Modelling Heterogeneity Nakul Chitnis Workshop on Mathematical - PowerPoint PPT Presentation

Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Modelling Heterogeneity Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental Change and Infectious Diseases Trieste, Italy 10 May 2017

Outline Heterogeneity Rabies in N’Djamena Discrete-Time Population-Based Models Individual-Based Models OpenMalaria 2017-05-10 2

Outline Heterogeneity Rabies in N’Djamena Discrete-Time Population-Based Models Individual-Based Models OpenMalaria 2017-05-10 3

Heterogeneity In the basic SIR-type models, we have assumed that the population is homogeneous, that is, everyone is considered to be identical (except for disease status) and to have random contacts. In general, disease systems contain many heterogeneities such as, ◮ Population heterogeneity ◮ Spatial heterogeneity ◮ Temporal heterogeneity (seasonality). 2017-05-10 4

Population Heterogeneity Populations can be heterogeneous in terms of (model parameters) Susceptibility and infectivity Rate of recovery and acquired immunity Contact rates Mortality and disease-induced mortality. Heterogeneity (in model parameters) may depend on demographic characteristics such as Age Socioeconomic status Occupation Degree of contacts Gender Species Individual characteristics. 2017-05-10 5

Population Heterogeneity Populations can be heterogeneous in terms of (model parameters) Susceptibility and infectivity Rate of recovery and acquired immunity Contact rates Mortality and disease-induced mortality. Heterogeneity (in model parameters) may depend on demographic characteristics such as Age Socioeconomic status Occupation Degree of contacts Gender Species Individual characteristics. 2017-05-10 5

Modeling Population Heterogeneity We determine parameter values for each demographic category. We can do this in three ways. 1. We can replicate the SIR model equations for different groups representing different values of a demographic parameter leading to high-dimensional ODE models with multiple groups. 2. We can assume that the demographic parameter is continuous leading to partial integrodifferential equations. 3. We can assign a particular value for a number of different demographic parameters to each individual in the population and numerically simulate the interactions of the individuals within the population and the subsequent progress of the disease, leading to an individual-based model. 2017-05-10 6

Modeling Population Heterogeneity We determine parameter values for each demographic category. We can do this in three ways. 1. We can replicate the SIR model equations for different groups representing different values of a demographic parameter leading to high-dimensional ODE models with multiple groups. 2. We can assume that the demographic parameter is continuous leading to partial integrodifferential equations. 3. We can assign a particular value for a number of different demographic parameters to each individual in the population and numerically simulate the interactions of the individuals within the population and the subsequent progress of the disease, leading to an individual-based model. 2017-05-10 6

Modeling Population Heterogeneity We determine parameter values for each demographic category. We can do this in three ways. 1. We can replicate the SIR model equations for different groups representing different values of a demographic parameter leading to high-dimensional ODE models with multiple groups. 2. We can assume that the demographic parameter is continuous leading to partial integrodifferential equations. 3. We can assign a particular value for a number of different demographic parameters to each individual in the population and numerically simulate the interactions of the individuals within the population and the subsequent progress of the disease, leading to an individual-based model. 2017-05-10 6

SIR Model with Two Age Groups I J I A r JJ β JJ N J + r JA β JA γ J Λ N A S J I J R J µ J µ J µ J ϕ ϕ ϕ I J I A r AJ β AJ N J + r AA β AA γ A N A S A I A R A µ A µ A µ A S J : Susceptible Juveniles S A : Susceptible Adults I J : Infectious Juveniles I A : Infectious Adults R J : Recovered Juveniles R A : Recovered Adults 2017-05-10 7

SIR Model with Two Age Groups Λ : Recruitment rate of new juveniles. r kl : Number of contacts per time between an individual in group k with individuals in group l . β kl : Probability of disease transmission per contact between an in- fectious in group l with a susceptible in group k . γ k : Recovery rate of individuals in group k . ϕ : Development rate (from juveniles to adults). µ k : Death rate of individuals in group k . N k : Total population of group k . N k = S k + I k + R k . for k = J or k = A and l = J or l = A . 2017-05-10 8

SIR Model with Two Age Groups � � d S J I J I A d t = Λ − r JJ β JJ + r JA β JA S J − ( ϕ + µ J ) S J N J N A d I J � I J I A � d t = r JJ β JJ + r JA β JA S J − ( γ J + ϕ + µ J ) I J N J N A d R J = γ J I J − ( ϕ + µ J ) R J d t d S A � I J I A � = ϕS J − r AJ β AJ + r AA β AA S A − µ A S A d t N J N A � � d I A I J I A d t = ϕI J + r AJ β AJ + r AA β AA S A − ( γ A + µ A ) I A N J N A d R A = ϕR J + γ A I A − µ A R A d t 2017-05-10 9

Next Generation Matrix for Two Age Group Model � K JJ K JA � K = K AJ K AA K JJ : Number of new juvenile individuals infected by one infectious ju- venile individual assuming a fully susceptible population through the duration of the infectious period. K JA : Number of new juvenile individuals infected by one infectious adult individual assuming a fully susceptible population through the duration of the infectious period. K AJ : Number of new adult individuals infected by one infectious juve- nile individual assuming a fully susceptible population through the duration of the infectious period. K AA : Number of new adult individuals infected by one infectious adult individual assuming a fully susceptible population through the duration of the infectious period. 2017-05-10 10

R 0 for Two Age Group Model R 0 is the spectral radius of K (eigenvalue with the maximum absolute value). R 0 = 1 �� � K 2 JJ + K 2 AA − 2 K JJ K AA + 4 K AJ K JA + K JJ + K AA 2 where r JJ β JJ K JJ = γ J + ϕ + µ J K JA = r JA β JA γ A + µ A r AJ β AJ K AJ = γ J + ϕ + µ J K AA = r AA β AA γ A + µ A 2017-05-10 11

Multi-Group ODE Models We can divide the population into any number of groups. We could include more than one demographic parameter but the equations then become complicated. We need to determine the contact matrix for each model. However, the population is still assumed to be homogeneous within each group. 2017-05-10 12

SIR Model with Continuous Age �� ∞ 0 r ( a, ˜ a ) α ( a ) β (˜ a ) I (˜ a, t ) d ˜ a ∂a + ∂S ∂S � ∂t = − + µ ( a ) S, � ∞ 0 N (˜ a, t ) d ˜ a �� ∞ 0 r ( a, ˜ a ) α ( a ) β (˜ a ) I (˜ a, t ) d ˜ a � ∂I ∂a + ∂I ∂t = S − ( γ + µ ( a )) I, � ∞ 0 N (˜ a, t ) d ˜ a ∂R ∂a + ∂R ∂t = γI − µ ( a ) R, where r ( a, ˜ a ) = r (˜ a, a ) is the contact rate between hosts of age a and ˜ a ; α ( a ) is the susceptibility of hosts of age a ; β ( a ) is the infectivity of hosts of age a ; and N ( a, t ) = S ( a, t ) + I ( a, t ) + R ( a, t ) ; with specified initial conditions, and boundary conditions, � ∞ S (0 , t ) = f ( a ) N ( a, t ) da, 0 � ∞ and I (0 , t ) = 0 , R (0 , t ) = 0 . And, N ( t ) = 0 N ( a, t ) da . Adapted from Hethcote (2000) 2017-05-10 13

Spatial Heterogeneity Modeling spatial heterogeneity is similar to modeling population heterogeneity but we divide the population by spatial location instead of by demographic characteristics. We can model space as continuous or discrete. ◮ Continuous space leads to partial integrodifferential equations. ◮ Discrete space leads to multi-group ODE models (for example, patch models) or cellular automata. We can model movement of hosts or of infection. 2017-05-10 14