On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach

Figure 1.  Schematic illustration of the distribution of traits around dominant steady states

Figure 2.  Dose rates (left) for the control $u_{\tau}$ defined by equation (26) for $\tau=1$ (top row), $\tau=1.75$ (middle row) and $\tau =2$ (bottom row), and corresponding time evolutions of the concentrations $c$ (middle) and the states $S$ and $R$ (right)

Figure 3.  Evolution of the states S and R if the therapy horizon is extended to 42 days for the control strategies u_τ for τ = 1:75 (left) and τ = 2 (right)

Figure 4.  An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \bar{\beta} = \frac{1}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)

Figure 5.  An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \frac{1}{21}$, $\bar{\beta} = \frac{8}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)