Abstract
The previous models described oncolytic virus dynamics in the context of specific ordinary differential equations. However, the same biological assumptions can be expressed with alternative mathematical expressions, and the arbitrary formulations that are chosen can sometimes determine the behavior of the model. It is therefore desirable to formulate a model such that its properties do not depend on arbitrary mathematical terms that are used to describe biological processes. This can be achieved through the formulation of axiomatic models, where biological processes are described by general functions that are subject to reasonable constraints. This chapter formulates an axiomatic model of oncolytic virus dynamics and identifies different classes of models that share common properties. The different model classes and their properties are discussed, focusing on the conditions required for virus-mediated tumor control.
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Notes
- 1.
This requirement fixes the scaling of the time-variable. In general, if the initial growth-rate \(\lim _{z\rightarrow 0}F(z)=r\), we scale time \(t'=tr\), and also use \(a'=a/r\) and \(\beta '=\beta /r\). The primes are dropped for convenience.
- 2.
Here we assume that the functions \(F\) and \(G\) are differentiable at zero. Non-differentiable functions are handled similarly by using generalized expansions.
- 3.
For very small system sizes, the proportion of cells participating in infection is formally zero because of the lack of uninfected cells, therefore the graph of the function \(G\) starts at zero, reaches a peak, and then declines for high values of \(x\).
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Komarova, N.L., Wodarz, D. (2014). Axiomatic Approaches to Oncolytic Virus Modeling. In: Targeted Cancer Treatment in Silico. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8301-4_13
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DOI: https://doi.org/10.1007/978-1-4614-8301-4_13
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