Introduction to SEIR Models

Nakul Chitnis

Workshop on Mathematical Models of Climate Variability,Environmental Change and Infectious Diseases

Trieste, Italy8 May 2017

Department of Epidemiology and Public Health

Health Systems Research andDynamical Modelling Unit

Outline

SI Model

SIS Model

The Basic Reproductive Number (R0)

SIR Model

SEIR Model

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Mathematical Models of Infectious Diseases

Population-based modelsI Can be deterministic or stochasticI Continuous time

• Ordinary differential equations• Partial differential equations• Delay differential equations• Integro-differential equations

I Discrete time• Difference equations

Agent-based/individual-based modelsI Usually stochasticI Usually discrete time

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Mathematical Models of Infectious Diseases

Population-based modelsI Can be deterministic or stochasticI Continuous time

• Ordinary differential equations• Partial differential equations• Delay differential equations• Integro-differential equations

I Discrete time• Difference equations

Agent-based/individual-based modelsI Usually stochasticI Usually discrete time

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Outline

SI Model

SIS Model

SIR Model

SEIR Model

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SI Model

Susceptible-Infectious Model: applicable to HIV.

rβI/N

dt= −rβS I

dt= rβS

S: Susceptible humansI: Infectious humansr: Number of contacts per unit timeβ: Probability of disease transmission per contactN : Total population size: N = S + I.

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Analyzing the SI Model

The system can be reduced to one dimension,

dt= rβ(N − I)

with solution,

I(t) =I0N

(N − I0)e−rβt + I0,

for I(0) = I0.Equilibrium Points:

Idfe = 0

Iee = N

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Analyzing the SI Model

dt= rβ(N − I)

with solution,

I(t) =I0N

(N − I0)e−rβt + I0,

Idfe = 0

Iee = N

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Numerical Solution of SI Model

0 5 10 15 200

Time (Years)

With r = 365/3 years−1, β = 0.005, N = 1000, and I(0) = 1.

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Definition of Transmission Parameters

Note that in some models, usually of diseases where contactsare not well defined, rβ (the number of contacts per unit timemultiplied by the probability of disease transmission percontact) are combined into one parameter (often also called β— the number of adequate contacts per unit time).

For diseases where a contact is well defined (such as sexuallytransmitted diseases like HIV or vector-borne diseases likemalaria), it is usually more appropriate to separate the contactrate, r, and the probability of transmission per contact, β.

For diseases where contacts are not well defined (such asair-borne diseases like influenza) it is usually more appropriateto combine the two into one parameter.

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Outline

SI Model

SIS Model

SIR Model

SEIR Model

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SIS Model

Susceptible-Infectious-Susceptible Model: applicable to thecommon cold.

rβI/N

dt= −rβS I

N+ γI

dt= rβS

N− γI

γ: Per-capita recovery rate

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Analyzing the SIS Model

dt= rβ(N − I)

N− γI,

with solution,

I(t) =

Nrβ · (rβ − γ)

1 +(Nrβ

(rβ−γ)I0

− 1)e−(rβ−γ)t

Idfe = 0

Iee =(rβ − γ)N

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Analyzing the SIS Model

dt= rβ(N − I)

N− γI,

with solution,

I(t) =

Nrβ · (rβ − γ)

1 +(Nrβ

(rβ−γ)I0

− 1)e−(rβ−γ)t

Idfe = 0

Iee =(rβ − γ)N

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Numerical Solution of SIS Model

0 10 20 30 40 500

Time (Days)

With rβ = 0.5 days−1, γ = 0.1 days−1, N = 1000, and I(0) = 1.

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Outline

SI Model

SIS Model

SIR Model

SEIR Model

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Anew swine-origin influenza A (H1N1) virus, ini-tially identified in Mexico, has now caused out-breaks of disease in at least 74 countries, with decla-

ration of a global influenza pandemic by the World HealthOrganization on June 11, 2009.1 Optimizing public healthresponses to this new pathogen requires difficult decisionsover short timelines. Complicating matters is the unpre-dictability of influenza pandemics: planners cannot basetheir decisions solely on pre-pandemic factors or on experi-ence from earlier pandemics. We suggest that mathematicalmodelling can inform and optimize health policy decisionsin this situation.

Uses of models

Mathematical models of infectious diseases are useful toolsfor synthesizing the best available data on a new pathogen,comparing control strategies and identifying important areasof uncertainty that may be prioritized for urgent research.

As an example of synthesizing data, consider the processof developing a mathematical model of the effectiveness ofinfluenza vaccines: modellers must draw together informationon influenza epidemiology (including patterns of spread indifferent age groups), the natural history of influenza, theeffectiveness of vaccines in randomized trials and the dura-tion of immunity following vaccination or natural infection,2,3

which cannot all be derived from a single study. Once themodel is developed, rapid and inexpensive “experiments” canbe performed by simulating alternative vaccination strategies(e.g., vaccinating children most likely to transmit influenza,or vaccinating older adults most likely to have severe compli-cations of influenza).2

The uncertainty involved in this process can be assessedthrough sensitivity analysis (in this case, by varying estimatesof vaccine effectiveness across plausible ranges) to examinewhether such variation results in markedly different out-comes. Uncertain model inputs that are extremely influentialin determining the best course of action should be prioritizedfor future research.

Elements of models

Elements of epidemic models often include “compart-ments” or “states” that describe the susceptibility, infec-tiousness or immunity of individuals in a population, and“parameters” (numbers) that describe how individualsmove between these states.

A key model parameter is the basic reproductive num-ber, referred to as R0.4 This is the number of new, secondaryinfections created by a single primary infectious case intro-duced into a totally susceptible population (Figure 1). Theimportance of R0 relates to the information it provides toplanners: R0 determines the potential of a new pathogen toD

Pandemic Influenza Outbreak Research Modelling Team (Pan-InfORM)

Modelling an influenza pandemic: A guide for the perplexed

Key points

• When a new infectious disease emerges, mathematicalmodels can project plausible scenarios, guide controlstrategies and identify important areas for urgent research.

• Models of influenza pandemics suggest roles for antiviraldrugs and vaccines. Models also raise concerns about anti viral resistance.

• Knowledge translation is a key part of modelling activitiesthat aim to optimize policy decisions for containment ofnew infectious diseases.

CMAJ Analysis

Disease is endemic (R = 1)Initial phase of epidemic (R0 = 3)

Generation

Figure 1: The number of new infections generated when thebasic reproductive number (the number of new cases createdby a single primary case in a susceptible population) is 3. Casesof disease are represented as dark circles, and immune individ-uals are represented as open circles. When there is no immu-nity in the population (left) and the basic reproductive number(R0) is 3, the initial infectious case generates on average 3 sec-ondary cases, each of which in turn generates 3 additionalcases of disease. Once the disease becomes endemic owing toacquired immunity (right), each case generates on average1 additional case (effective reproductive number [R] = 1).

@@ See other H1N1 articles: Editorial, page 123; Research, page 159

Pan-InfORM (2009)

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Definition of R0

The basic reproductive number, R0, is the number ofsecondary infections that one infected person would producein a fully susceptible population through the entire duration ofthe infectious period.

R0 provides a threshold condition for the stability of thedisease-free equilibrium point (for most models):

I The disease-free equilibrium point is locally asymptoticallystable when R0 < 1: the disease dies out.

I The disease-free equilibrium point is unstable when R0 > 1:the disease establishes itself in the population or an epidemicoccurs.

I For a given model, R0 is fixed over all time.

This definition is only valid for simple homogeneousautonomous models.

Can define similar threshold conditions for more complicatedmodels that include heterogeneity and/or seasonality but thebasic definition no longer holds.

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Definition of R0

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Definition of R0

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Definition of R0

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Evaluating R0

R0 can be expressed as a product of three quantities:

Number ofcontacts

per unit time

Probability oftransmissionper contact

( Duration ofinfection

For SIS model:

R0 = r × β × 1

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Reproductive Numbers

The (effective) reproductive number, Re, is the number ofsecondary infections that one infected person would producethrough the entire duration of the infectious period.

Typically, but not always, Re is the product of R0 and theproportion of the population that is susceptible.

Re describes whether the infectious population increases ornot. It increases when Re > 1; decreases when Re < 1 and isconstant when Re = 1. When Re = 1, the disease is atequilibrium.

Re can change over time.

The control reproductive number, Rc, is the number ofsecondary infections that one infected person would producethrough the entire duration of the infectious period, in thepresence of control interventions.

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Evaluating Re

Re(t) =

Number ofcontacts

per unit time

Proportion ofsusceptiblepopulation

For SIS model:

Re(t) = R0 ×S(t)

=rβS(t)

γN(t).

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http://www.cameroonweb.com/CameroonHomePage/NewsArchive/Ebola-How-does-it-compare-316932

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Outline

SI Model

SIS Model

SIR Model

SEIR Model

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SIR Model

Susceptible-Infectious-Recovered Model: applicable to measles,mumps, rubella.

S I RrβI/N γ

dt= −rβS I

dt= rβS

N− γI

dt= γI

R: Recovered humanswith N = S + I +R.

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Analyzing the SIR Model

Can reduce to two dimensions by ignoring the equation for Rand using R = N − S − I.

Can no longer analytically solve these equations.

Infinite number of equilibrium points with I∗ = 0.

Perform phase portrait analysis.

Estimate final epidemic size.

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R0 for the SIR Model

Number ofcontacts

per unit time

R0 = r × β × 1

If R0 < 1, introduced cases do not lead to an epidemic (thenumber of infectious individuals decreases towards 0).

If R0 > 1, introduced cases can lead to an epidemic(temporary increase in the number of infectious individuals).

Re(t) =rβ

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Phase Portrait of SIR ModelTHE MATHEMATICS OF INFECTIOUS DISEASES 605

0 0.2 0.4 0.6 0.8 1susceptible fraction, s

infective

fraction,

smax↑= 1

σ = 3

Fig. 2 Phase plane portrait for the classic SIR epidemic model with contact number σ = 3.

passively immune class M and the exposed class E are omitted. This model usesthe standard incidence and has recovery at rate γI, corresponding to an exponentialwaiting time e−γt. Since the time period is short, this model has no vital dynamics(births and deaths). Dividing the equations in (2.1) by the constant total populationsize N yields

ds/dt = −βis, s(0) = so ≥ 0,

di/dt = βis − γi, i(0) = io ≥ 0,(2.2)

with r(t) = 1 − s(t) − i(t), where s(t), i(t), and r(t) are the fractions in the classes.The triangle T in the si phase plane given by

T = {(s, i) |s ≥ 0, i ≥ 0, s + i ≤ 1}(2.3)

is positively invariant and unique solutions exist in T for all positive time, so that themodel is mathematically and epidemiologically well posed [96]. Here the contact num-ber σ = β/γ is the contact rate β per unit time multiplied by the average infectiousperiod 1/γ, so it has the proper interpretation as the average number of adequate

Hethcote (2000)

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Numerical Solution of SIR Model

0 20 40 60 80 1000

Time (Days)

SusceptibleInfectiousRecovered

With rβ = 0.3 days−1, γ = 0.1 days−1, N = 1000, andS(0) = 999, I(0) = 1 and R(0) = 0.

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Numerical Solution of SIR Model

0 20 40 60 80 1000

Time (Days)

With rβ = 0.3 days−1, γ = 0.1 days−1, N = 1000, andS(0) = 580, I(0) = 20 and R(0) = 400.

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Human Demography

Need to include human demographics for diseases where thetime frame of the disease dynamics is comparable to that ofhuman demographics.

There are many different ways of modeling humandemographics

I Constant immigration rateI Constant per-capita birth and death ratesI Density-dependent death rateI Disease-induced death rate.

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Endemic SIR Model

S I RrβI/N γ

Death Death Death

µ µ µ

dt= Λ − rβS

N− µS

dt= rβS

N− γI − µI

dt= γI − µR

N = S + I +R

Λ: Constant recruitment rateµ: Per-capita removal rate

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Analyzing the Endemic SIR Model

Can no longer reduce the dimension or solve analytically.

There are two equilibrium points: disease-free and endemic

Sdfe =Λ

µSee =

Λ(γ + µ)

Idfe = 0 Iee =Λ(rβ − (γ + µ))

rβ(γ + µ)

Rdfe = 0 Ree =γΛ(rβ − (γ + µ))

rβµ(γ + µ)

Can perform stability analysis of these equilibrium points anddraw phase portraits.

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R0 for the Endemic SIR Model

Number ofcontacts

per unit time

R0 = r × β × 1

γ + µ

If R0 < 1, the disease-free equilibrium point is globallyasymptotically stable and there is no endemic equilibriumpoint (the disease dies out).

If R0 > 1, the disease-free equilibrium point is unstable and aglobally asymptotically stable endemic equilibrium point exists.

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Numerical Solution of Endemic SIR Model

0 20 40 60 80 1000

Time (Days)

With rβ = 0.3 days−1, γ = 0.1 days−1, µ = 1/60 years−1,Λ = 1000/60 years−1, and S(0) = 999, I(0) = 1 and R(0) = 0.

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Numerical Solution of Endemic SIR Model

0 20 40 60 80 100 1200

Time (Years)

With rβ = 0.3 days−1, γ = 0.1 days−1, µ = 1/60 years−1,Λ = 1000/60 years−1, and S(0) = 999, I(0) = 1 and R(0) = 0.

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Outline

SI Model

SIS Model

SIR Model

SEIR Model

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SEIR Model

Susceptible-Exposed-Infectious-Recovered Model: applicable tomeasles, mumps, rubella.

S E I RrβI/N ε γ

Death Death Death Death

µ µ µ µ

E: Exposed (latent) humansε: Per-capita rate of progression to infectious state

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SEIR Model

dt= Λ − rβS

N− µS

dt= rβS

N− εE

dt= εE − γI − µI

dt= γI − µR

withN = S + E + I +R.

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R0 for the Endemic SEIR Model

Number ofcontacts

per unit time

Probabililty ofsurviving

exposed stage

R0 = r × β × 1

γ + µ× ε

ε+ µ

=rβε

(γ + µ)(ε+ µ)

If R0 < 1, the disease-free equilibrium point is globallyasymptotically stable and there is no endemic equilibriumpoint (the disease dies out).

If R0 > 1, the disease-free equilibrium point is unstable and aglobally asymptotically stable endemic equilibrium point exists.

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Extensions to Compartmental Models

Basic compartmental models assume a homogeneouspopulation.

Divide the population into different groups based on infectionstatus:M : Humans with maternal immunityS: Susceptible humansE: Exposed (infected but not yet infectious) humansI: Infectious humansR: Recovered humans.

Can include time-dependent parameters to include the effectsof seasonality.

Can include additional compartments to model vaccinated andasymptomatic individuals, and different stages of diseaseprogression.

Can include multiple groups to model heterogeneity, age,spatial structure or host species.

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References

O. Diekmann, H. Heesterbeek, and T. Britton, Mathematical Tools forUnderstanding Infectious Disease Dynamics.Princeton Series in Theoretical and Computational Biology. Princeton UniversityPress, Princeton, (2013).

H. W. Hethcote, “The mathematics of infectious diseases”, SIAM Review 42,599–653 (2000).

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans andAnimals.Princeton University Press, Princeton, (2007).

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