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This study proposes a nabla discrete fractional-order system of differential equations to model lung cancer and its interactions with lung epithelial cells, mutated cells, oncogenes, tumor suppressor genes, immune cells, cytokines, growth factors, angiogenic factors, and extracellular matrix. The proposed model can help predict cancer growth, metastasis, and response to treatment. Analytical results show the system is stable with a unique solution, and the model predicts that the immune system responds to cancer cells but eventually becomes overpowered. The numerical analysis employed the forward and backward Euler method and demonstrated that changes in parameter values have significant effects on the steady-state solution. The findings show that the growth of lung epithelial cells or their interaction with immune cells can cause an increase in the number of lung cancer cells. Conversely, an increase in cell death or a reduction in the interaction between lung epithelial cells and immune cells can decrease the number of lung cancer cells. The study highlights the usefulness of the nabla discrete fractional model in studying lung cancer dynamics.

Objective: The Mediterranean island of Cyprus has a high rate of human transit as a result of tourism, human trafficking, and migration, which could contribute to the rate of HIV infection. The island is divided into 2 states - the northern portion is mainly populated by Turkish-Cypriots, and the southern portion is populated by Greek-Cypriots. The aim of this study was to assess the dynamics of HIV infection in both segments of Cyprus using a mathematical model. Materials and Methods: Data of HIV-positive individuals diagnosed during the period 1997-2018 in northern Cyprus were obtained from the Turkish Republic of Northern Cyprus Ministry of Health database, yielding a total of 129 cases, and data for the southern region, the Republic of Cyprus, were retrieved from a European Centre for Disease Prevention and Control (ECDC) 2013 report and ECDC/World Health Organization 2017 data. The total number of reported HIV infections was 1057. A mathematical model was used to evaluate the current and future HIV infection rate. Results: Stability analysis of the equilibrium point - disease-free or endemic - was conducted using the Lyapunov function. The basic reproduction number, represented as R0, is a measure of the potential for disease spread and serves as a threshold for stability. The R0 value was 0.83 in northern Cyprus and 0.040 in southern Cyprus. An R0 indicated a disease-free equilibrium. Conclusion: The calculations suggest that there is no current HIV epidemic on either part of the island; however, the model predicted a significant increase in the near future. [ABSTRACT FROM AUTHOR]