Models documentation

Metapopulation SIR model specified with differential equations

The epidemic spreads from one node to another through a network. The base network is given in the file "equation_disease_network.vpz" in the surveillance repository.

The epidemic dynamics in each node "i" is given by the followings equations:

( {dS_i \over dt} = -\beta S_i I_i + \sum_j m_{j \rightarrow i} S_j - \sum_j m_{i \rightarrow j} S_i)

( {dI_i \over dt} = \beta S_i I_i - \gamma I + \sum_j m_{j \rightarrow i} I_j - \sum_j m_{i \rightarrow j} I_i)

( {dR_i \over dt} = \gamma I_i + \sum_j m_{j \rightarrow i} R_j - \sum_j m_{i \rightarrow j} R_i)

with (S_i), (I_i) and (R_i) the density of suceptible, infectious and recorvered individuals in node i, ( \beta ) and (\gamma) are intra-node epidemic parmaters, and (m_{i \rightarrow j}) are inter-node flows.

    Note: equation_disease_network.vpz gives basic structure for this
    first model of the disease with nodes dynamics specified as
    a set of differential equations. DESS extention mus be used to code 
    the integration in DEVS models.

Metapopulation SIR model specified with DEVS

The epidemic spreads from one node to another through a network. The base network is given in the file "/exp/automaton-sir.vpz" in the surveillance repository.

The epidemic dynamics in each node "i" is a coupled model containing two DEVS model: Infection_model and transmission_model.

DEVS specification of both models

** The transmission model** is specified as follows (see DEVS state chart to have global picture): ( Transmission = \left< X, Y, S, \delta_{int}, \delta_{ext}, \delta_{con}, \lambda, ta \right> )

• State

(S = { phase, (t_i){a\leq i \leq No}, (port_i){a\leq i\leq No} } ) with ( phase \in {INIT, Idle, Infecting} ), No is the number of output ports towards other nodes,((t_i){1\leq i \leq No}) a list of times, ((port_i){a \leq i \leq No}) a list of ports names and "a" is a number (1 \leq a \leq No).

• Exernal transition

( \delta_{ext}(phase=Infecting, (t_i){a\leq i\leq No}, (port_i){a\leq i\leq No}) = (phase= Idle))

( \delta_{ext}(phase=Idle, (t_i){a\leq i\leq No}, (port_i){a\leq i\leq No}) = (phase= Infecting, (t_i){1\leq i\leq No}, (port_i){1\leq i\leq No}) )

where ( (t_i)_{1\leq i\leq No} ) is computed the following way:

                    \( \forall 1\leq i\leq No, d_i \sim E(1/p) \) where E(p) is the Negative exponential distribution of parameter p 
                    
                    and p represent the probability that infection event occurs from an infecting node to a given neighboor during 1 day.

                    \( \forall 1\leq i\leq No, t_i = d_i - d_{i-1} \) with \(d_0=0\).

and ((port_i){1\leq i\leq No} = (port_names{\sigma(j)})_{1\leq j\leq No}) where (\sigma ) operator randomly arranges output port names.

• Internal transition

( \delta_{int} (Infecting, (t_i){a\leq i\leq No}, (port_i){a\leq i\leq No})) = (Infecting, (t_i){a+1\leq i\leq No}, (port_i){a+1\leq i\leq No}) )

• Time advance

( ta(Idle)=\infty )

( ta(Infecting, (t_i)_{a\leq i\leq No})=t_a )

• Output function

( \lambda(Infecting) = (infection_evt ) on ( port_a) )

The Infection model is entirely specified by its DEVS state chart (see below)

    Note: automaton_disease_network.vpz gives basic structure for this
    first model of the disease with nodes dynamics specified as
    automaton.