Abstract
The human body’s immune response to bacterial challenge, even when successful in controlling the infection, can result in negative consequences for the host, including reduced functionality of associated tissues. We present and analyze a low-dimensional mathematical model of this immune response to pathogen invasion, incorporating the coordinated actions of active immune cells, and both pro- and anti-inflammatory cytokines. The model simulates both the positive (pathogen reduction) and negative (local tissue dysfunction) effects of the immune response and includes the important role of immunologic memory in the process of a return to stasis. This differential equation-based model is sufficiently general to be applicable to a wide range of human tissues and organs.
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Appendix
Appendix
Proof of Theorem 4
Letting \(\epsilon >0\), we will use the shorthand notation \(P_\epsilon \equiv P_{\mathrm{max}}-\epsilon \), \(\phi _\epsilon \equiv \phi (P_\epsilon )\), \(\delta _\epsilon \equiv \delta (P_\epsilon )\), \(\psi _\epsilon \equiv \psi (P_\epsilon ,0)\), and \(\beta _\epsilon \equiv \beta (P_\epsilon ,0)\).
Note that \(P_{\mathrm{null}}(P)\) attains an absolute minimum \(M_\epsilon >0\) on \(0\le P\le P_\epsilon \). By Property 3(b) of Lemma 3, it suffices to demonstrate the existence of \(q>0\) and \(c>0\) for which
To this end, we can choose q to satisfy \(\displaystyle {0< q < \frac{r}{\lambda ^{\frac{3}{2}}}\left( M_\epsilon +B\right) }\), from which it follows that
and, consequently,
so that
thus making it possible to choose c to satisfy \(\displaystyle {0< c < \frac{1}{\sqrt{\lambda }}\left( \frac{K+P_\epsilon }{P_\epsilon }\right) M_\epsilon \left( \phi _\epsilon + \frac{\delta _\epsilon }{M_\epsilon +B}\right) }\). Then, we have
Taking positive square roots yields
so that
as desired. \(\square \)
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Caudill, L., Lynch, F. A Mathematical Model of the Inflammatory Response to Pathogen Challenge. Bull Math Biol 80, 2242–2271 (2018). https://doi.org/10.1007/s11538-018-0459-6
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DOI: https://doi.org/10.1007/s11538-018-0459-6