Continuous Time Models

1
Lecture 3

• Continuous Time Models

2
Mathematical Epidemiology

• The simplest model of this type has 3
  compartments that individuals may belong to
• S susceptible
• I infected
• R recovered
• These models are compartmental models

3
Infectious Disease Dynamics

• ? - birth rate
• d natural death rate
• a disease induced death rate
• ? - infection rate
• ? - recovery rate
• w waning rate
• The SIR model can be used to model epidemics and
  endemic diseases
• Some of the earliest work on theory of epidemics
  is due to Kermack and McKendrick (1927)

4
SIR Model

• Lets analyze an SIR model
• Write the model and parameters
• Assumptions
• Find the steady states
• Find the eigenvalues for each steady state
• Stability
• The stability conditions are related to an
  important term in Mathematical Epidemiology
• The Basic Reproductive Ratio (R0)

5
SIR Model

• Lets analyze an SIR model
• Write the model and parameters
• Assumptions
• Find the steady states
• Find the eigenvalues for each steady state
• Stability
• The stability conditions are related to an
  important term in Mathematical Epidemiology
• The Basic Reproductive Ratio (R0)

6
Basic Reproductive Ratio

• R0
• The number of infected individuals produced by
  one infected individual introduced into a
  population of susceptibles
• A fundamental concept in both within-host and
  epidemiological models of pathogen dynamics
• Used to
• Gauge risk of a novel pathogen
• Estimate degree of control necessary to contain
  and outbreak

7
Basic Reproductive Ratio

• Examples of use
• Emerging diseases
• SARS, avian influenza
• Livestock diseases
• BSE, Foot and Mouth disease
• In-host models
• HIV, HPV, Hepatitis, influenza
• Vector-borne diseases
• Dengue, Malaria, West Nile Virus

8
Basic Reproductive Ratio

• R0
• The number of infected individuals produced by
  one infected individual introduced into a
  population of susceptibles
• What does this really mean?
• How do we calculate this?
• From models
• From data
• What conclusions can we make from calculating R0 ?

9
Basic Reproductive Ratio

• R0gt1
• Infected makes more than itself on average
• So disease grows
• R0lt1
• Infected makes less than itself on average
• So disease dies out
• A goal in mathematical epidemiology is to
  determine how to decrease R0 using controls (i.e.
  vaccine, drug therapy, quarantine, isolation,
  etc) so that it is less than 1

10
SIR model with controls

• Lets add vaccination to our SIR model
• What happens to our eigenvalues?
• Lets add isolation to our SIR model
• What happens to our eigenvalues?
• Lets add drug therapy to the SIR model
• What happens to our eigenvalues?

11
SIR Model

• Example
• Influenza infection in an English boarding school
• Read attached handout
• Make a model to determine the size of the
  outbreak using the first 3-4 data points
• Does your model agree with the chart data and the
  description of the outbreak in the text?

12
Markov Processes

• Sequence of random variables x0, x1, x2, taking
  values is a set S of states
• Such that for any n, the distribution of xn
  depends only on xn-1
• This is the Markov Property
• Where you go next depends only on where you are
  now
• Sentimental and historical core of stochastic
  processes

13
Continuous Time

• We have a set X(t), t ? 0,8)
• This process satisfies the Markov Property if
• For any finite set 0 t t2 lt lt tn lt tn1 of
  times and corresponding state ii, i2, , I, j of
  states we have

14
Continuous Time

• In addition, if for all s and t such that 0 s
  t
• depends only on t-s and not on s and t
  themselves, then the process is called homogenous
  and has stationary transition probabilities
• We will assume this

15
Continuous Time

• We usually think of this Pr as a transition
  probability equal to some value p
• Here we think of it in terms of a transition
  function instead

16
Continuous Time

• W need a whole matrix of functions to find the
  process
• And a vector of starting probabilities PrX(0)i

17
Useful Definitions

• Directed graph
• State diagram with nodes and connecting arrows
• Sometimes, converges to
  as n ? 8
• is called the long run distribution or
  the invariant distribution
• If this happens, then the process P is ergotic
• Absorbing state
• Pik0 for k?I
• No way to leave this state

18
Useful Definitions

• Closed set
• C is a closed set if pik0 for all i in C and k
  not in C
• no way to leave this set of states, one you get
  there
• Periodic states repeat at regular intervals
• Aperiodic opposite
• Persistent states
• If you leave you will always come back eventually
• Transient states - opposite

19
Useful Definitions

• Two states intercommunicate if
• Can get between them (and back) in n(m) steps
• Irreducible set
• No closed subset exists (except the empty set)
• True if and only if all pairs in the set
  intercommunicate

20
Continuous Time

• Note that
• Chapman Kolmorgorov for cts time

21
Continuous Time

• Now let qij right derivative of pij(t) at t0
• Graph p11 and q11 is the slope
• Graph p12 and q12 is the slope
• For i?j
• qij is the intensity of the transition from i
  to j
• qij is not a probability (i.e. can be gt 1)
• qij is 0 or positive for all i ? j
• For ij
• qij is the intensity of the passage from i to
  somewhere else
• qij is 0 (absorbing state) or negative

22
Continuous Time

• How are the qijs related to the pijs?
• For small time increment h
• So qij h pij(h), h small

23
Continuous Time

• Similarly, qii related
• So qii h 1-pii(h)
• For stationary Markov processes, the transition
  intensities are constants
• Can use Qqij to define the process and get
  long run distribution, etc

24
Continuous Time

• So if the process is ergotic, the invariant
  distribution is found by solve for pi
• Intensity in intensity out
• i.e.

25
Continuous Time

• So we have matrix Q with
• Diagonal entries ve or 0
• Off diagonals are ve or 0
• Row sums 0 if process is honest (conservative)
• Sojourn
• Let Ti be the sojourn in state i
• i.e.

26
Continuous Time

• What this means is
• You stay in state I for mean time -1/qii than
  move to state j with probability qij/qii

27
Continuous Time

• Another way to state the Markov Property is