SEIS and SEISd

Susceptible-Exposed-Infected-Susceptible Models

Source: vignettes/human_seis.Rmd

xde

ramp.xds includes the standard SEIS model and the delay SEIS model. The dependent state variables for both models are:

  • SS the density of susceptible hosts

  • EE the density of exposed hosts who are infected but not yet infectious

  • II the density of infectious hosts

In the basic versions of the model – without demographic changes – population density is constant, so H=S+E+I.H = S+E+I.

ramp.xds includes the standard SEIS model and the delay SEIS (SEISd) model.

Compartment Model as an ODE

The SEIS model is a human model modified from the SIS model to include the Exposed group of individuals (EE). It is incorporated within the ‘ramp.xds’ with the fulfillment of the generic interface of the human component.

The model has three parameters:

  • bb is the fraction of infective bites that cause an infection;

  • ν\nu is the transition rate from exposed to infectious: the duration of the latent period is 1/ν1/\nu

  • rr is the clearance rate for infections: the average duration of infection in this model is 1/r1/r

These are coupled systems of ordinary differential equations forced by the force of infection, denoted h(t)h(t), where h=Fh(E);h = F_h(E); here, we assume that exposure is Poisson, so we let:

h=bEh = bE

$$ \frac{dS}{dt} = - h S + r I \\ \frac{dE}{dt} = h S - \nu E \\ \frac{dI}{dt} = \nu E - r I\\ $$

HPop = 1000
MYZo = list(MYZm = 2000/365, f=1, q=1)
Xo = list(b= 0.5, H=HPop)
seis = xds_setup(Xname = "SEIS", MYZname = "trivial", MYZopts = MYZo, Xopts=Xo, HPop=HPop)
seis = xds_solve(seis, 3650, 15)
unlist(list_Xvars(seis$outputs$last_y, seis, 1)) -> seis_inf
seis_inf
#>           S           E           I           H 
#>  911.577028    6.381039   82.041933 1000.000000

This model has the steady state…

$$ \bar E = H \frac {hr}{h(r+\nu) + r \nu} \\ \bar I = H \frac {h\nu}{h(r+\nu) + r \nu} \\ \bar S = H - \bar E - \bar I $$

xde_steady_state_X(1/365, 1000, seis$Xpar[[1]]) -> seis_ss
seis_ss
#>          S          E          I          H 
#>  652.95170   25.04472  322.00358 1000.00000
sum((seis_inf-seis_ss)^2) < 1e-9
#> [1] FALSE

Compartment Model as an DDE

In the delay differential equation model, we let ν\nu denote the duration of the incubation period, and we let hν=h(t−ν).h_\nu = h(t-\nu).

$$ \frac{dS}{dt} = - h S + r I \\ \frac{dE}{dt} = h S - h_\nu S_\nu \\ \frac{dI}{dt} = h_\nu S_\nu - r I\\ $$ and H=S+E+I.H = S+E+I. At steady state, h=hvh=h_{v}, so

$$ \bar S = \frac{H}{1+h \nu + h/r} \\ \bar E = h \bar S \nu \\ \bar I = \frac hr \bar S \\ $$

SEISd = xds_setup(Xname = "SEISd", MYZname = "trivial", MYZopts = MYZo, Xopts=Xo, HPop=HPop)
SEISd = xds_solve(SEISd, Tmax = 3650, dt=15)
unlist(list_Xvars(SEISd$outputs$last_y, SEISd, 1))[1:4] -> seisd_inf
seisd_inf
#>           S           E           I           H 
#>  911.577042    6.381024   82.041934 1000.000000
xde_steady_state_X(1/365, 1000, SEISd$Xpar[[1]]) -> seisd_ss
seisd_ss
#>          S          E          I          H 
#>  652.95170   25.04472  322.00358 1000.00000
sum((seisd_inf-seisd_ss)^2) < 1e-5
#> [1] FALSE