Bifurcation analysis of an existing mathematical model reveals novel treatment strategies and suggests potential cure for type 1 diabetes (Q2922509)

The focus of this paper is the mathematical model proposed early by \textit{A. F. M. Marée} et al. [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 364, No. 1842, 1267--1282 (2006; Zbl 1152.92324)]. The model is studied by bifurcation analysis. Giving the biological reasoning behind the basic model, the authors present, then the DuCa model, which has the identical parameters as the Marée model, but has some added changes: \(J=a\), \(e^1=e^2=e\) and \(g=f^1\). The model consists of a system of 5 ODEs that describe the change in concentration of resting macrophages, \(M\), activated macrophages, apoptotic \(\beta\)-cells, \(B_a\), \(B_n\), cytokines \(C\).NEWLINENEWLINEFirst, it is proved that the equations are continuously differentiable; this provides existence and uniqueness of the solutions. Then, it is found the trapping region as an outflow-closed region: a region in the phase space that fulfills the condition that, once the solution is in this region, it cannot leave it.NEWLINENEWLINEProposition 1. For any \(\varepsilon>0\), \(\mathfrak{T}\mathfrak{R}^\varepsilon\) is a trapping region for the ODE system given in (3.1--3.5), with \(\mathfrak{T}\mathfrak{R}=\bigcap_{\varepsilon>0}\mathfrak{T}\mathfrak{R}^\varepsilon\). NEWLINENEWLINEProposition 2. Any point in \((\mathbb{R}_+\cup \{0\})^5\setminus\mathfrak{T}\mathfrak{R}^\varepsilon\) is attracted to \(\mathfrak{T}\mathfrak{R}^\varepsilon\), and it enters the trapping region in finite time. Hence, \(\mathfrak{T}\mathfrak{R}^\varepsilon\) is an attracting trapping region in the positive cone for any \(\varepsilon>0\).NEWLINENEWLINEAfter proving these two propositions, the authors define an attracting trapping region. From biological considerations, two phagocytosis parameters \((f_1,f_2)\) are selected as bifurcation parameters and the whole analysis based on the Newton-Raphson method of producing the bifurcation diagrams.NEWLINENEWLINEBy exploring these two parameters separately and simultaneously (co-dimension 1 and 2), it is found that by increasing the phagocytosis rates, either by themselves or simultaneously, the inflammation in NOD mice could be reversed.NEWLINENEWLINEThe authors expected ``that as their understanding of T1D and its precursors evolves it will become easier and safer to the pinpoint the opportune moment, or period, for administration of syngeneic macrophages.''NEWLINENEWLINETherefore, the results of their analysis give more deep medical insight to the disease research and produce four novel treatment strategies, which might may be proved after some years of human trials, but the results confirm that the mathematical modeling and analysis can be useful tools for designing of perspective medical strategy in treatment of different diseases.