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Modeling and optimal control of cancer-immune system 87

tumor cells infect some of the effector cells and therefore the population of
uninfected effector cells decreases at the rate μ. σ 0 is a treatment term that
represents the external source of the effector cells such as ACI. Furthermore,
in the absence of any tumor, the cells will die at a rate δ. The loss of tumor
cells is denoted by an immune-effector cell interaction, modeled also by
Michaelis-Menten kinetics to indicate the limited immune response to
ρEðtÞTðtÞ
tumors, so that FðE,TÞ¼ . In this term, ρ is the maximum immune
η + TðtÞ
response rate and η is the steepness of immune response. In the second equa-
tion, the rate of change of the tumor cells follows a logistic growth term
αTðtÞ 1 βTðtÞÞ.
ð
In model (1), one can incorporate a discrete time-lag τ to consider the
time needed by the IS to develop a suitable response after recognizing the
tumor cells. The new model with discrete time-lag takes the form

dEðtÞ ρEðt τÞTðt τÞ
¼ σ + μEðt τÞTðt τÞ δEðtÞ,
dt η + Tðt τÞ
t 0 (2)
dTðtÞ
¼ r 2 TðtÞ 1 βTðtÞð Þ nEðtÞTðtÞ:
dt
This model is called DDEs, in which we must provide initial functions:
E(t) ¼ ψ 1 (t) and E(t) ¼ ψ 2 (t), for all t [ τ, 0], instead of initial values.
In model (2), the presence of tumor cells stimulates the immune response,
represented by the positive nonlinear growth term for the immune cells
ρE(t τ)T(t τ)/(η + T(t τ)). ρ and η are positive constants, and τ
0 is the time-delay that presents the time needed by the IS to develop a suit-
able response after recognizing the tumor cells. The saturation term
(Michaelis-Menten form) with the E(t) compartment and logistic term with
the T(t) compartment are considered. The presence of the tumor cells vir-
tually initiates the proliferation of tumor-specific effector cells to reach a sat-
uration level parallel with the increase in the tumor populations. Hence, the
recruitment function should be zero in the absence of the tumor cells,
whereas it should increase monotonically toward a horizontal asymptote
(Villasana and Radunskaya, 2003). Of course, the solution of DDEs model
(2) should be bounded and nonnegative (Bodnar et al., 2011).

2.1 Boundedness and nonnegativity of the model solutions

To show that the solutions of model (2) are bounded and remain nonneg-
ative in the domain of its application for sufficiently large values of time t,we
recall the following lemma: