Markovian susceptible-infectious- susceptible (SIS) dynamics on finite networks: endemic prevalence and invasion probability Robert Wilkinson Kieran Sharkey University of Liverpool

Outline 1 Markovian SIS dynamics on networks (infection with no acquired immunity) 2 Invasion probability and endemic prevalence 3 Outline of our main results 4 Theory 5 The prevalence-invasion relationship 6 Numerical Illustration 7 Conclusion

Markovian SIS dynamics on networks Susceptible individualInfected individual Contact network the rate of infecting contacts from j to i 2 types of event for individual i Infection Recovery the rate at which i recovers when infectious population size

‘Invasion probability’ and ‘endemic prevalence’ Context 1: mean-field theory (large, evenly-mixed homogeneous population) Fraction of population infected in endemic situation (stationary solution) Probability of invasion from single initial infected (from theory of birth/death processes)

Issues: 1 Finite state space – extinction certain – no stationary distribution for the endemic situation. 2 Meaning of invasion is obscure. Context 2: finite heterogeneous networks Objectives: 1) To define invasion probability and endemic prevalence. 2) To determine the relationship between the quantities. First: Numerically investigate Invasion and Prevalence on a complex network: The network of contacts between sites in the British Poultry Flock ‘Invasion probability’ and ‘endemic prevalence’

Approx. 12,700 sites

Weighted Structured Approx. 12,700 sites

Look for dichotomised behaviour Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Invasion probability Average over apparently ‘stable’ behaviour Endemic prevalence ‘Invasion probability’ and ‘endemic prevalence’

Look for dichotomised behaviour Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Invasion probability For an individual node, prevalence is the probability that it is infectious in this “endemic state”. Average over apparently ‘stable’ behaviour Endemic prevalence ‘Invasion probability’ and ‘endemic prevalence’

Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Numerical illustration for a single individual in this complex network: Let T→kT for various k ‘Invasion probability’ and ‘endemic prevalence’

Outline of our main theoretical results: We can formulate precise mathematical quantifiers for both invasion probability and endemic prevalence, and demonstrate an exact relationship between them, by making use of: 1.‘Graphical representation’ of the dynamic process and the property of ‘duality’ (Harris 1976, Holley & Liggett 1975). 2.The quasi-stationary distribution (Daroch & Seneta 1967). Almost all work relating to the graphical representation and duality considers the contact network as an infinite square lattice (undirected, non- weighted). But, it is simple to apply these ideas to weighted and directed graphs. Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 ‘Invasion probability’ and ‘endemic prevalence’

Graphical representation i j k 0 t Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

Graphical representation 0 t TT 0 t Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

Duality Importantly, the property of duality for Markovian SIS dynamics on networks implies that: = where A is any set of individuals. Also: = Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

The quasi-stationary distribution (endemic situation) If the network is strongly connected such that the infection can travel, via some route, from any individual to any other individual then: 1 the transient states form a single commuting class 2 Conditional on non-absorption, the system will approach a unique quasi-stationary distribution (QSD) as t approaches infinity (assuming the system is initiated in a transient state) Therefore, for a subset (of individuals) A in a strongly connected network T, we define: = Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

The quasi-stationary distribution (endemic situation) Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

The quasi-stationary distribution (endemic situation) Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 !!!

Quasi-invasion (invasion probability) For a subset (of individuals) A in a strongly connected network T, we define: = The definition works because: 1) given non-extinction, the system converges to the unique QSD, irrespective of the initial transient state. 2) the all-infected state maximises the expected time to extinction (easily proved via graphical representation). Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

More generally, for any model of infection dynamics where there is a unique QSD that is independent of the initial transient state, we define: = Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Quasi-invasion (invasion probability)

The prevalence-invasion relationship Our main result can be stated as follows: = For any subset A of a strongly connected network T, conditional on Markovian SIS dynamics. This follows from the definitions and duality. Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

= Summing i over all N nodes and dividing by N: = Probability of invasion when seeded from a single individual chosen uniformly at random Fraction of sites infected in the QSD (endemic situation) Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 The prevalence-invasion relationship For a single individual i

i = Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Numerical illustration – small square lattice

i = Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 Numerical illustration – small square lattice

Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028

Conclusions Prevalence data can be used to inform invasion risk. The prevalence-invasion relationship lends support to the targeting of high risk individuals, in these kinds of (undirected) systems, as an effective strategy for the mitigation and control of emerging epidemics. We believe that it will now be possible to define invasion probability in a consistent way across a wide range of stochastic models, and that the prevalence-invasion relationship for more complicated models (extra compartments, non-Markovian) should be investigated. Wilkinson RR, Sharkey KJ (2013) PLoS ONE 8(7): e69028 = The End