MODELLING INFECTIOUS DISEASES

MODELLING INFECTIOUS DISEASES

Lorenzo Argante


GSK Vaccines, Siena

[email protected]

GSK IN A NUTSHELL

GSK VACCINES - GLOBAL PRESENCE

SIENA RESEARCH AND DEVELOPMENT (R&D) SITE

EXPLORATORY DATA ANALYTICS GROUP

➤ Mathematical modelling and computational simulations

➤ between host and within host

➤ Bioinformatics

➤ Reverse vaccinology

➤ Machine learning

Modeling Infectious

Diseases in Humans and Animals

M.J. Keeling and P. Rohani

Princeton University Press

BASIC QUESTIONS

• Understand observed epidemic

• How many cases? Temporal evolution?

• Management of epidemic? Prevention, control, treatment?

MODELLING EPIDEMICS

reality

abstraction, conceptualization

MODELLING EPIDEMICS

Aims Ingredients

Assumptions

Limitations

Validation

MODELLING EPIDEMICS

Aims Questions to be answered

Ingredients Relevant elements

Assumptions Elements to be neglected (impact?)

Limitations Not reality!

Validation Qualitative and quantitative 
 agreement to data

“ All models are wrong.

Some are useful.

-George E. P. Box

MODELLING EPIDEMICS

A wide spectrum of increasing complexity

BASIC COMPARTMENTAL MODELS

S I R

S I

Closed population of N subjects 
 divided in compartments

Susceptibles Infectious Recovered

“SIR” model

N=S+I+R

“SIS” model

N=S+I

• N = total population

• S(t) = no. of susceptible

• I(t) = no. of infectious

• R(t) = no. of recovered

• t = time

SIR MODEL

• Population is closed (no demographics, no migrations)

• Population is “well mixed” (no heterogeneities)

time

S I R

SIR MODEL - RECOVERY TRANSITION

S I µ R

Spontaneous transition: I R

Recovery rate (inverse of average infectious period) Average number of infected 


recovering during time : I = µI t µ = 1/⌧

t

SIR MODEL - INFECTION TRANSITION

• Infection rate depends on:

1. Transmission-given-contact rate 2. Number of contacts per unit time

3. Proportion of contacts that are infectious:

S ₌ ^I I R

N

Two-bodies interaction: S+I 2I

= I

N

}

I N

= I N

SIR MODEL - INFECTION TRANSITION

S I R

Infection rate: I N

Average number of susceptible


being infected during time : S = I

N S t

I

N t '

“Random” mixing, no social structure

➤ Statistically equivalent individuals

➤ probability of being infected

t

EVOLUTION OF S

S I R

Infected individuals “extracted” from S compartment

• Number of trials:

• Probability of success: p = I_t

N t S_t

S_{t+ t} = S_t Binom(S_t, I_t

N t)

EVOLUTION OF I

S I R

Number of trials


Probability 
 of success

p = I_t

N t S_t

p = µ t I_t

I_{t+ t} = I_t + Binom(S_t, I_t

N t) Binom(I_t, µ t)

STOCHASTIC SIR MODEL

S I R

✓ Constant population!

S_{t+ t} + I_{t+ t} + R_{t+ t} = S_t + I_t + R_t S_{t+ t} = S_{t S!I}

I_{t+ t} = I_t + _{S!I I!R} R_{t+ t} = R_t + _I!R

S!I = Binom(S_t, I_t

N t)

I!R = Binom(I_t, µ t)

Stochastic transitions:

Stochastic model:

STOCHASTIC SIR MODEL - SIMULATIONS

S I R

Stochastic SIR model pseudo-code

• set disease parameter values

• set initial conditions for S, I, R

• set number of runs

• set time step

• loop on runs r

๏ loop on time t

➤ get and

➤ update S, I, R

S!I = Binom(S_t, I_t

N t)

I!R = Binom(I_t, µ t)

Stochastic transitions:

p=0.2 I=50

100k runs

Binom(I, p)

Example: 1000000 


random binomial extractions

I!R

I!R S!I

EVOLUTION OF STOCHASTIC SIR MODEL

Single run,


one stochastic
 trajectory

Two stochastic
 trajectories

Three stochastic
 trajectories

Initial conditions:

Sstart=990 Istart=10 Rstart=990

Parameters:

= 0.1 = 0.3

µ

EVOLUTION OF STOCHASTIC SIR MODEL - MANY TRAJECTORIES

Same initial conditions and parameters, 100 runs —> 100 trajectories

DETERMINISTIC SIR MODEL

DETERMINISTIC SIR MODEL

• What’s the evolution of an epidemic? 


Deterministic model:

8>

<

>:

dS

dt = ^SI_N

dI

dt = ^SI_N µI

dR

dt = µI

S I µ R

• Set of ODEs 


(Ordinary Differential Equations)

• Continuous variables S, I, R

➡ Good only for large populations

• Continuous in time (limit dt→0)

• No analytical solution, has to be solved numerically

➡ Discretisation of time to numerically integrate the system (many

algorithms: Euler, Runge-Kutta, etc.)

I N

EPIDEMIC THRESHOLD AND BASIC REPRODUCTIVE NUMBER

S ^I I µ R

N

dI

dt =

✓ S

N µ

◆ I

R₀ > 1

Outbreak condition

• Deterministic model → study initial epidemic growth

✤ If → epidemic dies out

✤ Fully susceptible population: → 


• Basic reproductive number:


Average number of individuals 
 infected by an infectious subject 
 during his infectious period 


in a fully susceptible population

S/N < µ/

1 < µ/

S ' N

R₀ = /µ

APPLICATION

• Flu epidemic in a boarding school in England, 1978

(data from BMJ)

• Data can be fitted with SIR by least squares

• Estimated parameters:

• R₀ = 3.65

• infectious period = 2.2 days

BASIC REPRODUCTIVE NUMBERS

➤ In closed population, invasion only if fraction of S is larger than 1/R0

➤ Vaccination to reduce fraction of S and change epidemic threshold

VACCINATION

• We introduce a class of

vaccinated individuals, fully immune to the disease

• Vaccinated fraction =

➡ Susceptible population decreases

➡ New threshold for epidemic spreading

S I

SV

dI

dt =

 S

N (1 ) µ I

Critical vaccination fraction

c = 1 1/R₀

µ (1 ) > 1

Outbreak condition

VACCINATION

“Herd immunity”: To eradicate the infection, not all the individuals need to be vaccinated, depending on R⁰

c

SIS MODEL

• The disease persists as long as R0>1.

• The system reaches an endemic state, with:

I^⇤ =

✓

1 1

R₀

◆

N

S I

STOCHASTICITY

• Real world epidemics are stochastic processes

• The condition R₀>1 does not deterministically guarantee an epidemic to take off

• Individuals and contagion-recovery-vaccination
 events are discrete

Stochastic numerical simulations

MENINGOCOCCAL DISEASE MODELLING AND

VACCINES EFFECTIVENESS

N. MENINGITIDIS - COMPLEX INTERPLAY WITH HUMANS

➤ N. meningitidis is a bacterium, common human commensal

➤ Carried by humans only in respiratory tract

➤ No symptoms

➤ Long persistence (3-9 months)

➤ Transmission through oral secretions

➤ Highly common in adolescents (~20%)

➤ Classified in capsular serogroups:


A, B, C, X, W, Y, other

N. meningitidis or meningococcus

Age (years)

Carriage
 prevalence (%)

Human nasopharynx

INVASIVE MENINGOCOCCAL DISEASE

➤ 2-10 days after transmission, meningococci can enter blood and cause invasive meningococcal disease (IMD)

➤ Meningitis and sepsis most common

➤ Rare: 1-10 cases per 100000 pop., but often fatal (~10%)

➤ Easily misdiagnosed. Symptoms: headache, stiff neck, fever

➤ Swift: can kill in 24-48 
 hours, even if treated

➤ Serogroups B, C major 
 cause of IMD in US and 
 Europe during the last 
 100 years

Number of IMD cases in England per year

MENINGOCOCCAL VACCINES

Serogroup C (MenC) vaccine

• Protects from invasive disease

• Protects from carriage acquisition:

herd immunity

• Highly effective:


Vaccine Effectiveness (VE) > 90%

VE observational field studies

• Observe disease cases, than see if subject was vaccinated

• Rare disease —> “screening method”

• Formula: ^{VE = 1 −} # cases in vaccinated

# cases in not vaccinated

# not vaccinated

# vaccinated

MENINGOCOCCAL DISEASE AND VACCINATION MODELING

Ingredients of the model:

• England demography¹

• Contact patterns²

• Carriage prevalence³ and duration⁴

• Endemicity of carriage

• Progression to disease modalities

• Pre- and post-immunisation reported
 invasive disease cases⁵

• Vaccination schedules and coverage Parameters to be estimated:

• Direct VE: protection from IMD

• Indirect VE: protection from carriage → herd immunity

Age (years)

Reported
 IMD cases

Age (years) Carriage
 prevalence (%)

1: UK Gov. web site; 2: Mossong J, et al. PLoS Med. 2008; 


3: Christensen H, et al. Lancet Infect Dis. 2010; 4:Caugant, D. et al. Vaccine 2009 ; 5: PHE web site

MENINGOCOCCAL DISEASE AND VACCINATION MODELING

S = Susceptibles C = Carriers

J = number of infection events

V = Vaccinated I = Immune

Transmission model^1,2

1: Trotter CL, et al. Am J Epidemiol. 2005; 2: Christensen H, et al. Vaccine, 2013; 3: Ionides EL et al., PNAS 2006

Disease-observational³ model

MODEL-BASED INFERENCE OF VACCINE EFFECTIVENESS

Monte Carlo
 Maximum 
 Likelihood


inference

Data: cases reported during the first 2 years of MenC vaccination in England

ACCURATE AND PRECISE ESTIMATES OF VE (DIRECT AND INDIRECT)

1: Trotter CL, et al. Lancet. 2004; 2: Campbell H, et al. Clin Vaccine Immunol. 2010;


3: Maiden MC, et al. Lancet. 2002; 4 Maiden MC, et al. J Infect Dis. 2008

* Real MenC cases reported by Public Health England(PHE) 


† Synthetic MenC cases produced running the model in a predictive way,using MCML’s best estimates of VE as inputs

Real cases*

Model prediction^†

CONCLUSIONS

• Modelling approach to meningococcal VE estimation

• Simultaneous detectability of both direct and indirect effectiveness

• Increased power for Vaccine Effectiveness inference

• But assumptions must be correct

• Vaccine Effectiveness for MenC campaign in England estimated with high accuracy:

• Direct VE: 96.5% (95-98)_95%CI vs. 93% to 97%

• Indirect VE: 69% (54-83)_95%CI vs. 63% and 75%


• Smaller confidence intervals (higher precision)

➡ Faster evaluation of vaccines

Reference: Argante L., Tizzoni M., Medini D. “Fast and accurate dynamic estimation of field effectiveness of meningococcal vaccines” BMC

Medicine 2016

MORE GENERAL CONCLUSIONS

• Mathematical models are a framework

• to quantitatively evaluate infectious diseases and vaccines

• to predict evolution in time of outbreaks and immunisation campaigns

• Different approaches, depending on aims and data availability

• Continuous models

‣ Sometimes analytically solvable

• Discrete and stochastic models

‣ Almost never solvable, but easier to simulate

‣ Nearer to reality

THANK YOU!

Any questions?