���M.S.S. in Health Economics�HE 604: Public Health and Epidemiology (PHE)�Designing Infectious Disease Models���

Dr. Aninda Nishat Moitry

MBBS, MPH, MSc

04 April 2022

Overview of the presentation

• Models of infectious disease transmission
• Compartmental models
• Modelling an epidemic: SIR model
• Special cases: HIV, gonorrhoea

​

Models

• Simplification of a system, suitable for analysis
• Tools for thinking
• Needs to capture essential behaviour of interest and incorporate essential processes
• Everyone uses models in their head: making them explicit mathematically clarifies thinking and allows others to examine them
• Mathematical models allow precise, rigorous analysis and quantitative prediction

​

Models of infectious disease transmission

• We are interested in the population-level effect of processes occurring at the individual level
• An uninfected individual’s risk of becoming infected (the “force of infection”) depends upon
• the prevalence of infectious individuals (a population-level characteristic)
• (and rate of contact between individuals, infectiousness of infected individuals, etc.)
• So transmission of infection in a population is a dynamic process and the individual risk of infection can change over time
• Incidence depends upon prevalence and of course prevalence depends upon incidence

🡪Non-linearity, with time-varying positive and negative feedback

• It requires dynamic models for prediction and analysis of putative programmes

Modelling considerations

• Natural history of infection
• e.g., latency, infectious period, immunity

​

• Transmission of infection
• e.g., direct or indirect? What affects contact rate?

​

• Population structure and demography
• e.g., stratify by age, sex, geographic location, etc.?

​

• Interventions
• What parts of the disease-transmission process are targeted?

​

Depends on infection

Depends on population

Depends on intervention(s)

Creating a compartmental model

• Compartmental infectious disease models are constructed from two basic types of objects:

​

1. Compartments: containing people in each infection state (stage of infection)
1. These values are stored in state variables, which together describe the state of the system

(2) Rates of change of numbers in compartments:

e.g. birth rate, incidence of infection, recovery rate

​

• These rates usually depend upon the values of one or more of the state variables, so they change as the state of the system changes – i.e. there is feedback – e.g. population growth rates, disease transmission rates

​

​

Creating a compartmental model (Contd.)

• The relationships between the values of the state variables and their rates of change are expressed by functions (one for each rate of change)

​

• Each compartment has
• A state variable, keeping track of the number of individuals in that compartment; and
• A differential equation, describing the rate of change of its state variable, which is comprised of one or more of the functions

​

​

​

More complex natural histories

Examples:

• Maternal antibodies in young
• Incubation and latent periods
• Asymptomatic infection
• Infectious period with multiple stages
• Resolution of infection
• Death, immunity, return to susceptibility?
• Immunity
• Sterilizing?
• Waning or permanent?
• Strain-specific or general?

​

But let’s start simply!

• The point is to understand fundamental processes rather than to make models ‘realistic’ initially
• Realistic models are complicated and hard to understand unless fundamental processes that occur are understood first

• Each flow rate is the no. individuals entering or leaving a compartment per unit time

​

• It depends upon:
• The per-capita rate (the hazard); and
• The number of individuals subjected to that per-capita rate (i.e. exposed to the hazard)

​

• The population rate is the product of these
• i.e. per-capita rate x no. individuals

​

​

Flows between compartments

• Rate of recovery from infection
• Let the per-capita rate of recovery be 𝝈;

​

​

• Since the no. Infecteds is I,

​

​

​

• The population recovery rate is therefore 𝝈I

​

​

​

​

​

​

Flows between compartments (Contd.)

When we analyse the model we vary parameter values and see how this affects the state variables – i.e. how the output graphs change

State variable

• Varies intrinsically as model runs
• Not manipulated directly
• Model ‘outputs’

Parameter

• Specified extrinsically
• Only changes if/when we specify
• Model ‘inputs’

• Death rate of Susceptibles, Recovereds
• Let the per-capita “background” death rate be b;
• Since the no. Susceptibles is S, and Recovereds, R,
• The population death rate of Susceptibles is bS and of Recovereds is bR

​

• Death rate of Infecteds
• Infecteds experience the “background” death rate (b) + disease-induced death rate (⍺), so their per-capita death rate is (b + ⍺), so
• The population death rate of Infecteds is (b + ⍺)I

​

​

Flows between compartments (Contd.)

Flows between compartments (Contd.)

• Birth rate
• Let the per-capita birth rate (averaged over males and females) be a;
• Assuming individuals in all compartments give birth at the same rate, the no. giving birth is S+I+R, which we cam simplify by defining total population size, N = S+I+R, so
• The population birth rate is aN
• In practice, we often set the per-capita rate equal to the per-capita background death rate to maintain a constant population size in the absence of disease
• However, using different parameters means that we do not have to make them equal if we don’t want to

​

Flows between compartments (Contd.)

• Infection rate
• Per-capita rate of infection of Susceptibles (the “force of infection”) is not fixed;
• It depends upon:
• The number of Infectious at the particular point in time, and
• The rate of contact with Susceptible individuals, and
• The transmission probability
• Therefore the population rate of infection depends upon the number of Infecteds as well as the number of Susceptibles
• Hence it is non-linear

��Solving the equations: Integration��

• We are interested in plotting how the numbers in each compartment change over time
• In our differential equations we have specified derivatives- i.e. defined how the slopes of the lines of the numbers in each compartment relate to the state of the system (i.e. the numbers in each compartment) at any point in time
• To get the lines themselves, we solve the equations by integration
• Most models cannot be solved algebraically, so use computers for numerical integration

Example epidemic: SIR model

• Initially in an epidemic in a naïve population, the rate of spread accelerates as transmission ↑ Infecteds, which ↑ force of infection which further ↑ the rate of spread
• Then spreading slows as Susceptibles ↓ significantly, (even though force of infection continues to ↑)
• If Infecteds recover to become immune then the epidemic can fade out, unless
• New naïve individuals enter the population, or
• Immunity wanes, returning individuals to susceptibility

�Behaviour of simple models�

• The point is to understand fundamental processes rather than to make models ’realistic’ initially
• Realistic models are complicated and hard to understand unless fundamental processes that occur are understood first
• Analysis involves varying parameter values singly and in combination to see how model outputs change (but no time to do it here)

More complex natural histories

• Although models should be parsimonious, they need to capture essential details of the disease, so we often have more compartments
• Which details are included will depend on the questions addressed by the modelling, and the availability of data

Latent period

• In simple models, we assume individuals become infectious as soon as they are infected
• However, there may be a significant latent period between being infected and becoming infectious
• In modelling literature, latently-infected are often called “Exposed”

Incubation period

• This is the time between becoming exposed and becoming symptomatic
• Often infections are only treated when a person becomes symptomatic and so becomes aware that they have an infection
• For some diseases, symptoms occur before the person is infectious (e.g. SARS) whilst for others symptoms begin after the person is infectious (e.g. HIV, possibly influenza)
• For some diseases, symptoms and infectiousness occur together (e.g. pulmonary TB)

​

Branching natural histories

• Some infections have different natural histories in different people
• Example, with gonorrhoea, some people develop symptomatic infection, whilst others develop asymptomatic infection

Final remarks

• Modelling is not alchemy!
• It cannot turn base metal (poor quality* or non-existent data) into gold (an accurate, precise prediction)

[*NB. quality >> quantity]

• Where data are lacking or imprecise, modelling can examine scenarios based on varying a parameter within its plausible range
• This can determine the importance (or not) of measuring that parameter more accurately
• It might be possible to rule-out potential interventions even without quantifying them accurately- if the best case scenario is still unimpressive

A model is not a substitute for data - it is a tool to analyse data

Parameter estimation is the hardest part

• To model something mathematically its effect has to be quantified: models have parameters, which have numerical values – either:
1. Measured: usually hard – and requires a lot of high-quality data
2. Varied across plausible ranges – scenario analysis
• Models are data-hungry because lots of factors affect transmission- this is the price of realism
• Estimating parameters from data is typically the hardest part of modelling – not the designing or programming of the model
• It requires expertise in modelling and understanding of the data- multidisciplinary
• Most modelling analyses use models specially designed (or at least adapted) to the available data

Acknowledgement

My professors from Imperial College London

• Prof. Peter J White
• Prof. Christophe Fraser
• Dr. Thomas Hagenaars
• Prof. Thomas Churcher
• Dr. Patrick Walker

Thank you