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96 Control theory in biomedical engineering

cells, effector cells, and normal cells of each control strategy in 30 days. The
numerical simulations displayed in the figure validate the existence of opti-
mality of the control variables and show that the immuno-chemotherapy
protocol reduces the tumor load in a few months of therapy. Careful deter-
mination of the inclusion of the delay in the optimal control setting for the
forward- and backward-oriented system in the numerical setting is needed.
The numerical simulations validate the existence of optimality of the
control variables. The presence of chemotherapy protocol reduces the
tumor load in a few days of therapy. Of course, this model can be extended
to more powerful models for cancer treatment design with optimal combi-
nations, doses, and scheduling of treatments to speed up the development of
individualized therapies; see Rihan et al. (2019), Rihan and Velmurugan
(2020), and Kim et al. (2018). It is also useful to investigate how a small shift
(change) in the input parameters would change the stability of the tumor-
free equilibrium, and detect the most significant parameter that has a major
impact on the model dynamics.

Appendix
A.1 DDEs with optimal control
Mathematical modeling with DDEs is widely used for analysis and predictions
in epidemiology, immunology, and physiology (Rihan, 2000; Bocharov and
Rihan, 2000; Fowler and Mackey, 2002; Nelson and Perelson, 2002; Smith,
2011). Time-delays in these models take into account a dependence of the
present state of the modeled system on its past history. The delay can be related
to the duration of certain hidden processes like the stages of the life cycle, the
time between infection of a cell and the production of new viruses, the dura-
tion of the infectious period, the immune period and so on. In real life, things
are rarely so instantaneous. There is usually a propagation delay before the
effects are felt. This situation can be modeled using a DDE.

y ðtÞ¼ fðyðtÞ,yðt τ 1 Þ,yðt τ 2 Þ,…,yðt τ d Þ,tÞ,t t 0 , (A.1)
0
where all of the time-lags, τ i , are assumed to be none negative functions of
the current time t. Because of these delay terms it is no longer sufficient to
supply an initial value, at time t ¼ t 0 , to completely define the problem.
Instead, it is necessary to define the history of the state vector, y(t), suffi-
ciently far enough back in time from t 0 to ensure that all of the delayed state
terms, y(t τ i ) are always well defined. Thus, it is necessary to supply an
initial state profile of the form: