Bifurcation analysis of a mathematical model for the activated complement-mediated response to bacterial infection in humans: the complement system as part of the innate immune system

Abstract Bacteria and other environmental entities pose constant threats to human health, often causing allergies and inflammation. The complement system (CS), a vital component of the immune system primarily found in the liver, acts as the first line of defense against such invaders. In response to a pathogenic invasion, the complement system rapidly initiates a series of events to mount a defense. While activated complement components can eliminate threats, not all activation leads to complete elimination; some prime the pathogen for a higher-order defense in a process known as opsonization. We propose a nonlinear deterministic ordinary differential equation model and use it to study the dynamics of activated complement system’s interaction with bacterial infection, focusing on key elements of complement-mediated attacks. We analyze the model, investigating the qualitative changes in the disease dynamics, as well as the quantitative changes in the level and severity of the infection, under parameter variation. Using different functional response functions, including the nonlinear Hill hazard function (HHF), we explore scenarios where the activated complement system interactions coexist with, or eliminate, the invading bacteria. From our results, we show that a robust complement system response can efficiently eliminate free-floating bacteria. In other scenarios, depending on the strength of the activated complement system, it can only slow the bacterial infection without eliminating it, while it can also be overwhelmed by the infection, highlighting the need for higher immune responses. Bifurcation points, where a change in the dynamics occurs as certain parameters are perturbed, are illustrated both analytically and numerically. The work presented is a first step in understanding the innate immune response as well as the adaptive immune response, when necessary, to disease infection.