In [1] Forys et al. proposed a simple ODE model describing interactions between tumour and immune system. Later on Piotrowska generalised that model by considering a general form of the stimulus function $f$ and by incorporating a discrete time delay ($\tau$) representing time lag in the immune system response to presence of antigens, [2].

We further generalise that model by considering different types of distributed delays instead of discrete one. In particular, we replace the term $f(y(t-\tau))$ by

$f\left(\int_{0}^{\infty}K(s)y(t-s)ds\right)$,

where $K\colon[0,+\infty)\to \mathbb{R}_{\ge}$ is a probability density function with finite expectation.

During the presentation we are going to discuss basic mathematical properties of generalised model with distributed delays, the existence and stability of steady states depending on the forms of considered probability densities (with compact or not support, possibly separated from zero).

We also show how to evaluate the model with distributed delays with available experimental data by fitting model solutions to two data sets reporting the development of mice B-cell lymphoma in the spleens of BALB/c and chimeric mice, [3]. That is an essential part of our work since according to our knowledge appropriate software numerically solving the systems of equations with different probability densities does not exist.

This research was partially supported by National Science Centre, Poland, grant no. 2015/19/B/ST1/01163.

References

[1] U. Forys, J. Waniewski, and P. Zhivkov, Anti-tumor immunity and tumor anti-immunuty in a mathematical model of tumor immunotherapy, J. Biol. Sys., 14(1) (2006), 13--30.

[2] M.J. Piotrowska, An immune system---tumour interactions model with discrete time delay: Model analysis and validation, Commun. Nonlinear. Sci. Numer. Simul., 34, (2016), 185--198.

[3] J. W. Uhr, T. Tucker, R. D. May, H. Siu, and E. S. Vitetta, Cancer dormancy: Studies of the murine BCL$_1$ lymphoma., Cancer Res. (Suppl.), 51 (1991), 5045s--5053s.