# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 1 Introduction  
The Chikungunya virus (CHIKV) belongs to the Togaviridae family of the genus Alphavirus [1, 2]. It takes the form of an enveloped spherical viral particle with a diameter of 65 nm, containing a single RNA of positive polarity, encoding two polyproteins. The RNA is directly infectious and, therefore, serves as both a genome and messenger RNA. The 11,000 to 12,000 bases of its genome ultimately allow the synthesis of nine proteins, obtained after cleavage of the polyproteins by viral and cellular proteases. At the end of the viral cycle, which includes protein synthesis, replication, and assembly of viral particles, the viral proteins bud on the plasma membrane of the infected cell, and then penetrate the membrane. The virus is transmitted to humans when anticoagulant saliva is injected into the blood of the person bitten by an infected mosquito [1, 2]. We now know that the virus does not infect circulating blood cells, but rather, macrophages and adherent cells (endothelial, epithelial, fibroblast). In [3], [PERSON] et al. first adapted tools (flow cytometry, immunofluorescence, electron microscopy, etc.) to specify and quantify the CHIKV. Therefore, they proved in vitro that CHIKV does not replicate in circulating blood cells (lymphocytes, monocytes), but that it replicates in macrophages (phagocytic cells which are of blood origin but localized in tissues). These cells itself infect tissues, such as muscles and joints. The CHIKV also infects the so-called \"adherent\" cells, such as endothelial cells, epithelial cells, and fibroblasts. A study conducted by [PERSON] at al. [4] has shown that, in infected people, certain cells present in muscle tissue are targets of the CHIKV. Their work was based on the study of patient biopsies. They found that, in a biopsy taken at an acute stage of the disease from one patient, and in another taken at a later stage from another patient, the precursor cells of muscle cells--the satellite cells--were infected with the virus. In addition, these cells have been found, in cell culture, to be very permissive to the virus. The authors are nowtrying to discover if these cells play the role of a \"reservoir\" of the virus, which would explain the recurrence of muscle pain observed in some patients. [5] stated that like any virus, the CHIKV enters its host's cells and uses the cellular machinery there to replicate. In particular, it is shown that this arbovirus replicates in macrophages, cells of the immune system specializing in phagocytosis and present in several tissues and organs. On the other hand, it does not reproduce in the T and B lymphocytes which are circulating in the blood. CHIKV also infects epithelial cells (organ wall), endothelial cells (inner wall of blood vessels), as well as cells of fibrous tissues (fibroblasts). The contribution of mathematics is made initially through modelling [6; 7; 8; 9; 10; 11; 12; 13; 14]. This stage of mathematical modelling makes it possible to test, without wasting time or expense, the control measures that are envisaged: preventive measures, isolation of patients, treatments, vaccinations, and so forth. The model, nevertheless, does not completely reflect reality and is not intended to reproduce it in full. It must reproduce the characteristics of the phenomenon studied according to the objectives set for the framework of the study as well as possible. Modelling such a phenomenon consists of applying mathematical tools to a fragment of reality. It is transforming a need into equations, trying as much as possible to account for the constraints identified. Modelling is the most delicate, the longest, and often the most perilous step. Indeed, it is necessary to successfully understand the real problem to try to propose a suitable model. The first proposed attempt only very rarely meets expectations, and several modifications then follow, until a model is reached that groups together and reflects the maximum number of constraints that the real phenomenon must observe.