SIR model

Basic SIR model

Proposed model

A transmission model consists of 3 compartments: susceptible (S), infected (I), recovered (R)

With the following assumptions:

• Individuals are born into susceptible group (exposure time is age of the individual) then transfer to infected class and recovered class

• Recovered individuals gained lifelong immunity

• Age homogeneity

And described by a system of 3 differential equations

\[ \begin{cases} \frac{dS(t)}{dt} = B(t) (1-p) - \lambda(t)S(t) - \mu S(t) \\ \frac{dI(t)}{dt} = \lambda(t)S(t) - \nu I(t) - \mu I(t) - \alpha I(t) \\ \frac{dR(t)}{dt} = B(t) p + \nu I(t) - \mu R(t) \end{cases} \]

Where:

• \(B(t) = \mu N(t)\)
• \(\lambda(t) = \beta I(t)\) with \(\beta\) is the transmission rate
• \(\mu\) is the natural death rate
• \(\nu\) is the recovery rate
• \(\alpha\) is the disease related death rate
• \(p\) is the proportion of newborn vaccinated and moved directly to the recovered compartment

Fitting data

To fit a basic SIR model, use sir_basic_model() and specify the following parameters

• state - initial population of each compartment

• times - a time sequence

• parameters - parameters for SIR model

state <- c(S=4999, I=1, R=0)
parameters <- c(
  mu=1/75, # 1 divided by life expectancy (75 years old)
  alpha=0, # no disease-related death
  beta=0.0005, # transmission rate
  nu=1, # 1 year for infected to recover
  p=0 # no vaccination at birth
)
times <- seq(0, 250, by=0.1)
model <- sir_basic_model(times, state, parameters)
model$parameters
#>         mu      alpha       beta         nu          p 
#> 0.01333333 0.00000000 0.00050000 1.00000000 0.00000000
plot(model)

SIR model with constant Force of Infection at Endemic state

Proposed model

A transmission model consists of 3 compartments: susceptible (S), infected (I), recovered (R)

With the following assumptions:

• Time homogeneity

• Age heterogeneity

Described by a system of 3 differential equations

\[ \begin{cases} \frac{ds(a)}{da} = -\lambda s(a) \\ \frac{di(a)}{da} = \lambda s(a) - \nu i(a) \\ \frac{dr(a)}{da} = \nu i(a) \end{cases} \]

Where:

• \(s(a), i(a), r(a)\) are proportion of susceptible, infected, recovered population of age group \(a\) respectively
• \(\lambda\) is the force of infection
• \(\nu\) is the recovery rate

FItting data

To fit an SIR model with constant FOI, use sir_static_model() and specify the following parameters

• state - initial proportion of each compartment

• ages - an age sequence

• parameters - parameters for the model

state <- c(s=0.99,i=0.01,r=0)
parameters <- c(
  lambda = 0.05,
  nu=1/(14/365) # 2 weeks to recover
)
ages<-seq(0, 90, by=0.01)
model <- sir_static_model(ages, state, parameters)
model$parameters
#>   lambda       nu 
#>  0.05000 26.07143
plot(model)

SIR model with sub populations

Proposed model

Extends on the SIR model by having interacting sub-populations (different age groups)

With K subpopulations, the WAIFW matrix or mixing matrix is given by

\[ C = \begin{bmatrix} \beta_{11} & \beta_{12} & ... & \beta_{1K} \\ \beta_{21} & \beta_{22} & ... & \beta_{2K} \\ \vdots & \vdots & ... & \vdots \\ \beta_{K1} & \beta_{K2} & ... & \beta_{KK} \\ \end{bmatrix} \]

The system of differential equations for the i\(th\) subpopulation is given by

\[ \begin{cases} \frac{dS_i(t)}{dt} = -(\sum^K_{j=1}\beta_{ij}I_j(t)) S_i(t) + N_i\mu_i - \mu_i S_i(t) \\ \frac{dI_i(t)}{dt} = (\sum^K_{j=1}\beta_{ij}I_j(t)) S_i(t) - (\nu_i + \mu_i) I_i(t) \\ \frac{dR_i(t)}{dt} = \nu_i I_i(t) - \mu_i R_i(t) \end{cases} \]

FItting data

To fit a SIR model with subpopulations, use sir_subpops_model() and specify the following parameters

• state - initial proportion of each compartment for each population

• beta_matrix - the WAIFW matrix

• times - a time sequence

• parameters - parameters for the model

k <- 2 # number of population
state <- c(
  # proportion of each compartment for each population
  s = c(0.8, 0.6), 
  i = c(0.2, 0.4),
  r = c(  0,   0)
)
beta_matrix <- c(
  c(0.05, 0.00),
  c(0.00, 0.05)
)
parameters <- list(
  beta = matrix(beta_matrix, nrow=k, ncol=k, byrow=TRUE),
  nu = c(1/30, 1/30),
  mu = 0.001,
  k = k
)
times<-seq(0,10000,by=0.5)
model <- sir_subpops_model(times, state, parameters)
model$parameters
#> $beta
#>      [,1] [,2]
#> [1,] 0.05 0.00
#> [2,] 0.00 0.05
#> 
#> $nu
#> [1] 0.03333333 0.03333333
#> 
#> $mu
#> [1] 0.001
#> 
#> $k
#> [1] 2
plot(model) # returns plot for each population
#> $subpop_1

#> 
#> $subpop_2

MSEIR model

Proposed model

Extends on SIR model with 2 additional compartments: maternal antibody (M) and exposed period (E)

And described by the following system of ordinary differential equation

\[ \begin{cases} \frac{dM(a)}{da} = -(\gamma + \mu(a))M(a) \\ \frac{dS(a)}{da} = \gamma M(a) - (\lambda(a) + \mu(a)) S(a) \\ \frac{dE(a)}{da} = \lambda(a) S(a) - (\sigma + \mu(a)) E(a) \\ \frac{dI(a)}{da} = \sigma(a) E(a) - (\nu + \mu(a)) I(a) \\ \frac{dR(a)}{da} = \nu I(a) - \mu(a) R(a) \end{cases} \]

Where

• \(M(0)\) = B, the number of births in the population

• \(\gamma\) is the rate of antibody decaying

• \(\lambda(a)\) is the force of infection at age \(a\)

• \(\mu(a)\) is the natural death rate at age \(a\)

• \(\sigma\) is the rate of becoming infected after being exposed

• \(\nu\) is the recovery rate

Fitting data

To fit a MSEIR, use mseir_model() and specify the following parameters

• a - age sequence

• And model parameters including gamma, lambda, sigma, nu

model <- mseir_model(
  a=seq(from=1,to=20,length=500), # age range from 0 -> 20 yo
  gamma=1/0.5, # 6 months in the maternal antibodies
  lambda=0.2,  # 5 years in the susceptible class
  sigma=26.07, # 14 days in the latent class
  nu=36.5      # 10 days in the infected class
)
model$parameters
#> $gamma
#> [1] 2
#> 
#> $lambda
#> [1] 0.2
#> 
#> $sigma
#> [1] 26.07
#> 
#> $nu
#> [1] 36.5
plot(model)