Bacterial quorum sensing (QS) refers to the process of cell-to-cell bacterial communication enabled through the production and sensing of the local concentration of small molecules called autoinducers to regulate the production of gene products (e.g. enzymes or virulence factors). Through autoinducers, bacteria interact with individuals of the same species, other bacterial species, and with their host. Among QS-regulated processes mediated through autoinducers are aggregation, biofilm formation, bioluminescence, and sporulation. Autoinducers are therefore "master" regulators of bacterial lifestyles. For over 10 years, mathematical modelling of QS has sought, in parallel to experimental discoveries, to elucidate the mechanisms regulating this process. In this review, we present the progress in mathematical modelling of QS, highlighting the various theoretical approaches that have been used and discussing some of the insights that have emerged. Modelling of QS has benefited almost from the onset of the involvement of experimentalists, with many of the papers which we review, published in non-mathematical journals. This review therefore attempts to give a broad overview of the topic to the mathematical biology community, as well as the current modelling efforts and future challenges.B Judith Pérez-Velázquez

Infectious diseases are a serious problem for public health and spark the interest in interdisciplinary studies. In this paper, we present two mathematical models describing a possible scenario for infectious diseases. The first model considers the dynamics of the disease among adults and emphasizes the role of carriers in the SIR model and the second model assumes that the disease is transmitted to children by adults. We state the equilibria for each model and study the local stability of the equilibria. Furthermore, we perform simulations using a parameter set that explains the spread of a specific infectious disease (meningococcal disease) and interpret the possible cases of transmission via simulations.

Although policy makers recommend or impose various standard measures, such as social distancing, movement restrictions, wearing face masks and washing hands, against the spread of the SARS-CoV-2 pandemic, individuals follow these measures with varying degrees of meticulousness, as the perceptions regarding the impending danger and the efficacy of the measures are not uniform within a population. In this paper, a compartmental mathematical model is presented that takes into account the importance of personal cautiousness (as evidenced, for example, by personal hygiene habits and carefully following the rules) during the COVID-19 pandemic. Two countries, Turkey and Italy, are studied in detail, as they share certain social commonalities by their Mediterranean cultural codes. A mathematical analysis of the model is performed to find the equilibria and their local stability, focusing on the transmission parameters and investigating the sensitivity with respect to the parameters. Focusing on the (assumed) viral exposure rate, possible scenarios for the spread of COVID-19 are examined by varying the viral exposure of incautious people to the environment. The presented results emphasize and quantify the importance of personal cautiousness in the spread of the disease.

Quorum sensing, a special kind of cell-cell communication, has originally been described for well-mixed homogeneous bacterial cultures. However, recent perception supports its ecological relevance for spatially heterogeneous distributed cells, like colonies and biofilms. New experimental techniques allow for single cell analysis under these conditions, which is crucial to understanding the effect of chemical gradients and intercell variations. Based on a reaction-diffusion system, we develop a method that drastically reduces the computational complexity of the model. In comparison to similar former approaches, handling and scaling is much easier. Via a suitable scaling, this approach leads to approximative algebraic equations for the stationary case. This approach can be easily used for numerical situations.

Hepatitis B infection is one of the serious viral infections that is treating the global health. Turkey has an intermediate endemicity for hepatitis B. In this study, a classical SIR model for hepatitis B virus (HBV) transmission is proposed and analyzed. Based on the available data of Republic of Turkey Ministry of Health, associated parameters are estimated and the …tted model is shown by appropriate simulations. The basic reproductive number is obtained by using the estimated parameters. Finally, we discuss the sensitivity of parameters and the e¤ect of changes of parameters in the spread of disease.

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