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In this paper, we propose a reaction–diffusion mathematical model augmented with self/cross-diffusion in 2D domain which describes the oncolytic virotherapy treatment of a tumor with its growth following the logistic law. The tumor cells are divided into uninfected and infected cells and the virus transmission is supposed to be in a direct mode (from cell to cell). In the absence of cross-diffusion, we establish well posedness of the problem, non-negativity and boundedness of solutions, nonexistence of positive solutions, local and global stability of the nontrivial steady-state and the nonoccurrence of Turing instability. In the presence of cross-diffusion, we prove the occurrence of Turing instability by using the cross-diffusion coefficient of infected cells as a parameter. To have an idea about different patterns, we derive the corresponding amplitude equation by using the nonlinear analysis theory. In the end, we perform some numerical simulations to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a delayed mathematical model describing oncolytic virotherapy treatment of a tumour that proliferates according to the logistic growth function, incorporating viral lytic cycle. The tumour population cells are divided into uninfected and infected cell sub-populations and the virus spreading is supposed to be in a direct mode (i.e. from cell to cell). Depending on the time delay, we analyze the positivity and boundedness of solutions and the stability of tumour, infected and uninfected free equilibria (TFE, IFE, UFE) and uninfected–infected equilibrium (UIE) is established. We prove that, delay can lead to "Jeff's phenomenon" observed in a laboratory which causes oscillations in tumour size whose phase and period change over time. With nonlinear dependence of UIE equilibrium on time delay, we develop a more general algorithm determining the stability/instability of the oscillating periodic solutions bifurcating from the UIE equilibrium. Finally, we present numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]