Epidemics in Networks Part 2 Compartmental Disease Models

Epidemics in Networks Part 2 Compartmental Disease Models Joel C. Miller & Tom Hladish 18 20 July 2018 1 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 2 / 35

Recall our key questions For SIR: P, the probability of an epidemic. A, the attack rate : the fraction infected if an epidemic happens (better named the attack ratio). R 0, the average number of infections caused by those infected early in the epidemic. I (t), the time course of the epidemic. For SIS: P I ( ), the equilibrium level of infection R 0 I (t) 3 / 35

Simple Compartmental Models The most common models are compartmental models. Continuous time or Discrete time SIR or SIS 4 / 35

Simple Compartmental Models The most common models are compartmental models. Continuous time or Discrete time SIR or SIS The major assumptions: Every individual is average. Every interaction of u is with a randomly chosen other individual. Throughout: S + I + R = N [That is, we look at absolute number rather than proportions of the population. Unfortunately this is standard across much of the field and it causes our equations and initial conditions to be littered with Ns that do nothing to help us understand what is happening. I haven t used this convention in past years notes, so there may be typos occassionally. I ve given up fighting this.] 4 / 35

Continuous time: Kermack McKendrick SIR: S I R SIS: S I Assumptions: 5 / 35

Continuous time: Kermack McKendrick SIR: γi S I R SIS: Assumptions: Individuals recover with rate γ. S γi I 5 / 35

Continuous time: Kermack McKendrick SIR: γi S I R SIS: S Assumptions: Individuals recover with rate γ. Infected individuals transmit to others at rate β K (usually we combine these into a single parameter). β represents the transmission rate per partnership. K represents the typical number of partners. γi I 5 / 35

Continuous time: Kermack McKendrick SIR: β K IS N γi S I R SIS: β K IS N S Assumptions: Individuals recover with rate γ. Infected individuals transmit to others at rate β K (usually we combine these into a single parameter). β represents the transmission rate per partnership. K represents the typical number of partners. The proportion of transmissions that go to susceptible individuals is S/N. γi I 5 / 35

Stochastic simulation SIR case What behavior do we see with β K = 2, γ = 1? Cumulative infections Cumulative infections 50 40 30 20 10 0 0 5 10 15 20 25 30 t N = 50 6 / 35

Stochastic simulation SIR case What behavior do we see with β K = 2, γ = 1? Cumulative infections Cumulative infections 200 150 100 50 0 0 5 10 15 20 25 30 t N = 250 6 / 35

Stochastic simulation SIR case What behavior do we see with β K = 2, γ = 1? Cumulative infections Cumulative infections 1000 800 600 400 200 0 0 5 10 15 20 25 30 t N = 1250 6 / 35

Stochastic simulation SIR case What behavior do we see with β K = 2, γ = 1? Cumulative infections 4000 3000 2000 1000 0 Cumulative infections 5000 0 5 10 15 20 25 30 t N = 6250 6 / 35

Stochastic simulation SIR case What behavior do we see with β K = 2, γ = 1? Cumulative infections 5000 N=6250 N=1250 N=250 4000 N=50 3000 2000 1000 0 0 5 10 15 20 25 30 The distinction between small and large outbreaks becomes clear as N increases. 6 / 35

Stochastic simulation SIR case What does the final size distribution look like? 7 / 35

Stochastic simulation SIR case What does the final size distribution look like? Probability 10 1 10 2 10 3 N=6250 N=1250 N=250 N=50 10 4 10 5 10 0 10 1 10 2 10 3 Final size 7 / 35

Stochastic simulation SIR case What does the final size distribution look like? 10 1 N=6250 N=1250 N=250 N=50 10 2 Probability 10 3 10 4 10 5 0 20 40 60 80 100 Final size In large populations: Small outbreaks affect the same number of individuals. 7 / 35

Stochastic simulation SIR case What does the final size distribution look like? 10 1 N=6250 N=1250 N=250 N=50 20.0 17.5 15.0 N=6250 N=1250 N=250 N=50 Probability 10 2 10 3 10 4 Probability density 12.5 10.0 7.5 5.0 10 5 0 20 40 60 80 100 Final size 2.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Final proportion In large populations: Small outbreaks affect the same number of individuals. Epidemics affect approximately the same proportion of the population 7 / 35

Stochastic simulation SIS case What behavior do we see with β K = 2, γ = 1? Infections 40 Infections 30 20 10 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t N = 50 8 / 35

Stochastic simulation SIS case What behavior do we see with β K = 2, γ = 1? Infections Infections 160 140 120 100 80 60 40 20 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t N = 250 8 / 35

Stochastic simulation SIS case What behavior do we see with β K = 2, γ = 1? Infections Infections 700 600 500 400 300 200 100 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t N = 1250 8 / 35

Stochastic simulation SIS case What behavior do we see with β hk i = 2, γ = 1? Infections 3000 Infections 2500 2000 1500 1000 500 0 0.0 2.5 5.0 7.5 10.0 t 12.5 15.0 17.5 20.0 N = 6250 8 / 35

Stochastic simulation SIS case What behavior do we see with β hk i = 2, γ = 1? Infections Infections 2500 Proportional Infections N=50 N=250 N=1250 N=6250 3000 2000 1500 1000 500 0 0.0 2.5 5.0 7.5 10.0 t 12.5 15.0 17.5 20.0 N=50 N=250 N=1250 N=6250 0.8 0.6 0.4 0.2 0.0 0.0 2.5 5.0 7.5 10.0 t 12.5 15.0 17.5 20.0 Typically extinction occurs either early or after exponentially long time. The proportion infected at equilibrium is approximately the same for different population sizes. 8 / 35

Stochastic simulation SIS case What does the equilibrium distribution look like? 9 / 35

Stochastic simulation SIS case What does the equilibrium distribution look like? 10 1 N=50 N=250 N=1250 N=6250 Probability 10 2 10 3 10 4 10 0 10 1 10 2 10 3 Size at t = 20 9 / 35

Stochastic simulation SIS case What does the equilibrium distribution look like? 20.0 17.5 15.0 N=50 N=250 N=1250 N=6250 Probability density 12.5 10.0 7.5 5.0 2.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportional size at t = 20 The equilibrium proportion infected is about the same for different population sizes. 9 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 10 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = β K IS/N 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = β K IS/N I = 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = β K IS/N I = β K IS/N γi 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = β K IS/N I = β K IS/N γi Ṙ = 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: Ṡ = β K IS/N I = β K IS/N γi Ṙ = γi 12 / 35

SIR: β K IS N γi S I R SIS: β K IS N S γi I SIR equations are: SIS equations are Ṡ = β K IS/N I = β K IS/N γi Ṙ = γi Ṡ = β K IS/N + γi I = β K IS/N γi 12 / 35

Comparison N=1000 N=5000 N=25000 N=125000 N=625000 theoretical prediction 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.5 0.4 0.3 0.2 0.1 10 0 10 20 t 30 40 20 10 0 10 t 20 30 40 N=1000 N=5000 N=25000 N=125000 N=625000 theoretical prediction 0.40 Proportion infected N=1000 N=5000 N=25000 N=125000 N=625000 theoretical prediction 0.6 0.0 20 SIS: Proportion recovered SIR: Proportion infected We compare our differential equations predictions with simulations having transmission rate β hk i IS/N and recovery rate γi. 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 20 10 0 t 10 20 30 13 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 14 / 35

R 0 An important quantity in disease modeling is R 0. 16 / 35

R 0 An important quantity in disease modeling is R 0. R 0 is the expected number of infections caused by an individual infected early in the epidemic. 16 / 35

R 0 An important quantity in disease modeling is R 0. R 0 is the expected number of infections caused by an individual infected early in the epidemic. If R 0 < 1 epidemics cannot occur. If R 0 > 1 epidemics can occur, but are not guaranteed. 16 / 35

R 0 calculation The calculation of R 0 is the same for SIR and SIS: Under our assumptions, every interaction an infected individual has is with a new randomly chosen individual. Early in the epidemic, the probability it is with a susceptible individual is S/N 1. The typical infection duration is 1/γ. The transmission rate during infection is β K. So the number of new infections is R 0 = β K /γ. 17 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 18 / 35

Epidemic probability i 0 Consider an individual u who becomes infected at time t = t 0. Define i m (t) and r m (t) to be the probability u has transmitted to m individuals and is infectious or recovered. 20 / 35

Epidemic probability i 0 K βi 0 i 1 γi 0 r 0 Consider an individual u who becomes infected at time t = t 0. Define i m (t) and r m (t) to be the probability u has transmitted to m individuals and is infectious or recovered. The probability of transmitting at least once before recovering is K β K β+γ. 20 / 35

Epidemic probability K βi 0 K βi 1 i 0 i 1 i 2 γi 0 γi 1 r 0 r 1 Consider an individual u who becomes infected at time t = t 0. Define i m (t) and r m (t) to be the probability u has transmitted to m individuals and is infectious or recovered. The probability of transmitting at least once before recovering K β is K β+γ. The [ probability ] the first m events are transmisions is K β m. K β+γ 20 / 35

Epidemic probability K βi 0 K βi 1 K βi 2 i 0 i 1 i 2 γi 0 γi 1 γi 2 r 0 r 1 r 2 Consider an individual u who becomes infected at time t = t 0. Define i m (t) and r m (t) to be the probability u has transmitted to m individuals and is infectious or recovered. The probability of transmitting at least once before recovering K β is K β+γ. The [ probability ] the first m events are transmisions is K β m. K β+γ For exactly m transmissions before recovery it is ( ) K β m γ r m ( ) = K β + γ K β + γ 20 / 35

We have r m ( ) = ( K β ) m γ K β + γ K β + γ 21 / 35

We have r m ( ) = ( K β ) m γ K β + γ K β + γ For reasons that will be clear later, we define f (x) = m r m ( )x m 21 / 35

We have r m ( ) = ( K β ) m γ K β + γ K β + γ For reasons that will be clear later, we define f (x) = r m ( )x m γ = ( ) m ( K β + γ) 1 K βx K β+γ 21 / 35

We have r m ( ) = ( K β ) m γ K β + γ K β + γ For reasons that will be clear later, we define f (x) = r m ( )x m γ = ( ) = m ( K β + γ) 1 K βx K β+γ γ K β(1 x) + γ 21 / 35

We have r m ( ) = ( K β ) m γ K β + γ K β + γ For reasons that will be clear later, we define f (x) = r m ( )x m γ = ( ) = m ( K β + γ) 1 K βx K β+γ γ K β(1 x) + γ The probability of extinction after the first individual is the probability it recovers without transmitting. f (0) = r 0 ( ) The probability of reaching no further than the first generation of offspring is the probability that none of the first generation individuals causes a transmission. r m ( )[f (0) ] m = f (f (0)) = f (2) (0) m [The superscript with parentheses denotes function iteration] 21 / 35

The probability of reaching no further than generation g is the probability that none of the first generation individuals causes a transmission chain of length longer than g 1. r m ( )[f (g 1) (0)] m = f (g) (0) m The probability the outbreak goes extinct in a finite number of generations (in an infinite population) is lim g f (g) (0). 22 / 35

Cobweb diagrams Consider f (x) = 1 + x 3. We can keep iterating f on 0. 2 1.0 0.8 0.6 f(0) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 23 / 35

Cobweb diagrams Consider f (x) = 1 + x 3. We can keep iterating f on 0. 2 1.0 0.8 0.6 f (2) (0) f(0) 0.4 0.2 0.0 0.0 0.2 0.4 f(0) 0.6 0.8 1.0 23 / 35

Cobweb diagrams Consider f (x) = 1 + x 3. We can keep iterating f on 0. A more 2 direct way is with a cobweb diagram. 1.0 0.8 0.6 f (2) (0) f(0) 0.4 0.2 0.0 0.0 0.2 0.4 f(0) 0.6 0.8 1.0 23 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 24 / 35

SIS equilibrium size The SIS equations are Ṡ = β K IS N + γi I = β K IS N γi 26 / 35

SIS equilibrium size The SIS equations are Ṡ = β K IS N + γi I = β K IS N γi Setting Ṡ = 0, we conclude that either I eq = 0 (the disease has gone extinct) or β K S eq /N = γ. 26 / 35

SIS equilibrium size The SIS equations are Ṡ = β K IS N + γi I = β K IS N γi Setting Ṡ = 0, we conclude that either I eq = 0 (the disease has gone extinct) or β K S eq /N = γ. Taking the second case gives S eq = γn/β K and I eq = N(1 γ/β K ). If γ > β K the disease must die out. If γ < β K at equilibrium the disease has died out or reaches an equilibrium having a fraction γ/β K susceptible and the rest infected. 26 / 35

Alternate equations for SIR Before deriving the final size relation, we derive an alternate system of equations. The system has some important properties: 27 / 35

Alternate equations for SIR Before deriving the final size relation, we derive an alternate system of equations. The system has some important properties: There is a single governing ODE. It will make the final size relation trivial. It has a useful alternate interpretation that gives insight into equations for disease on networks. 27 / 35

Deriving alternate equations Starting from Ṡ = β K I S N 28 / 35

Deriving alternate equations Starting from Ṡ = β K I S N We write Ṡ + β K I N S = 0 28 / 35

Deriving alternate equations Starting from We write Multiplying by e ξ(t) we have Ṡ = β K I S N Ṡ + β K I N S = 0 Ṡe ξ + β K I N Seξ = 0 28 / 35

Deriving alternate equations Starting from We write Multiplying by e ξ(t) we have Ṡ = β K I S N Ṡ + β K I N S = 0 Ṡe ξ + β K I N Seξ = 0 We choose ξ so that ξ = β K I /N. Then e ξ is an integrating factor. We can arbitrarily assume ξ(0) = 0. 28 / 35

Deriving alternate equations So Se ξ = S(0) and thus S(t) = S(0)e ξ. 29 / 35

Deriving alternate equations So Se ξ = S(0) and thus S(t) = S(0)e ξ. We have ξ = β K I /N and Ṙ = γi. So ξ = β K Ṙ Nγ 29 / 35

Deriving alternate equations So Se ξ = S(0) and thus S(t) = S(0)e ξ. We have ξ = β K I /N and Ṙ = γi. So ξ = β K Ṙ Nγ And so or ξ = β K R R(0) R R(0) = R 0 γ N N R = Nξ R 0 + R(0) 29 / 35

Deriving alternate equations So Se ξ = S(0) and thus S(t) = S(0)e ξ. We have ξ = β K I /N and Ṙ = γi. So ξ = β K Ṙ Nγ And so or Finally ξ = β K R R(0) R R(0) = R 0 γ N N R = Nξ R 0 + R(0) I = N R S = N Nξ R 0 R(0) S(0)e ξ 29 / 35

SIR final size Our full equations are thus ξ = β K I ( N = β K 1 ξ R(0) ) R 0 N S(0)e ξ N combined with S = S(0)e ξ, R = Nξ R 0 + R(0), I = N S R 30 / 35

SIR final size Our full equations are thus ξ = β K I ( N = β K 1 ξ R(0) ) R 0 N S(0)e ξ N combined with S = S(0)e ξ, R = Nξ R 0 + R(0), I = N S R So we find a single independent governing equation (for ξ) that depends only on parameters, initial conditions, and ξ. The final size is found using I ( ) = 0, so S( ) = N R( ) 30 / 35

So the final fraction A = R( )/N satisfies A = 1 e R 0A 31 / 35

So the final fraction A = R( )/N satisfies A = 1 e R 0A 1.0 0.8 0.6 iteration 0 iteration 1 iteration 2 iteration 3 iteration 4 iteration 100 A 0.4 0.2 0.0 0 1 2 3 4 5 R 0 We can solve this iteratively, starting from a guess A = 1. 31 / 35

So the final fraction A = R( )/N satisfies A = 1 e R 0A 1.0 0.8 0.6 A 0.4 0.2 0.0 0 1 2 3 4 5 R 0 The results are in good agreement with simulation (subject to the simulation satisfying the assumptions made in the equation derivation). 31 / 35

Direct derivation of alternate equations Any time you derive an expression for some quantity, you should think about what it means. Peter Saeta (my freshman physics professor) 32 / 35

Direct derivation of alternate equations Any time you derive an expression for some quantity, you should think about what it means. Peter Saeta (my freshman physics professor) We will give a direct derivation of these new equations, and later use this approach to derive SIR equations for diseases in networks. 32 / 35

Derivation Define ˆξ: the expected number of transmissions a random individual has received by time t. 33 / 35

Derivation Define ˆξ: the expected number of transmissions a random individual has received by time t. Then S(t) = S(0)e ˆξ(t) is the probability an individual was initially susceptible and has not received any further transmissions. The transmission rate per infected individual is β K, The population-wide transmission rate is β K I. Of these, a fraction 1/N are expected to go to any given individual. 33 / 35

Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review 34 / 35

Review We can write down fairly simple equations for mass-action mixing of SIS and SIR disease. The sizes of epidemics are proportional to the population size. The sizes of small outbreaks are independent of (large enough) population size. We can calculate epidemic probability and final size through iteration. The mass-action SIR model can be reduced to a single ODE. 35 / 35