Delay-induced bifurcations and chaos in a two-dimensional model for the immune response

Nonlinear dynamics in tumor-immune system interaction models with delays

“…In order to study the original delay model (86), Buric et al [22] and Yu and Wei [126] expressed the variable of f as a combination of T (t) and T (t − δ), that is f (aT (t) + (1 − a)T (t − δ)) with a(0 ≤ a ≤ 1) being a constant, and allowed the function g to depend on the combination of E(t) and E(t − ∆), that is g(bE 1+T (t−δ) u , g(E(t − ∆)) = sE(t−∆) n 1+E(t−∆) n , recently Mendonça et al [90] performed detailed linear stability analysis of original delayed model (86) to investigate possible stability switches induced by the existence of characteristic delay times of the dynamical processes and showed that stability switches of stable periodic cycle solutions can be induced by enforcing appropriated time delays in the tumor cell reproduction as well as in the cooperative immune response. They also provided numerical simulations of the corresponding set of delayed differential equations to support the analytical results, showing bifurcations and quasi-chaotic behavior.…”

“…Delayed responses are very crucial and important for the tumor and immune system interaction, just as Asachenkov et al [10] and Mayer et al [87] pointed out that the delays should be taken into account to describe the times necessary for molecule production, proliferation, differentiation of cells, transport, etc. In fact, tumor and immune system interaction models with delay have been studied considerably, in particular two-dimensional delay differential equations model, see Abdulrashid et al [1], Asachenkov et al [10], Banerjee and Sarkar [11], Barbarossa et al [12], Bi et al [13,14,15,16], Bodnar and Foryś [18], Buric [22], Dong et al [36], d'Onofrio [37,38,39], d'Onofrio and Gandolfi [40], d'Onofrio et al [42], Galach [56], Grossman and Berke [57], Khajanchi and Banerjee [69], Liu et al [81], Mayer et al [87], Mendonça et al [90], Piotrowska [96], Piotrowska and Foryś [97], Rordriguez-Perez et al [101], Villasana and Radunskaya [115], Yu and Wei [126], Yu et al [127,128], and the references cited therein.…”