MODELLING INFECTIOUS DISEASES Lorenzo Argante GSK Vaccines, Siena lorenzo.x.argante@gmail.com
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 Validation Limitations
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 Closed population of N subjects divided in compartments Susceptibles Infectious Recovered SIR model N=S+I+R S I R N = total population SIS model N=S+I S I S(t) = no. of susceptible I(t) = no. of infectious R(t) = no. of recovered t = time
SIR MODEL S I R Population is closed (no demographics, no migrations) Population is well mixed (no heterogeneities) time
SIR MODEL - RECOVERY TRANSITION S I µ R Spontaneous transition: I R Recovery rate µ =1/ Average number of infected recovering during time t: I = µi t (inverse of average infectious period)
SIR MODEL - INFECTION TRANSITION S I R Two-bodies interaction: S+I 2I Infection rate depends on: 1. Transmission-given-contact rate 2. Number of contacts per unit time 3. Proportion of contacts that are infectious: } I N = I N
SIR MODEL - INFECTION TRANSITION S I R I Infection rate: N Average number of susceptible being infected during time t: S = I N S t Random mixing, no social structure Statistically equivalent individuals I N t ' probability of being infected
EVOLUTION OF S S I R Infected individuals extracted from S compartment Number of trials: S t Probability of success: p = I t N t S t+ t = S t Binom(S t, I t N t)
EVOLUTION OF I S I R S t Number of trials I t p = I t N t Probability of success p = µ t I t+ t = I t + Binom(S t, I t N t) Binom(I t,µ t)
STOCHASTIC SIR MODEL S I R Stochastic model: Stochastic transitions: S!I = Binom(S t, I t N t) I!R = Binom(I t,µ t) S t+ t = S t S!I I t+ t = I t + S!I I!R R t+ t = R t + I!R Constant population! S t+ t + I t+ t + R t+ t = S t + I t + R t
STOCHASTIC SIR MODEL - SIMULATIONS S I R Stochastic SIR model pseudo-code set disease parameter values Stochastic transitions: S!I = Binom(S t, I t N t) I!R = Binom(I t,µ t) set initial conditions for S, I, R set number of runs set time step Example: 1000000 I!R random binomial extractions loop on runs r p=0.2 loop on time t I=50 100k runs get S!I and I!R update S, I, R Binom(I, p)
EVOLUTION OF STOCHASTIC SIR MODEL Initial conditions: Sstart=990 Istart=10 Rstart=990 Parameters: Single run, one stochastic trajectory Two stochastic trajectories µ = 0.1 = 0.3 Three stochastic trajectories
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 = di dt = dr dt = µi SI N SI N µi 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 S I R I N µ solved numerically Discretisation of time to numerically integrate the system (many algorithms: Euler, Runge-Kutta, etc.)
EPIDEMIC THRESHOLD AND BASIC REPRODUCTIVE NUMBER S I R I N µ di dt = Deterministic model study initial epidemic growth S/N < µ/ If epidemic dies out S N µ Fully susceptible population: S ' N 1 <µ/ I Basic reproductive number: R 0 = /µ Average number of individuals infected by an infectious subject during his infectious period in a fully susceptible population Outbreak condition R 0 > 1
APPLICATION Flu epidemic in a boarding school in England, 1978 (data from BMJ) Data can be fitted with SIR by least squares Estimated parameters: R 0 = 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 S I We introduce a class of vaccinated individuals, fully immune to the disease SV di dt = apple S N (1 ) µ I Vaccinated fraction = Susceptible population decreases New threshold for epidemic spreading Outbreak condition µ (1 ) > 1 Critical vaccination fraction c =1 1/R 0
VACCINATION Herd immunity : To eradicate the infection, not all the individuals need to be vaccinated, depending on R0 c
SIS MODEL S I The disease persists as long as R0>1. The system reaches an endemic state, with: I = 1 1 R 0 N
STOCHASTICITY Real world epidemics are stochastic processes The condition R 0 >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) N. meningitidis or meningococcus Transmission through oral secretions Highly common in adolescents (~20%) Classified in capsular serogroups: A, B, C, X, W, Y, other Carriage prevalence (%) Human nasopharynx Age (years)
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 demography 1 Contact patterns 2 Carriage prevalence 3 and duration 4 Endemicity of carriage Progression to disease modalities Pre- and post-immunisation reported invasive disease cases 5 Vaccination schedules and coverage Parameters to be estimated: Direct VE: protection from IMD Age (years) Indirect VE: protection from carriage herd immunity Carriage prevalence (%) Reported IMD cases Age (years) 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 Transmission model 1,2 Disease-observational 3 model S = Susceptibles V = Vaccinated C = Carriers I = Immune J = number of infection events 1: Trotter CL, et al. Am J Epidemiol. 2005; 2: Christensen H, et al. Vaccine, 2013; 3: Ionides EL et al., PNAS 2006
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) Real cases* Model prediction * 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 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
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
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