Basic concepts of math epi - Models in one population and their basic properties

4 April 2022

Julien Arino

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)

* The University of Manitoba campuses are located on original lands of Anishinaabeg, Cree, Oji-Cree, Dakota and Dene peoples, and on the homeland of the Métis Nation.

Outline

  • The (epidemic) Kermack and McKendrick SIR model
  • SLIAR extension of the KMK model
  • Incidence functions
  • The (endemic) SIS model
  • SLIRS model with constant population
  • Effect of vaccination - Herd immunity
  • SLIRS model - Global properties

The (epidemic) Kermack and McKendrick SIR model

This is a particular case!

K-MK formulated the model that follows in a much more general work

Really worth taking a look at this series of papers!

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?

The SIR model without demography

  • The time interval under consideration is sufficiently small that demography can be omitted (we say there is no vital dynamics)
  • Individuals in the population can be susceptible () or infected and infectious with the disease (). Upon recovery or death, they are removed from the infectious compartment ()
  • Incidence is mass action

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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)

Second case is a problem: usual linearisation does not work as EP are not isolated! (See Practicum 02)

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 ,

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Trajectories in phase plane corresponding to IC and

The basic reproduction number

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

Define

Let be a solution to - in proportions and be defined as in . If , then tends to 0 when If , then first reaches a maximum

then tends to 0 when

The proportion of susceptibles is a nonincreasing function and its limit is the unique solution in of the equation

SLIAR extension of the KMK model

Extensions of the KMK model

Many many works (especially since COVID-19) have used KMK-type models

An extension of KMK - SLIAR models

SIR is a little too simple for many diseases:

  • No incubation period
  • A lot of infectious diseases (in particular respiratory) have mild and less mild forms depending on the patient

model with SIR but also L(atent) and (A)symptomatic individuals, in which I are now symptomatic individuals

Arino, Brauer, PvdD, Watmough & Wu. Simple models for containment of a pandemic (2006)

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Basic reproduction number

where

Final size relation

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 part 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

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

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

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

Asymptotic contact

where is one of the functions we just described

When , contacts are proportionnal to , whereas when , contacts are independent from

Asymptomatic transmission

where is a constant. E.g.,

with the function describing the contact rate and the function describing disease spread, assumed here to be of negative binomial incidence-type

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 and the aim of modelling is to describe evolution of numbers in each compartment

  • 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

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The model

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

Remarks

  • In what follows, assume to keep population constant
  • - 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

For system -, the following alternative holds

  • If , then
    • , the disease goes extinct
  • If , then
    • , the disease becomes endemic

as a function of

The higher , the higher the proportion of infectious individuals in the population

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Further remarks about

  • determines the propensity of a disease to become established in a population
  • The aim of control policies is therefore to reduce to values less than 1
  • The "verbal" definition of is the average number of secondary infections produced by the introduction of an infectious individual in a completely naive population
  • 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)

Infection Place Period
Measles Cirencester, England 1947-50 13-14
England & Wales 1950-68 16-18
Kansas, USA 1918-21 5-6
Ontario, Canada 1912-3 11-12
Willesden, England 1912-3 11-12
Ghana 1960-8 14-15
East Nigeria 1960-8 16-17

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

The main result of PvdD and Watmough (2002)

Suppose the DFE exists. Let then

with matrices and obtained as indicated. Assume conditions (A1) through (A5) hold. Then

  • if , then the DFE is LAS
  • if , the DFE is unstable

(We make conditions (A1)-(A5) explicit in Practicum 02 and discuss why it can be important to check they do hold true in Lecture 09)

Summary thus far

  • An SIR epidemic model (the KMK SIR) in which the presence or absence of an epidemic wave is characterised by the value of
  • An SLIAR epidemic model extending the KMK SIR
  • 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 the KMK SIR and the SIS are integrable in some sense. This is an exception!!!

SLIRS model with constant population

Incubation time

  • SIS and SIR: instantaneous progression from S to I
  • Some incubation periods (time from infection to symptoms - often matches beginning of infectiousness):
Disease Incubation
Yersinia Pestis 2-6 days
Ebola hemorragic fever (HF) 2-21 days
Marburg HF 5-10 days
Lassa HF 1-3 weeks
Tse-tse weeks, months
HIV/AIDS months, years

Hypotheses

  • Suppose there is demography. New individuals are born at a constant rate independent of the population
  • Disease is not transmitted to newborns (no vertical transmission): all births are to the S compartment
  • Disease does not cause additional mortality
  • New infections occur at the rate
  • There is an incubation period
  • After recovery, individuals are immune to the disease for some time

SLIRS model

  • average duration of the incubation period

  • average time until recovery

  • average duration of immunity

Behaviour of the total population

has dynamics

We can either solve explicitly or "study" qualitatively; either way,

total population asymptotically constant

Note that this is an easy way to set the total population to what you may want it to be: is known (average lifetime), so set so equals what you want..

DFE

Assume

Then (always a reasonable assumption)

From this,

DFE has and , denoted

Classic computation of

Computation of

Mathematically

bifurcation parameter aggregating system parameters, such that the DFE loses local asymptotic stability as crosses 1 from left to right

  • Obtained by considering the linearisation of the system at the DFE
  • Quickly becomes unmanageable (matrix size) and we get a non unique form

Jacobian matrix of - at an arbitrary point

where . Since (asymptotically) constant, , so

At the DFE, (in most cases) and , so

Eigenvalues are , and those of

If , then there is a single positive real root, while if , either both roots are real and negative or they are complex conjugate

Computing using the next generation matrix method

Example of the SLIRS model

Variations of the infected variables described by

Thus

Then compute the Jacobian matrices of vectors and

We have

Also, when constant, , then

and thus,

Let

Then

  • if , the DFE is LAS
  • if , the DFE is unstable

Important to stress local nature of stability that is deduced from this result. We will see later that even when , there can be several positive equilibria

Application : most frequently used incidence functions

  • Frequency-dependent contacts (mass action incidence)

  • Proportion-dependent contacts (standard incidence)

Remark

Method shown here is easily applied to more complex models

We will see some cases later

Links between model types

  • SLIR is SLIRS where
  • SLIS limit of an SLIRS when
  • SLI is SLIR where
  • SIRS limit of an SLIRS when
  • SIR is SIRS where
  • SIS limit of an SIRS when
  • SIS limit of an SLIS when
  • SI is SIS where

Expressions of

the DFE, denote

Model Model Model
SLIRS SIRS SIS
SIR SLIS SI
SLI SLIR

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

center

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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.,

By vaccinating a fraction of newborns, we thus are in a situation where the disease is eventually eradicated

This is herd immunity

SLIRS model - Global properties

  • As already mentioned, global stability properties are important in a mathematical context, less so in public health
  • Understanding that a DFE or an EEP are GAS is however important: this precludes the existence of other types of behaviours
  • In the following: models were SEIRS originally, I just wrote as SLIRS

SLIRS - Mukherjee, Chattopadhyay & Tapaswi

SLIRS with constant birth rate , per capita death rate and incidence function

After establishing uniform boundedness of the system, they define

Mukherjee, Chattopadhyay & Tapaswi. Global stability results of epidemiological models with nonlinear incidence rates. Mathematical and Computer Modelling 18 (1993)

Defining matrix

with functions and s.t.

they obtain the following result

Function is s.t. if

Clearly, this is hard to use in practice so the system was studied in other ways

Li & Muldowney (1995) - SLIR model

with

Li & Muldowney. Global stability for the SEIR model in epidemiology. Mathematical Biosciences 125 (1995)

Since

the 3-dim. simplex

is positively invariant, - dissipative and the global attractor is contained in

So they work on the simplex

If or and , the endemic equilibrium is globally asymptotically stable in the interior of

Proof uses compound matrix results

Li, Muldowney & PvdD - SLIRS model

SEIRS with incidence

where is s.t. , for and

They normalise the total population, so . They make an additional hypothesis about :

  • (H) ; when , for sufficiently small

Li, Muldowney & van den Driessche. Global stability of SEIRS models in epidemiology. CAMQ 7 (1999)

We have

Since

The local stability results already established are valid here, since is a particular case of the function used to establish these results

The system is uniformly persistent if there exists s.t. any solution of SEIRS with initial condition satisfies

If satisfies hypothesis (H), then system with incidence is uniformly persistent if and only if

Suppose that incidence satisfies (H) and that

Suppose additionnally that and that one of the conditions

is satisfied, where

and defined by . Then there is no closed rectifiable curve that is invariant with respect to SEIRS. Furthermore, each semi-trajectory of SEIRS in converges to an equilibrium

The proof uses compound matrices (see Practicum 02)

Lyapunov function for SLIR and SLIS

Consider an SLIR with constant population normalised to 1 and vertical transmission

  • proportion of newborns from who are at birth
  • proportion of newborns from who are at birth
  • does not influence the dynamics of the system, so not shown

Korobeinikov. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Mathematical Medicine and Biology 21 (2004)

Equilibria

  • DFE:
  • EEP: with

where

is the basic reproduction number with vertical transmission. We have . On the other hand, when

is biologically relevant only when

Use the function

  • If , then system - has the globally asymptotically stable equilibrium
  • If , then system - has the globally asymptotically stable DFE and is not biologically relevant