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Dynamical Fever: under the hood���Juliet Pulliam, PhD�Department of Biology and�Emerging Pathogens Institute�University of Florida��

Notes for ‘the reveal’

MMED 2016

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Under the hood

  • This model world represents my ideal world
    • Everyone has a dog (ie, same population size for dogs and humans)
    • All dogs & all people get along great and spend most of their time at the dog park (ie, contacts are well mixed)
    • For the most part, everyone is supper happy in DAIDD county so the population doesn’t change throughout the year; however, there is one dog whose owner takes it to visit grandma every year over Christmas break, and that dog always comes back on 1 Jan infected with DF

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More detail…

  • Dogs can infect dogs and people, and contact/transmission rates are the same to both
  • People can’t infect dogs or people
  • Infected/infectious period is 1 week; no incubation or latency
  • The vaccine is perfectly efficacious
  • Immunity (natural or vaccine-derived) is short-lived; everyone’s always completely susceptible again by 1 Jan
  • No differences of any kind between individuals, other than the one that always starts the epidemic

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Definitely not the real world…

  • If people ask about symptoms:
    • DF doesn’t do any long-term damage to humans or pups; it just turns them blue for a week. So it’s not desirable… but not killing anyone!

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Model specification

  • The model world is implemented as:
    • Stochastic
    • Discrete time
    • Compartmental model

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  • Specifically, a Reed-Frost-like chain binomial:
    • Non-overlapping generations
    • R0 = 2

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Model taxonomy

continuous time

  • Gillespie algorithm

discrete time

  • Chain binomial type models

(eg, Stochastic Reed-Frost models)

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Stochastic

continuous time

  • Stochastic differential equations

discrete time

  • Stochastic difference equations

​

​

​

Discrete treatment of individuals

​

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

​

continuous time

  • Ordinary differential equations
  • Partial differential equations

discrete time

  • Difference equations

(eg, Reed-Frost type models)

​

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NOTE TO AW

  • The remaining slides include more technical material on Reed-Frost and R-F-like models (both deterministic and stochastic versions)
  • You will probably not want to use them, but the stochastic R-F model (ie, a chain binomial w non-overlapping generations) is the basis for the simulation code, which is available here:

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https://github.com/ICI3D/RTutorials/blob/master/dynamicalFeverModelScript.R

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Model taxonomy

continuous time

  • Gillespie algorithm

discrete time

  • Chain binomial type models

(eg, Stochastic Reed-Frost models)

​

Stochastic

continuous time

  • Stochastic differential equations

discrete time

  • Stochastic difference equations

​

​

​

Discrete treatment of individuals

​

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

​

continuous time

  • Ordinary differential equations
  • Partial differential equations

discrete time

  • Difference equations

(eg, Reed-Frost type models)

​

🡸

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The Reed-Frost model

Abbey, H (1952) An examination of the Reed-Frost theory of epidemics. Hum Biol 24: 201-233. [As quoted in Fine, PEM (1977) Am J Epi 106(2): 87-100.]

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The Reed-Frost model

  • Time unit is roughly time from infection to end of infectiousness
  • Generations of cases do not overlap
  • If p=1-q is the probability of any two individuals coming into “adequate contact” during a time unit, 1-q^C is the probability a susceptible individual becomes infected during a time unit, so the expected number of cases in the next time unit is

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The Reed-Frost model

  • The full set of equations describing the deterministic population update is:

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​

​

​

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  • If N=S+C+R is the total population size, the basic reproductive number for this model is

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The Reed-Frost model

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Model taxonomy

continuous time

  • Gillespie algorithm

discrete time

  • Chain binomial type models

(eg, Stochastic Reed-Frost models)

​

Stochastic

continuous time

  • Stochastic differential equations

discrete time

  • Stochastic difference equations

​

​

​

Discrete treatment of individuals

​

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

​

continuous time

  • Ordinary differential equations
  • Partial differential equations

discrete time

  • Difference equations

(eg, Reed-Frost type models)

​

🡸

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The stochastic R-F model

  • The stochastic formulation of the Reed-Frost model is a type of chain binomial model with non-overlapping generations

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​

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  • For small populations (eg, households), final size distributions can be calculated

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The stochastic R-F model

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Chain binomial models

  • Chain binomial models can also be formulated based on the same parameters we used in the ODE models and with overlapping generations
  • As before, instantaneous hazard of infection for a individual susceptible individual is
  • For a susceptible at time t, the probability of infection by time is

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Chain binomial models

  • Similarly, for an infectious individual at time t, the probability of recovery by time t+ dt is

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  • The stochastic population update can then be described as

where

and

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Chain binomial models

  • For this model, if D is the average duration of infection, the basic reproductive number is:

​

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  • Non-generation-based chain binomial models can be adapted to include many variations on the natural history of infection
  • Discrete-time simulation of chain binomials is far more computationally efficient than event-driven simulation in continuous time

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Chain binomial simulation

while (I > 0 and time < MAXTIME)

Calculate transition probabilities

Determine number of transitions for each type

Update state variables

Update time

end