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MODELLING INFECTIOUS DISEASES
Lorenzo Argante GSK Vaccines, Siena
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” modelN=S+I+R
“SIS” modelN=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
µS I 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 R� = �I
N
Two-bodies interaction: S+I 2I
� = �I
N
} �I
N
� = �I
N
SIR MODEL - INFECTION TRANSITION
S I R
Infection rate: � I
NAverage number of susceptible being infected during time : �S = �
I
NS�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 = �
ItN
�t
St
St+�t = St �Binom(St,�It
N
�t)
EVOLUTION OF I
S I R
Number of trials
Probability of successp = �
ItN
�t
St
p = µ�t
It
It+�t = It +Binom(St,�It
N
�t)�Binom(It, µ�t)
STOCHASTIC SIR MODEL
S I R
✓ Constant population!
St+�t + It+�t +Rt+�t = St + It +Rt
St+�t = St ��S!I
It+�t = It +�S!I ��I!R
Rt+�t = Rt +�I!R
�S!I = Binom(St,�It
N
�t)
�I!R = Binom(It, 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(St,�It
N
�t)
�I!R = Binom(It, 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><
>:
dSdt = �� SI
NdIdt = � SI
N � µIdRdt = µI
S I Rµ
• 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 Rµ�I
N
dI
dt=
✓�S
N� µ
◆I
R0 > 1Outbreak 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)
Car
riage
prev
alen
ce (%
) 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)
Rep
orte
dIM
D c
ases
Age (years)
Car
riage
prev
alen
ce (%
)
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 model1,2
1: Trotter CL, et al. Am J Epidemiol. 2005; 2: Christensen H, et al. Vaccine, 2013; 3: Ionides EL et al., PNAS 2006
Disease-observational3model
MODEL-BASED INFERENCE OF VACCINE EFFECTIVENESS
Monte CarloMaximum Likelihoodinference
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?