Mathematical models for bacteriophage dynamics applicable to phage therapy

Date

2013-08

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Abstract

Bacteriophages, more commonly known as phages, are viruses that kill bacteria. Phages are used to treat food or animals infected with bacteria. Phages are classified into two groups: lytic and lysogenic. The lytic phages kill the host bacteria. Phages attack to a bacterium, inject their DNA or RNA, multiply inside the bacterium, then burst from the bacterium, releasing many new phage particles. We formulate a host-phage model, a system of differential equations for susceptible bacteria, latently infected bacteria, actively infected bacteria, and phage, that includes two important features to model phage therapy: bacteria carrying capacity and latent period due to phage replication. In addition, this model accounts for multiple phage attachment and phage loss due to attachment to latently and actively infected bacteria. Based on this deterministic model we derive two stochastic models, a continuous-time Markov chain model (CTMC), and an It^o stochastic differential equation (SDE) model that accounts for the variability due to phage transmission, entry into host bacteria, births, deaths, and release strategy. Both local and global stability of the phage-free non-trivial equilibrium are verified and the basic reproduction number, R0, is computed. In addition, it is verified that the same threshold applies to the stochastic models. That is, if R0 < 1, the phage population does not grow in either the deterministic or stochastic formulations. We analyse the CTMC model using multitype continuous-time branching process approximations both when the burst number is fixed and variable. The branching process provides an estimate of probability of successful phage growth when the threshold exceeds one. Numerical results applied to a particular bacteriophage system, Escherichia coli and T4, illustrate the importance of initial phage density, phage burst number, phage death rate, bacteria carrying capacity, and bacteria-phage adsorption rate in controlling bacterial infections. Numerical simulations using the SDE model illustrate the effects of a variable burst number for the establishment of phage in a bacteria population, for parameter values selected for bacteriophage interaction, E. coli and T4. Finally, we generalize our initial deterministic model by introducing additional state variables. The first extension is to include bacteriophage complexes and the second extension is to include resistant bacteria and bacteria debris. The model with ressistant bacteria and bacteria debris accounts for multiple phage attachment, phage detachment, and phage loss due to attachment to debris. Based on these generalized models, CTMC models and SDE models were derived and analysed.

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Keywords

Bacteriohage models, Lytic phage, Lysogenic phage, Resistant bacteria, Bacteria debris, Burst number, Phage extinction

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