MATHEMATICAL MODELLING OF INFECTIOUS DISEASES

Infectious disease transmissionSusceptible => Infected => Removed Immune Entirely susceptible population... Infectious disease transmissionSusceptible => Infected => Removed Immune One

St Luke’s International University,

Graduate School of Public Health Planning Office, St Luke’s

International University

Something about me Zoie Shui-Yee WONG, PhD

Graduate School of Public Health Planning Office, St Luke’s

International University, Japan (2016- Present)

Community Medicine, University of New South Wales,

Australia (2015-2016)

Systems Informatics Engineering, City University of Hong

Kong (2013-2015)

Institute (PARI), The University of Tokyo, Japan (2011-2013)

Cross disciplinary research in health domain

Research Interests: Health Informatics, Infectious disease

modelling, Simulation method

Modelling overview

Emergence/re-emergence of human pathogens

•An

infectious disease

infectious disease is a clinically evident illness resulting from the presence of

pathogenic microbial agents with potential of transmission from one person or

species to another

Human

Human lives lives

•Spanish Flu (1918): 10 - 50 M deaths

•SARS (2003): 811 deaths in 8 months

•H1N1/09 virus: ~14,000 as of Jan2010 (ECDC,2010)

•Ebola west Africa: 11,310 (as of 10 Jan 2016)

Economics

•SARS: Worldwide: approx US$ 50B

•Avian Flu: approx US$ 30B

•Foot & Mouth Disease: US$ 20B

Travel restrictions and alerts

Epidemiologic Triangle

• An epidemic of a communicable disease

is aninterplay among the pathogen, and the host, &

the environment

• Transmissibility of a communicable disease depends

– Pathogen (disease epidemiology),

– Characteristics of the host population,

• E.g contact patterns and immunity among individuals

– Environmental factors

• E.g climate conditions and animal reservoirs

• Complicated interplay – what do you observe (data)

may not reflect what is happening

– E.g hidden disease compartments may result

Ebola flare-up cases (Abbate, Murall et al 2016)

• Computational epidemiology aims to model the

establishment and spread of pathogens

– Allows us to examine some assumptions

Host

Environment Agent

Disease

Epidemic Models

• Infectious disease models are useful for understanding mechanisms of disease spread, predicting disease parameters, projecting the future course of outbreaks and evaluating control strategies

• Nature of epidemic modelling studies

• Estimation Studies – e.g disease parameters estimation, R0, fatality rate

• Projection Studies – e.g forecast future number of cases, effectiveness of interventions

Ongoing Infectious Disease Surveillance

DNA sequencing data

Infectious Disease Epidemiology and historical outbreak data

Early outbreak investigation, serology

data

Modelling infectious disease dynamics

in the complex landscape of global health (Heesterbeek et al., 2015)

Fit

Adapt

Build

Available Data

Data Collection

Scientific Understanding

Scientific Insight Policy

Question

Policy Advice

Cycles of model testing and analysis thus lead to policy advice and improved scientific understanding

Policy questions define the

model ’ s purpose

Initial model design is based

on current scientific

understanding and the

available relevant data

Model validation and fit to disease data may require further adaptation; sensitivity and uncertainty analysis can point to requirements for collection of additional specific data

What are epidemic models?

• A conceptual tool that explains how an object (or system of objects) will

behave A model that

– Allows us to translate between behavior at various scales, or extrapolate

from a known set of conditions to another

– Predicts the population-level epidemic dynamics from an individual-level

knowledge of epidemiological factors,

– Predicts the long-term behavior from the early invasion dynamics

– Predicts the impact of vaccination on the spread of infection

• Which sort of model is the most appropriate depends on

– the precision or generality required

– the available data

– the time frame in which results are needed

• By definition, all models are “wrong,” - even the most complex will make

some simplifying assumptions

– Considering the absence of data and uncertainty of data quality

– i.e models that capture the essential features of a system

Formulating a model for a particular problem

Trade-off between three important & conflicting elements:

• Accuracy - the ability to reproduce the observed data and reliably

predict future dynamics

– improves with increasing model complexity and the inclusion of more

heterogeneities and relevant biological detail

– The accuracy of any model is always limited

• Transparency - being able to understand (either analytically or more

often numerically) how the various model components influence the

dynamics and interact.

– achieved by adding or removing components and building upon general intuitions from simpler models

– # of model components increases, it becomes more difficult to assess the role of each component and its interactions with the whole

– Transparency is, therefore, often in direct opposition to accuracy

• Flexibility - measures the ease with which the model can be adapted

to new situation A good fit is necessary if the model is used to advise

on future control policies

– Esp important when the model is to evaluate control policies or predict future

disease levels

– This is usually compromised by

• the computational power,

• the mechanistic understanding of disease natural history, and

• the availability of necessary parameters

Building models

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Entirely

susceptible

population

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

One infected

person is

introduced

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Transmission

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

More

transmissions

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Prevalence

increase

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

R0 is about

1.5

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Disease

propagation

stops

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

The primary

case

recovers and

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Everybody

recovers and

develops

immunity

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

Another

infected

person is

introduced

Infectious disease transmission

Susceptible => Infected => Removed (Immune)

No transmission ;

herd immunity

HIT = 1-1/R

~=33%

Considerations of disease modelling

• Natural History of Infection

– Latency period, infectious period, case fatality rate

• Route/Mode of Transmission

– Direct or indirect? Aerosol/droplets or sexual or vectors?

• Population structure and demography

– Age-structured, risk-group structured, geographic location?

Natural history of infection

Infectious

Symptoms

Susceptible Infectious Removed

Immunity Virus

Infection

Decades

Divide population into compartments and contain individuals in different stages of infection

Modelling terms definitions (Wong et al., 2017)

Homogenous mixing Homogenous model assumes that all hosts have identical rate of disease-causing contacts.

The simple susceptible-(exposed)-infected-removed model without consideration of additional population heterogeneity are considered as homogenous mixing in this study

Heterogeneous mixing Heterogeneous model sub-divides population into different groups, depending upon

characteristics that may influence the risk of receiving and transmitting an infection Models considering any host heterogeneities or included additional compartments (such as hospital and/or funeral) are considered as heterogeneous mixing here

Basic reproduction number Basic reproduction number is the expected number of secondary cases generated by one

infected individual over the course of their infection in a fully susceptible population (i.e before interventions are put in place or immunity develops)

Serial interval Serial interval is defined as the time between illness onset in the primary case to illness onset

in the secondary case Understanding serial interval and their moment generating function would help shaping the relationship between epidemic growth rates and reproductive numbers

Latency period Latency period is the time between infected individual to become infectious This metric can

be converted to the rate at which an exposed becomes infective as a modelling parameter for those models considering exposed stage

Infectious period Infectious period is defined as the period that an infected person transmits disease to a

Concepts of building a mathematical

modelling

• Compartmental models consists of two types of

objects:

• Individuals in different stage of infection

• State variables – keep track of the number (proportion) of individuals

• Such as incidence of infection, recovery rate, birth/death rates

• These rates usually are determined by the values of 1 or multiple state variables (e.g disease transmission rate would depend on the number of infected and susceptible) – change as the state of the system changes

• Differential equations describes the rate of change of

its state variable that compose of function(s) that

express(es) the relationship between state variables

Simple SIR model with birth and death events

Infectious,

I

Removed, R

Transmission events

Recoveries

Births Deaths of S

Deaths of R

Deaths of I

Flows between compartments

• Each flow rate is the number of individuals entering or escaping

from a compartment per unit time, depends on

– Per-capita rate

– Number of individuals exposed to the hazard

– i.e population recovery rate: σ * I

– Per-capita rate of infection of susceptible (i.e force of infection) is a

variable depending on:

• # of infectious at the particular time

• Rate of contact with susceptible

• Transmission probability (per-contact)

Formulating transmission rate

For 1 susceptible individual,

• Rate contacting other individuals, contact rate: c

• Probability of transmission when an infectious individual contacts a

susceptible (per-contact): p

• Proportion of population infectious = I/N

• Rate of contacting infectious individual = c*I/N

• Rate of transmission from infectious individuals = p*c*I/N (force of

infection)

For all susceptible individuals,

• Population transmission rate = p*c*S*I/N

Usually β is used to replace p*c and S, I, N are state variables that

change intrinsically.

Simple SIR model with birth and death events

• N,S,I,R are state parameters

• a,b, α, β and σ are parameters

• Solve the equations by integration to understand fundamental processes of infection

• Epidemic can fade out if infected becomes immune –closed population, herd immunity threshold : 1- 1/R0 (i.e measles: 1-1/14 ~= 93%)

Susceptible, S

Infectious,

I

Removed, R

Transmission events

Recoveries

Births Deaths of S

Deaths of R

Deaths of I

Susceptible Infected

Susceptible Infected

Susceptible Infected Removed

Susceptible Exposed Infected Removed Susceptible Exposed Infected Removed

e.g Gonorrhoea, curable STD

e.g Influenza, immunizing

infections

Conceptual diagrams illustrating EVD models of historical Ebola virus

outbreaks

SEIR, removed and SEIHFR,

susceptible-exposed-infectious-hospitalized-funeral-removed Risk associated with unsafe burial contributes to the total daily risk for transmission if effective isolation is not in place

=> Wong ZSY, Bui C, Chughtai A, MacIntyre R A Systematic Review of Early Modeling Studies of Ebola Virus Disease in

Removed

Transmission

Incubation

Recover ed

Community Infection

Hospitalized and infectious

Dead but not buried

Died

Safety Buried Died

Hospitaliz ed

Other modelling complications

• Multiple hosts model

– Many diseases are host-specific, many others can infect multiple and often highly

diverse species

– Shared Hosts: 2 host species and 1 single disease that can be transmitted both within

and between species Such as Food-and-mouth virus in livestock species

– Vectored Transmission: require a secondary “host” to spread infection between primary

hosts Such as Malaria, dengue fever (secondary host: female mosquito)

• Risk-structure model (Host heterogeneity)

– Understand how practices/behaviours dictate the prevalence of infection and how to

combat their increase

– Require transmission matrices -Who Acquires Infection From Whom matrices.Such as

sexually transmitted infections

• Temporally forced model - Seasonality in other systems

– Change in transmission rates through time in early disease outbreak

• Stochastic Dynamics

– A stochastic model is more realistic than a deterministic one! (in principle)

– The relative magnitude of stochastic fluctuations reduces as the number of cases

increases

– Example: If we are interested in eradication of a disease (e.g measles), or if irregular

Individual based Model – example of modelling complex

individual-level behavior (Wong et al., 2016)

scenarios (closure length: 1 week) (Wong et al., 2016)

• The peak would be lower

• For instance, the NCP3-KPS scenario (Day 197: 86,961 cases) achieved a 27%

decrease in the unmitigated peak (Day 89: 119,038).

• Economic viable?

Estimating reproduction numbers for epidemic outbreaks

Basic Reproduction numbers

• Basic reproduction number, R0, is the

expected number of secondary cases

generated by one infected individual over

the course of their infection in a fully

susceptible population (i.e before

interventions are put in place or immunity

develops)

– If R0 > 1 then (on average) the pathogen

will invade that population

– implications: control measure

necessary to prevent (delay) an

epidemic

• If R0 < 1 then infection cannot invade a

population

Generatio n

0 1 2

Initial phase of epidemic (R0

= 3) Exponential increase of case over time

Effective reproduction number

• The growth rate declines once a substantial proportion of

contacts for each infected cases have been

infected/blocked.

• It = I0 exp (r*t)

Effective reproduction number,

• Re = s* R0, where s is proportion still susceptible

• Rule of thumbs: s<1/ R0, i.e R0<1, epidemic goes into

How do we determine R 0

Transmission probability per exposure, p – depends on the disease itself

• use gloves, condoms

Number of contacts per time unit, c – relevant contact depends on infection

• Isolation, sexual abstinence

Duration of infectious period, d

• may be reduced by medical interventions (e.g TB)

R0 = p • c • d

Probability of transmission per contact

Duration of infectiousness

Difficulties in determining R 0

Ideally

• Full information about who infected whom (closely monitored epidemics)

– to construct an infection network - cases are connected if one person

infected the other

• Estimation of R involves simply counting the number of secondary infections

per case

Practically

• Only the epidemic curve is observed

• No information about

– who infected whom,

– no contact information (when and how),

– missing cases

• When only times of symptom onset are available, we approximate R

– by assuming an exponential increase / growth rate in the number of

cases over time

• These counts increase exponentially in the initial phase of an epidemic

– by fitting a specific model that summarizes assumptions about the

epidemiology of the disease