Appl. Math. Inf. Sci. 9, No. 6, 2775-2782 (2015) 2775 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090603 A Delayed Mathematical Model for the Acute Inflammatory Response to Infection Carlo Bianca 1,2,∗ , Luca Guerrini 3 and Julien Riposo 1,2 1 Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee, Sorbonne Universit´ es, UPMC Univ Paris 06, UMR 7600, 75252 Paris cedex 05, France 2 CNRS, UMR 7600 LPTMC, Paris, France 3 Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121 Ancona, Italy Received: 18 Dec. 2014, Revised: 25 Apr. 2015, Accepted: 30 Apr. 2015 Published online: 1 Nov. 2015 Abstract: This paper deals with further developments on a mathematical model recently proposed for the modeling of the acute inflammatory response to infection or trauma. In particular in order to take into account that some interactions have not an immediate effect, we introduce time delays. Specifically the paper deals with the existence of steady states, determining the parameter regimes where the equilibrium points are stable, and the onset of Hopf bifurcation appears. Numerical simulations are performed with the main aim of supporting the analytical results and investigate further dynamics. Keywords: Wound healing, Time delay, Hopf bifurcation, Asymptotic analysis, Fixed points 1 Introduction The problem of abnormal organ repair has gained much attention considering that there is a significant shortage of organs available for transplantation. In this context the normal repair process, i.e. wound healing process, assumes an important role. Wound healing is an complex process by which the skin or organ repairs itself after injury [1, 2]. Specifically wound healing comprises three sequential, overlapping, phases: the inflammation phase (hemostasis and the actual inflammation), the proliferation phase, and the maturation (remodeling) phase. Hemostasis occurs immediately after tissue injury and can be compared with the acute phase reaction of the innate immune system during infection. The first cells to appear in the wound area are neutrophils which start with the critical task of phagocytosis in order to destroy and remove bacteria, foreign particles and damaged tissue. Phagocytotic activity is crucial for the subsequent processes, because acute wounds that have a bacterial imbalance will not heal. The macrophage becomes the predominant inflammatory cell type in clean noninfected wounds. Every phase of the healing process consists of complex interactions between cells and mediators which tend to regulate the process. Cells participating in wound healing must be activated, i.e. undergo phenotypic alterations of cellular, biochemical, and functional properties. During the inflamation phase, the immune system performs a fundamental action, see [3, 4, 5]. The response of the immune system to an infectious agent is subdivided into two main categories: Innate (non-specific) immunity response, which is mediated by granulocytes, macrophages, and NK cells [6]; Adaptive (specific, acquired) immune response, which is mediated by the lymphocytes [7]. The innate immune system is constitutively active and reacts immediately to infection. The adaptive immune response to an invading organism takes some time to develop. Different mathematical models have been proposed for the modeling of immune system response [8, 9]. Specifically mathematical models based on ordinary differential equations [10, 11, 12, 13], partial differential equations [14, 15], kinetic theory approach [16, 17] and continuum mechanics approach [18]. In the pertinent literature computational models have been also proposed, see [19] and the review paper [20]. However the previous cited models are based on instantaneous interactions thus avoiding to take into account that various phenomena occur with some delay. In order to overcome this issue, ∗ Corresponding author e-mail: carlo.bianca@polito.it c  2015 NSP Natural Sciences Publishing Cor.