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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]

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the qualitative analysis of the travelling waves solutions of a reaction diffusion model that refers to the competition between the predator and prey with modified Leslie–Gower and Holling type II schemes. The well posedeness of the problem is proved. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of the forth degree exponential polynomial characteristic equation. We also prove the existence of a Hopf bifurcation which leads to periodic oscillating travelling waves by considering the diffusion coefficient as a parameter of bifurcation. Numerical simulations are given to illustrate the analytical study. [Copyright &y& Elsevier]

Abstract: The model analyzed in this paper is based on the model set forth by [M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069–1075, A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., in Press.] with time delay, which describes the competition between predator and prey. This model incorporates a modified version of Leslie–Gower functional response as well as that of the Holling-type II. In this paper, we consider the model with one delay and a unique non-trivial equilibrium and the three others are trivial. Their dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation at for small and large delays and at the third trivial equilibrium that is proven to exist as the delay (taken as a parameter of bifurcation) crosses some critical values. We illustrate these results by numerical simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an ordinary differential equations (ODE) for tumor–immune system with positive immune response and a unique nontrivial positive equilibrium. Its dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation which is proven to exist as the parameter of the normal rate of the flow of adult ECs into the tumor site crosses some critical values. We illustrate these results numerically. [Copyright &y& Elsevier]

This paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a system of differential equations with one delay. It is shown that the dynamics depends crucially on the time delay parameter. By using the time delay as a parameter of bifurcation, the analysis is focused on the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state. The obtained results depict the oscillations, given by simulations (see [M. Galach, Int. J. Appl. Math. Comput. Sci., 13 (2003), pp. 395–406]), which are observed in reality (see [D. Kirschner and J. C. Panetta, J. Math. Biol., 37 (1998), pp. 235–252]). It is suggested to examine by laboratory experiments how to employ these results for control of tumor growth. [ABSTRACT FROM AUTHOR]