Epidemiological modelling

Question 1

Deterministic models - characteristics, advantages (1), disadvantages (1)

Answer 1

Describe what happens “on average” in a population. In these models, the input parameters are fixed and therefore the models predictions are predetermined (model performs the same way for a given set of initial conditions). Majority of deterministic models are compartmental models (individuals in population are divided into “compartments” such as susceptible, infectious and recovered and the model tracks the number (difference equations) or rate of change (differential equations) of the number of individuals in each compartment over time. Fractions of the population are assigned to a particular state at any given time.

Advantages:

1. Enables assessment of the sensitivity of a systems behavior to changes in certain parameters

Disadvantages:

1. Only permit evaluation of the average outcome (i.e. not able to show the probability of that outcome occurring or the range of possible outcomes)

Question 2

Stochastic models - characteristics, advantages (1), disadvantages (2)

Answer 2

Input parameters aren’t fixed but rather take on a range of values according to some (assigned) probability distribution. Thus, stochastic models incorporate elements of random variation and chance. Used when modelling transmission in small populations or when it is important to provide estimates of the range of possible outcomes. Random number generators can be used at each step to determine if an individual become infected or not.

Advantages:

1. The results can be expressed as confidence intervals and expected values, not just point estimates.

Disadvantages:

1. More complicated to set up
2. More data intensive (require estimates for probability distributions)

Question 3

Mathematical models - advantages (1), disadvantages (1)

Answer 3

Method for deriving a solution depends on mathematical manipulation i.e. closed-form solution to the state of the system at some equilibrium.

Advantages:

1. Can be evaluated rigorously and stability criteria can be determined

Disadvantages:

1. Less realistic (many assumptions)

Question 4

Simulation models - method

Answer 4

Method for deriving a solution depends on numerical substitution according to model-defined rules (e.g. probability distribution) to find expected outcome. Typically, simulation models are stochastic.

Question 5

Stages of model development and assessment (6)

Answer 5

1. Specify the objectives of the model
2. Specify model input parameters - review data available, collect new data as needed, estimate unknown parameters by fitting the model to available data
3. Set up the model - describe model equations
4. Model validation - check model outputs against independent data sets
5. Model optimization -
6. Use of model in decision support - e.g. predicting impact of control strategies

Question 6

Reed-Frost Model - method, advantages (2), disadvantages (4)

Answer 6

Example of a chain binomial model, with event measured at discrete points in time. The following parameters are set initially:

1. Size of the population
2. Number of individuals already immune
3. Number of cases (usually set at 1)
4. Probability of adequate contact

Simple mathematical formula describes how many become infected and how many become immune in each successive time interval.

Advantages:

1. Possible to predict how an epidemic will behave in a population (time course, numbers affected);
2. Predict how time course, numbers affected are influenced by specific control measures which affect probability of adequate contact

Disadvantages (assumptions):

1. Homogenenous mixing;
2. Direct contact transmission (not transmitted in any other way);
3. Cases become infectious then immune over successive time intervals;
4. Closed population

Question 7

Kermack-McKendrick (SIR) model - advantages (2), disadvantages (4)

Answer 7

Example of a deterministic (compartmental) model - SIR. Used differential equations to describe rate of movement between compartments. Demonstrated that there is a threshold density of susceptibles below which epidemic will not occur (herd immunity).

Advantages:

1. Possible to predict how an epidemic will behave in a population (time course, numbers affected);
2. Predict how time course, numbers affected are influenced by specific control measures which affect probability of adequate contact

Disadvantages (assumptions):

1. Homogenous mixing (at rate betaSI - law of mass action);
2. Cases are instantly infectious (no incubation period - no exposed category);
3. Cases recover with full immunity at a constant rate (1/duration of illness)
4. Closed population (no births, no deaths) so population size is constant