Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control

Figures & data

Table 1. Parameters description and values.

Parameter	Description	Estimate	Scaling values
$a_0$	Growth rate of tumour size population	0.1946	0.01946
$1/c_0$	Carrying capacity of tumour size	369	0.00271
${\delta }_0$	Natural death rate for all tumours cells	1 $\times {10}^{-5}$	1 $\times {10}^{-6}$
${\delta }_1$	Natural death rate for all effector cells	1 $\times {10}^{-5}$	$\times {10}^{-6}$
$a_1$	Maximum rate of TGF-$\beta$ production	0.3	0.03
$c_2$	Critical tumour cell population	300	3
$d_1$	Decay rate of TGF-$\beta$	7 $\times {10}^{-4}$	7 $\times {10}^{-4}$
r	Tregs differentiate from effector T cell rate	0.01	0.001
$p_1$	Maximum rate of effector cell proliferation	0.1245	0.01245
$g_1$	Half-saturation constant	2 $\times {10}^7$	2 $\times {10}^2$
$q_1$	Maximum rate of anti-proliferative effect of TGF-$\beta$	0.1121	0.01121
$q_2$	Half-saturation constant	2 $\times {10}^6$	20
$p_3$	Growth rate of IL-2 × days	5 $\times {10}^3$	0.000025
$g_4$	Half-saturation constant	1 $\times {10}^3$	0.01
${\mu }_2$	Decay rate of IL-2	10	Estimated

Figure 1. Sensitivity solutions of tumour cell population from (19) to (23) w.r.t a₀, δ₀, δ₁, r and p₁.

Figure 2. Plots the characteristic equation of (25(25) $\begin{array}{rl}& {\lambda }^2-0.01845899975\lambda -0.0001947945975+{\mathrm{e}}^{-13\lambda }(-3.112422190\times {10}^{-7}\lambda \\ & +6.056773505\times {10}^{-9})=0,\end{array}$(25) ) for different τ values (Table 1).

Table 2. Parameters description and values.

Parameter	Mouse data	Human data	Source
$p_1$	0.1245	0.1245	[15]
$g_1$	2 $\times {10}^7$	2 $\times {10}^7$	[15]
${\mu }_2$	10	10	Estimated
$a_0$	2 $\times {10}^{-3}$	4.31$\times {10}^{-1}$	Estimated
$c_0$	1.02 $\times {10}^{-9}$	2.17 $\times {10}^{-8}$	[9]
${\delta }_0$	3.23 $\times {10}^{-7}$	7.13 $\times {10}^{-10}$	[9,10]
${\mu }_3$	10	10	[15]
$p_3$	6 $\times {10}^{-1}$	6 $\times {10}^{-1}$	Estimated
$g_4$	1 $\times {10}^5$	1 $\times {10}^5$	[9]
β	9 $\times {10}^{-1}$	9 $\times {10}^{-1}$	[5]
$a_0$	0.1946	0.1946	[28]
r	0.01	0.01	[14]

Figure 3. The dynamics of the tumour size in three treatment regimes. Shown are the results of the numerical simulations for Mouse data corresponding from Table 2 with initial condition $T\left(t\right)={10}^5$, $B\left(t\right)={10}^5$, $R\left(t\right)={10}^5$, $E\left(t\right)={10}^5$, $I\left(t\right)={10}^5$, $M\left(t\right)={10}^5$.

Figure 4. The dynamics of the tumour size in three treatment regimes. Shown are the results of the numerical simulations for human data corresponding from Table 2 with initial condition $T\left(t\right)={10}^5$, $B\left(t\right)={10}^5$, $R\left(t\right)={10}^5$, $E\left(t\right)={10}^5$, $I\left(t\right)={10}^5$, $M\left(t\right)={10}^5$.

Figure 5. The optimal control graph for the chemo therapeutic drug control (M) using the parameter values given in Table 2 with $B_{\rho }=1$ and $B_{\eta }=2$ for mouse and human data.

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