Mathematical modeling of cancer–immune system, considering the role of antibodies

Abstract

A mathematical model for the quantitative analysis of cancer–immune interaction, considering the role of antibodies has been proposed in this paper. The model is based on the clinical evidence, which states that antibodies can directly kill cancerous cells (Ivano et al. in J Clin Investig 119(8):2143–2159, 2009). The existence of transcritical bifurcation, which has been proved using Sotomayor theorem, provides strong biological implications. Through numerical simulations, it has been illustrated that under certain therapy (like monoclonal antibody therapy), which is capable of altering the parameters of the system, cancer-free state can be obtained.

Appendix: Sotomayor theorem (Perko 1991)

Consider the system of ordinary differential equations

$$\begin{aligned} \frac{{\mathrm{d}}x}{{\mathrm{d}}t}=f(x,\eta ).~~x\in {\mathbb {R}}^n, \end{aligned}$$

(10)

where $\eta \in {\mathbb {R}}$ is a system parameter. It is assumed that the function f is sufficiently differentiable so that all the derivatives appearing in that theorem are continuous on ${\mathbb {R}}^n\times {\mathbb {R}}$. We denote the matrix of partial derivatives of the components of the vector field f with respect to the components of x by Df and the vector of partial derivatives of the components of f with respect to parameter $\eta$ denoted by $f_{\eta }$.

Theorem 5

(Perko 1991) (Sotomayor):Suppose that $f(x_0,\eta _0)$ and that  $n\times n$ matrix $B\equiv Df(x_0,\eta _0)$ has a simple eigenvalue $\lambda =0$ with eigen vector v and that $B^T$ has an eigen vector u corresponding to the eigenvalue $\lambda =0.$ Furthermore, suppose that B has k eigenvalues with negative real parts and $(n-k-1)$ eigenvalues with positive real parts.

• (i) If the following conditions are satisfied

  $$\begin{aligned} u^Tf_\eta (x_0,\eta _0)\ne 0,~~~~~~~~u^T[D^2f(x_0,\eta _0)(v,v)]\ne 0 \end{aligned}$$

  (11)

  Then there is a smooth curve of equilibrium point of  (10) in ${\mathbb {R}}^n\times {\mathbb {R}}$ passing through  $(x_0,\eta _0)$ and tangent to the hyperplane ${\mathbb {R}}^n\times {\eta _0}$. Depending on the signs of the expressions in  (11), there are no equilibrium points of  (10)  near  $x_0$  when  $\eta <\eta _0$ (or $\eta >\eta _0$) and there are two equilibrium points of  (10) near  $x_0$  when $\eta >\eta _0$  (or $\eta <\eta _0$). The two equilibrium points of  (10)  near $x_0$ are hyperbolic and have stable manifolds of dimension  k  and  $k+1$  , respectively, that is, the system  (10) experiences a saddle-node bifurcation at the equilibrium point $x_0$  as the parameter $\eta$ passes through the bifurcation value $\eta =\eta _0$. The set of  $C^\infty$ -vector fields satisfying the above conditions is an open, dense subset in the Banach space of all $C^\infty$, one parameter vector fields with an equilibrium point at $x_0$ having a simple zero eigenvalue. (ii)If the following conditions are satisfied

  $$\begin{array}{r} u^Tf_\eta (x_0,\eta _0)= 0,\nonumber \\ u^T[Df(x_0,\eta _0)v\ne 0 \nonumber \\ u^T[D^2f(x_0,\eta _0)(v,v)]\ne 0 \end{array}$$

  (12)

  the system (10) experiences a transcritical bifurcation at the equilibrium point $(x_0,\eta _0)$ as the parameter  $\eta$ varies through the bifurcation value $\eta =\eta _0$.