Resumen de Mathematical models for bacteria-phage interaction experiments

Juan Jose Rivaud Gallardo

• In this work we present three different approaches that lead to models of ordinary and partial differential equations with delay for the in vitro dynamics of Salmonella enterica bacteria and some of their attacking phages.

  The experiments were designed, carried out and measured by a team of microbiologists and our work as mathematicians consisted in proposing and validating these mathematical models via comparisons versus experimental data.

  Under these circumstances, the mathematical work to do was not initially determined, but it was guided by the results of the comparisons that pointed out some required adjustments on the models. For this purpose we developed computer programs that allowed us to run numerical simulations of all the models we proposed.

  In this way, each approach became more complicated than its predecessor, in mathematical terms, at the same time it appeared more promising.

  We conclude our first approach with an ordinary differential equations system with delay for the interaction of m bacterial strains versus n distinct phages, based on the mass action law with fixed adsorption constants and regarding a mutation rates coefficient matrix. We can include resistant bacteria and divide bacterial strains into sub populations characterized by having the same parameter values. One important issue is that we manage to treat super infections in an appropriate way by allowing adsorptions on infected, dead and lysed bacteria.

  In the second approach we structured a susceptible and a resistant bacterial populations by the cell age and size, obtaining a system with three partial differential equations combined with three single variable integro differential equations where a delay term is included in one of them.

  The last approach deals with a physiologically structured system. We regard a susceptible bacterial population with a structure by the number of receptors on the cell membrane, together with the possibility of allowing a viral attach-detach mechanism as part of the absorption process.

  We also consider one phage kind and a resistant bacteria population. This idea is presented in a discrete and in a continuous fashion.

  One thing we learned throughout this process is that there is still much work to be done, but we have determined some paths to be followed that can produce good results in a near future whenever the computing power continues raising as it does nowadays.