## A5

Project A5: Algebra of Infections

Research Team: A. Hisi (INPE), E. Macau (INPE), L. Santos (CEMADEN), I. Sokolov (HUB)

Outline: It is now generally accepted that, except for very special cases of well-mixed populations or of the populations with extremely high site fidelity, the correct description of the infection spread has to be based on the investigation of the network of contacts. Most of approaches use time-aggregated networks, in which a link between nodes A and B is present if the contact between them existed during some interval of time. The network of contacts is then considered as the substrate on which the infection spreads, and the epidemics dynamics on this network is obtained by solving the corresponding problem on the individual or on the metapopulation level. This approach disregards the temporal structure of the contact network which is however of extreme importance. Thus, if at any time there was a link between A and B, and at any other time a link between B and C, a path of two steps between A and C exists on the aggregated network. If however the link between A and B was established later than the link between B and C disappeared, no direct propagation from A to C is possible, and the path between A and C cannot be traversed in the causal order.

Research Topic: The problems of accessibility on the temporal networks can be addressed algebraically. Thus, the set of adjacency matrices of the temporal snapshots allows for building the accessibility graph of a network, which contains a link wherever a causal path between the two nodes exists. Building an accessibility graph by consecutively adding paths of growing length (unfolding) gives information about the distribution of shortest paths and characteristic time scales. Moreover, by comparing the accessibility graphs of a temporal and of the aggregated networks, one can define a measure of goodness of their static representation. A nice feature of the algebraic representation is the fact that using the same algebraic structures on the level of adjacency matrices and redefining the arithmetic operations (i.e. using the “school arithmetic” with standard (+,×) operations, the Boolean (max,×) arithmetic, or the or Shimbel (min,+) arithmetic) one can obtain the results on such properties as number of paths, length of the shortest path, existence of the paths, etc. More complex metrics pertinent to weighted networks, like mean first passage times and unusual distances like the one introduced in have also to be considered on the equal footing. The model in corresponds to the SI (susceptible-infected) scheme on the individual level, and is not of immediate epidemiological interest. The project aims onto obtaining algebraic representations for more relevant epidemiological schemes (SIS, SIR, ect.) involving the time of infection. Such models have to resort to age classes reproduced by transfer matrices. The work in the group should concentrate onto the investigation of the corresponding models on the algebraic level, on building realistic mathematical models of such networks pertinent to epidemiological situations, and on the work with real data, as provided by the Brazilian side.