Basic concepts of mathematical epidemiology

11-14 October 2022

Julien Arino (julien.arino@umanitoba.ca)

Department of Mathematics & Data Science Nexus
University of Manitoba*

Canadian Centre for Disease Modelling
Canadian COVID-19 Mathematical Modelling Task Force
NSERC-PHAC EID Modelling Consortium (CANMOD, MfPH, OMNI/RÉUNIS)

Outline

• Compartmental models
• The (epidemic) Kermack and McKendrick SIR model
• Incidence functions
• The (endemic) SIS model
• Effect of vaccination - Herd immunity

Compartmental models

Compartmental models

• Compartmental models have become synonymous with epidemiological models
• While epidemic models have mostly been compartmental, most development in the 1970s-1980s was not about epidemiological models in particular
• See in particular the works of John Jacquez, Carl Simon, GG Walter
• Injustly forgotten to a large extent; there are some really nice results in the field

Compartment (Jacquez 1979)

A compartment is an amount of some material which acts kinetically like a distinct, homogeneous, well-mixed amount of material. A compartmental system consists of one or more compartments which interact by exchanging the material. There may be inputs into one or more compartments from outside the system and there may be excretions from the compartments of the system.

A compartment

• size of the compartment, i.e., amount of kinetically homogeneous material present in ;
• and fractional transfer coefficients/functions
• excretion coefficient/function
• input from outside the system

The (epidemic) Kermack and McKendrick SIR model

This is a particular case!

Kermack & McKendrick formulated the model that follows in a much more general work

Really worth taking a look at the series of papers!

• A contribution to the mathematical theory of epidemics (1927)

Followed by "Contributions to the mathematical theory of epidemics."

• II. The problem of endemicity (1932)
• III. Further studies of the problem of endemicity (1933)
• IV. Analysis of experimental epidemics of the virus disease mouse ectromelia (1937)
• V. Analysis of experimental epidemics of mouse-typhoid; a bacterial disease conferring incomplete immunity (1939)

The underlying question - What is the size of an epidemic?

• Suppose we consider the occurrence of an epidemic peak
  • Does it always take place?
  • When it occurs, how bad is it?
  • If an epidemic moves through a population, is everyone infected?

General setup

• Consider a closed population
• Assume individuals in the population can be in one of three states:
  • susceptible (to the disease) if they are not currently harbouring the pathogen
  • infectious (and infected) if they have contracted the disease and are actively spreading it
  • removed (from being infectious) once they have actively stopped spreading the disease, whether it is through recovery or death

there are three compartments and the aim of modelling is to describe evolution of numbers in each compartment

Types of compartments

• the number of susceptibles at time
• the number of infectious/infected at time
• the number removed at time
• the total population

The SIR model without demography

• Time interval under consideration sufficiently small that demography can be omitted (we say there is no vital dynamics)
• Incidence is mass action
• Incubation period is short or even non-existent
• Infection has limited duration for each individual

Model flow diagram

• Flow diagrams represent the different courses individuals may follow before infection and while infected with the disease
• They are extremely useful communication and working tools

The Kermack-McKendrick SIR model

Considered with initial conditions (IC) , and (often the latter is zero)

The Kermack-McKendrick SIR model

As often with ODE, write and omit time-dependence of state variables:

Considered with initial conditions (IC) , and (often the latter is zero)

Reducing the problem

3 compartments, but inspection shows that removed individuals do not influence the dynamics of or

Furthermore, total population satisfies

so is constant and dynamics of can deduced from that of

So now consider

Equilibria

Consider the equilibia of

From

• either
• or

Substitute into

• in the first case,
• in the second case, any is an equilibrium (continuum of EP)

Workaround - Study

What is the dynamics of ? We have

provided

Careful! Remember that and are and .. equation thus describes the relationship between and along solutions to the original ODE -

We can integrate equation , giving trajectories in phase space

with

The initial condition gives , and the solution to - is thus, as a function of ,

Trajectories in phase plane corresponding to IC and

The basic reproduction number

Suppose total population is normalised, i.e., . Then

Define

Incidence functions

Before proceeding further, worth spending some time discussing incidence functions, which describe how contacts between individuals take place and lead to new cases

See in particular McCallum, Barlow & Hone, How should pathogen transmission be modelled?, Trends in Ecology & Evolution 16 (2001)

Remark - Incidence function versus force of infection

Two different forms for the rate of movement of individuals from to whatever infected compartment they end up in:

• is an incidence function
• is a force of infection

The two are of course essentially equivalent, the context tends to drive the form used. Advanced PDE models that consider for instance an age-of-infection structure need to integrate over and thus often use force of infection, others are somewhat random..

Interactions - Infection

• Rate at which new cases appear per unit time is the incidence function

• Depends of the number of susceptible individuals, of infectious individuals and, sometimes, of the total population
• Incidence includes two main components:
  • a denumeration of the number of contacts taking place
  • a description of the probability that such a contact, when it takes place, results in the transmission of the pathogen
• Choosing an appropriate function is hard and probably one of the flunkiest parts of epidemic modelling

Two most frequently used functions

The two most frequently used incidence functions are mass action incidence

and standard (or proportional) incidence

In both cases, is the disease transmission coefficient

Mass action incidence

• There is homogenous mixing of susceptible and infectious individuals
• Strong hypothesis: each individual potentially meets every other individual

In this case, one of the most widely accepted interpretations is that all susceptible individuals can come across all infectious individuals (hence the name, by analogy with gas dynamics in chemistry/physics)

When population is large, the hypothesis becomes unrealistic

Standard (proportional) incidence

The other form used frequently:

Each susceptible individual meets a fraction of the infectious individuals

Or vice-versa! See, e.g., Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences (1976)

Case of a larger population

Constant population

When the total population is constant, a lot of incidence function are equivalent (to units)

Suppose , then

and if the right hand side is satisfied, then and identical

Keep in mind units are different, though

Units of

Recall that if is the population in compartment at time , then has units

In a differential equation, left and right hand side must have same units, so..

Mass action incidence

has units number/time if has units

Standard incidence

has units number/time if has units

General incidence

These functions were introduced with data fitting in mind: fitting to data, find the best matching the available data

Incidence with refuge

The following implements a refuge effect; it assumes that a proportion of the population is truly susceptible, because of, e.g., spatial heterogenities

Negative binomial incidence

For small values of , this function describes a very concentrated infection process, while when , this function reduces to a mass action incidence

Switching incidence

Arino & McCluskey, Effect of a sharp change of the incidence function on the dynamics of a simple disease, Journal of Biological Dynamics (2010)

Scale population so switch occurs at and suppose

In SIS with non-constant population

and disease-induced death rate, periodic solutions found

The (endemic) SIS model

• Consider a closed population
• Assume individuals in the population can be in one of two states:
  • susceptible (to the disease) if they are not currently harbouring the pathogen
  • infectious (and infected) if they have contracted the disease and are actively spreading it

there are two compartments

• the number of susceptibles at time
• the number of infectious/infected at time
• the total population total

The following hypotheses describea disease for which the incubation period is short or even non-existent

We also assume infection has limited duration for each individual

Type of compartments

Susceptible individuals

• Are born at a per capita rate proportional to the total population
• Die at the per capita rate , proportional to the susceptible population
• Newborns are susceptible (vertical transmission is ignored)

Infected and infectious individuals

• Die at the per capita rate , proportional to the infected/infectious population
• Recover at the per capita rate
• We do not take into account disease-induced death

Model flow diagram

The model

Consider initial value problem (IVP) consisting of this system together with initial conditions and

Remarks

• - is an SIS (Susceptible-Infectious-Susceptible) model
• If (no recovery), then model is SI
  • In this case, infected individual remains infected their whole life (but disease is not lethal since there is no disease-induced death)
• Diseases with these types of characteristics are bacterial diseases such as those caused by staphylococcus aureus, streptococcus pyogenes, chlamydia pneumoniae or neisseria gonorrhoeae

Birth and death are relative

Remark that the notions of birth and death are relative to the population under consideration

E.g., consider a model for human immunodeficiency virus (HIV) in an at-risk population of intravenous drug users. Then

• birth is the moment the at-risk behaviour starts
• death is the moment the at-risk behaviour stops, whether from "real death" or because the individual stops using drugs

Analysis of the system

System - is planar nonlinear

Typically, we would use standard planar analysis techniques

However, here we can find an explicit solution

NB: while this is useful illustration, this is a rare exception!!

Dynamics of

We have

As a consequence, for all ,

Proportions

Remark that . The derivative of is

Since ,

Substite the right hand term of in this equation, giving

The system in proportions

Since , we can use in the latter equation, giving . As a consequence, the system in proportion is

Since constant, solutions of - are deduced directly from those of - and we now consider -

Rewrite as

This is a Bernoulli equation and the change of variables gives the linear equation

so that finally

An integrating factor is

and thus

for , so finally

The initial condition takes the form . Thus,

which implies that

As a consequence, the solution to the linear equation is

and that of is

In summary, the solution to the system in proportions is given by

and

From these solutions, there are two cases:

• If , then , so and
• If , then ; thus, and

The basic reproduction number

Reformulate in epidemiological terms, using the basic reproduction number,

We have the following equivalencies

Also,

We have proved the following result

Further remarks about

• determines the propensity of a disease to become established in a population
• When establishment occurs (), also determines the level of endemicity of the disease ( and )
• The aim of control policies is therefore to reduce to values less than 1 (ideally) or reduce the value of in any case to lower endemicity
• Remark that for our basic model, is the average time of sojourn in the compartment before death or recovery and is the probability of infection

The basic reproduction number

Indicator often used in epidemiology. Verbally

average number of new cases generated when an infectious individual is introduced in a completely susceptible population

• If , then each infectious individual infects on average less than 1 person and the epidemic is quite likely to go extinct
• If , then each infectious individual infects on average more than 1 person and an epidemic is quite likely to occur

Some values of (estimated from data)

Classic way to compute

Take SIS - normalised to

DFE:

When eigenvalues and

LAS of the DFE is determined by the sign of

Find same as before

A more efficient way to : next generation matrix

Diekmann and Heesterbeek, characterised in ODE case by PvdD & Watmough (2002)

Consider only compartments with infected individuals and write

• flows into infected compartments because of new infections
• other flows (with sign)

Compute the (Frechet) derivatives and with respect to the infected variables and evaluate at the DFE

Then

where is the spectral radius

Here, there is a single infected compartment, so

So in turn

and thus . Thus, finally

The main result of PvdD and Watmough (2002)

I make conditions (A1)-(A5) explicit in these slides/this video and discuss why it can be important to check they do hold true in these slides/this video

Summary thus far

• The KMK SIR epidemic model in which the presence or absence of an epidemic wave is characterised by the value of
• An SIS endemic model in which the threshold is such that when , the disease goes extinct, whereas when , the disease becomes established in the population
• Both KMK SIR and SIS are integrable in some sense. This is an exception!!!
• By abuse of language, we often speak of epidemic models when referring to both epidemic and endemic models. We should instead say epidemiological models or models for disease propagation. That's one quixotic quest that I am not even considering...

Effect of vaccination - Herd immunity

Take SIR model and assume the following

• Vaccination takes susceptible individuals and moves them directly into the recovered compartment, without them ever becoming infected/infectious
• Birth = death
• A fraction is vaccinated at birth

Computation of

• DFE, SIR:

• DFE, SIR with vaccination

Thus,

• In SIR case

• In SIR with vaccination case, denote and

Herd immunity

Therefore

• if
• To control the disease, must take a value less than 1, i.e.,