Moment Approximation of Infection Dynamics in a Population of Moving Hosts

The modelling of contact processes between hosts is of key importance in epidemiology. Current studies have mainly focused on networks with stationary structures, although we know these structures to be dynamic with continuous appearance and disappearance of links over time. In the case of moving individuals, the contact network cannot be established. Individual-based models (IBMs) can simulate the individual behaviours involved in the contact process. However, with very large populations, they can be hard to simulate and study due to the computational costs. We use the moment approximation (MA) method to approximate a stochastic IBM with an aggregated deterministic model. We illustrate the method with an application in animal epidemiology: the spread of the highly pathogenic virus H5N1 of avian influenza in a poultry flock. The MA method is explained in a didactic way so that it can be reused and extended. We compare the simulation results of three models: 1. an IBM, 2. a MA, and 3. a mean-field (MF). The results show a close agreement between the MA model and the IBM. They highlight the importance for the models to capture the displacement behaviours and the contact processes in the study of disease spread. We also illustrate an original way of using different models of the same system to learn more about the system itself, and about the representation we build of it.


Appendix A IBM discretization and implementation A.1 Discretization
To simulate the IBM, we consider a regular time step ∆t and a discrete 2D-space (a square toroidal 8-neighbourhood lattice). We discretized the spatial kernels ω 1 and ω 2 for the movement and infection processes (Equations 1 and 3 of the paper). We used a uniform kernel function with a size equal to one discrete space step for the infection kernel ω 2 (See equation 1). This leads us to consider a contact per time step between two individuals if these individuals are located in a same cell during ∆t: We used a uniform kernel function with a size equal to three discrete space steps for the move kernel ω 1 (see Equation 2). The individual can only move in a neighbour square: A.2 Initialization and dynamics of the discrete model According to wether the individual infectious state is respectively S, E or I, we perform a random selection using probability P E , P I or P R computed with Equations 3, 4 or 5 of the paper to compute wether infectious state will be changed to respectively E, I or R. If the test succeeds, the state is changed, if not, it stays the same.
• Computing new position x t+∆t i We first compute wether an individual moves or stays in the same cell. We perform a random selection of probability P = P M (x t+∆t i = x t i ). If the random selection fails, the position is not changed, if it succeeds, the new position is computed as x t+∆t

A.3 Computer implementation
The discrete model was implemented as a discrete event dynamic system with regular time step using the virtual laboratory environment (VLE) [1]. The VLE is a set of tools based on discrete event system Specification (DEVS) and application programming interfaces in C++ for modelling and simulation. VLE provides a random number generator, is interfaced with the R statistical software [2] and enables us to distribute the simulations of an experimental plan on several processors. We used the R statistical software to generate the experimental plans and analyse simulation results.

Appendix B Development and discretization of the moment approximation
Here, we present the development of pair correlation dynamics. The pair correlation C SS (ξ, t) can be calculated for the susceptible-susceptible pairs as follows: The susceptible-susceptible pair correlation dynamics depends on the two processes in the above equation: • The movement terms (the first two) are computed in the same way as for the C SI pairs (see Subsection 2 in the "Moment Approximation" Section of the paper).
• The infection term computes the expected number of susceptible individuals in situation ξ that become infected. It depends on the triplet configuration T SSI and the interaction kernel ω 2 . If an infectious individual is located at distance ξ from an individual of a pair at distance ξ, it can destroy this pair.
There is a factor 2 before each term because the process can be applied on each individual of the pair.
The pair correlation C SE (ξ, t) can be calculated for the SE -pairs: The susceptible-exposed pair correlation dynamics depends on the three processes in the above equation: • The movement terms (the first two) are computed in the same way as the C SS pairs.
• The infection terms create and destroy some SE -pairs. The creation of a SE -pair depends on the T SSI configuration and the interaction kernel ω 2 . When the susceptible individual of a SS -pair is infected by an infectious individual located at distance ξ , it creates a new SE -pair. The destruction of a SE -pair depends on the T SEI configuration and the interaction kernel ω 2 . When the susceptible individual of a SE -pair is infected by an infectious individual located at distance ξ , it destroys the SE -pair.
• The latency term corresponds to the number of exposed individuals in a configuration at a distance of ξ with a susceptible individual and hence create a new SI -pair at a distance of ξ if they become infectious.
The dynamics of the second moments depends on the third moment T ijk (ξ, ξ ). This moment has to be closed to perform the calculation. The moment closure is dealt with in Section B.1.

B.1 Moment closure
To achieve a closed dynamic system, the highest spatial moment must be replaced by a function of lowerorder moments. This expression constitutes the moment closure. In our case, the third moment must be replaced by a function of the first and second moment. Moment closures are a key issue for moment approximation because the quality of the moment approximation is directly linked to the used moment closure. Several closures have been tested and studied in the literature [3]. The idea is to use different closures which have been tested by [3]. More details are available in these references, especially for the all properties linked to the different closures. We used power-2 closures already used by [3]. Power-2 closures are obtained by multiplying two of the three pair densities and by dividing by the density of the opposite corner. In our case, it is important to take into account the correlation that leads to the infection of susceptible individuals. This leads to the following approximation of the third moment: If we consider this closure and the third moment used in the approximation, the correlation involved in the infection process is taken into account.

B.2 Discretization
To implement the moment approximation, it is necessary to discretize the correlation functions. For this purpose, the same discretization as in the individual-based model is used. The movement kernel and the interaction kernel are the kernels described in appendix A. An infection of susceptible occurs only if an infectious individual is located in the same cell. The individuals move to an adjacent cell with a uniform probability. The current problem is considered as bidimensional. In other terms, we consider a correlation matrix. This leads to a simplification of the dynamics as follows: for the first moment dynamics. For the second moment dynamics, we have:

B.3 Computer implementation
The MA model was implemented with Matlab software (using Mathworks c ).