Analysis of a mathematical model of immune response to fungal infection

Abstract

Fungi are cells found as commensal residents, on the skin, and on mucosal surfaces of the human body, including the digestive track and urogenital track, but some species are pathogenic. Fungal infection may spread into deep-seated organs causing life-threatening infection, especially in immune-compromised individuals. Effective defense against fungal infection requires a coordinated response by the innate and adaptive immune systems. In the present paper we introduce a simple mathematical model of immune response to fungal infection consisting of three partial differential equations, for the populations of fungi (F), neutrophils (N) and cytotoxic T cells (T), taking N and T to represent, respectively, the innate and adaptive immune cells. We denote by $\lambda _F$ the aggressive proliferation rate of the fungi, by $\eta$ and $\zeta$ the killing rates of fungi by neutrophils and T cells, and by $N_0$ and $T_0$ the immune strengths, respectively, of N and T of an infected individual. We take the expression $I=\eta N_0 + \zeta T_0 - \lambda _F$ to represent the coordinated defense of the immune system against fungal infection. We use mathematical analysis to prove the following: If $I>0$, then the infection is eventually stopped, and $F \rightarrow 0$ as $t \rightarrow \infty$; and (ii) if $I<0$ then the infection cannot be stopped and F converges to some positive constant as $t\rightarrow \infty$. Treatments of fungal infection include anti-fungal agents and immunotherapy drugs, and both cause the parameter I to increase.