Section: New Results

A stochastic model for bacteriophage therapies

Participants: X. Bardina, D. Bascompte, C. Rovira, S. Tindel.

In the last years Bacteriophage therapies are attracting the attention of several scientific studies. They can be a new and powerful tool to treat bacterial infections or to prevent them applying the treatment to animals such as poultry or swine. Very roughly speaking, they consist in inoculating a (benign) virus in order to kill the bacteria known to be responsible of a certain disease. This kind of treatment is known since the beginning of the 20th century, but has been in disuse in the Western world, erased by antibiotic therapies. However, a small activity in this domain has survived in the USSR, and it is now re-emerging (at least at an experimental level). Among the reasons of this re-emersion we can find the progressive slowdown in antibiotic efficiency (antibiotic resistance). Reported recent experiments include animal diseases like hemorrhagic septicemia in cattle or atrophic rhinitis in swine, and a need for suitable mathematical models is now expressed by the community.

Let us be a little more specific about the (lytic) bacteriophage mechanism: after attachment, the virus' genetic material penetrates into the bacteria and use the host's replication mechanism to self-replicate. Once this is done, the bacteria is completely spoiled while new viruses are released, ready to attack other bacteria. It should be noticed at this point that among the advantages expected from the therapy is the fact that it focuses on one specific bacteria, while antibiotics also attack autochthonous microbiota. Roughly speaking, it is also believed that viruses are likely to adapt themselves to mutations of their host bacteria.

At a mathematical level, whenever the mobility of the different biological actors is high enough, bacteriophage systems can be modeled by a kind of predator-prey equation. Namely, set $S_t$ (resp. $Q_t$) for the bacteria (resp. bacteriophages) concentration at time $t$. Then a model for the evolution of the couple $(S,Q)$ is as follows:

where $\alpha$ is the reproducing rate of the bacteria and $k$ is the adsorption rate. In equation (1 ), $d$ also stands for the quantity of bacteriophages inoculated per unit of time, $m$ is their death rate, we denote by $b$ the number of bacteriophages which is released after replication within the bacteria cell, $\zeta$ is the delay necessary to the reproduction of bacteriophages (called latency time) and the coefficient $e^{-\mu \zeta }$ represents an attenuation in the release of bacteriophages (given by the expected number of bacteria cell's deaths during the latency time, where $\mu$ is the bacteria's death rate). A given initial condition $(S_0,Q_0)$ is also specified, and the term $\epsilon \mathrm{d}{W}_t$ takes into account a small external noise standing for both uncertainties on the measures and the experiment conditions (for similar modeling see e.g. [34] ). One should be aware of the fact that the latency time $\zeta$ (which can be seen as the reproduction time of the bacteriophages within the bacteria) cannot be neglected, and is generally of the same order (about 20mn) as the experiment length (about 60mn).

With this model in hand, our main results in this direction (see [15] ) have been the following:

• Quantification of the exponential convergence to a bacteria-free equilibrium of equation (1 ) when $d$ is large enough.

• Use of the previous result plus concentration inequalities in order to study the convergence of the noisy system to equilibrium in a reasonable time range.

• Simulation of the stochastic processes at stake in order to observe the convergence to equilibrium.