n, c) = α1 dτ 1+c # $ −n (4.43) and dc nc = g(n, c) = − dτ 1+c # $ − c + α2 (4.44) containing (dimensionless) parameters: α1 = VKmax C0 and α2 = F Kn 228 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: The Chemostat Revisited Eqns(4.43, 4.44) also contain dimensionless time, bacterial population & nutrient concentrations respectively: tF NαVKmax C , n = , c = V FKn Kn τ = The phase-plane plot for nVc is shown in Fig 4.11. It will be seen from the figure that when α1 = 3, α2 = 1 there are two equilibrium points: (n̄1 , c̄1 ) = # # α1 α2 − 1 1 , α1 − 1 α1 − 1 $ $ 3 1 = ( , ) 2 2 and (n̄2 , c̄2 ) = (0, α2 ) = (0, 1) as predicted. 229 / 230 Intro to the Topic Discrete Models Growth and Decay Linear & Non-Linear Interaction Models Introduction Linear Models Non-Linear Models Non-Linear Models Cont’d: The Chemostat Revisited Figure 4.11 : Chemostat Phase-Plane Plot, α1 = 3, α2 = 1 230 / 230