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This paper introduces a deterministic fractional-order epidemic model (FOEM) for studying the transmission dynamics of the human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS). The model highlights the substantial role of unaware and undetected HIV-infected individuals in spreading the disease. Control strategies, such as wielding condoms, level of preventive measures to avoid infection, and self-strictness of susceptibles in sexual contact, have been incorporated into the study. The basic reproduction number ℛ 0 α has been derived, which suggests the conditions for ensuring the persistence and elimination of the disease. Further, to validate the model, actual HIV data taken from Mexico and India separately have been used. The disease dynamics and its control in both countries are analyzed broadly. The values of biological parameters are estimated at which numerical solutions better match the actual data of HIV patients in the case of fractional-order (FO) instead of integer-order (IO). Moreover, in the light of ℛ 0 α , our findings forecast that the disease will abide in the population in Mexico, and at the same time, it will die out from India after a long time. [ABSTRACT FROM AUTHOR]

This paper studies the effect of habitat complexity on Greater one-horned rhinoceros (Rhinoceros unicornis) and tiger (Panthera tigris) population model in Kaziranga National Park (KNP), Assam, India. Based on the analysis of the data collected from PCCF, Wildlife, Assam, three mathematical models are formulated and studied. In view of ecology, the main objective of the study is to increase the size of rhino population in aforesaid park. The mathematical stability and the complex dynamical behavior of systems are analyzed here. In KNP, the immature rhinos are killed by tiger, so the first system describes a stage structured prey–predator interaction , where the rhino is the prey and is divided into immature (horn not developed) and mature (horn developed) category and the tiger is the predator. The immature rhino is killed by the tiger following Holling type-II functional response, while the mature rhino falls prey to human induced poaching activities [Source: PCCF Wildlife, Assam, India]. Though Kaziranga is famous all over the world for the Greater one-horned rhinoceros, many other herbivores are also present in really good number and the tiger preys on them too. Thus, the previous system is modified by modifying the Holling type-II functional response incorporating the effect of additional food availability. To increase the number of rhino and to maintain the ecological balance of KNP, the second system is further extended by introducing habitat complexity in the Holling type-II functional response. In each system, equilibrium points have been obtained and their stability are discussed. Finally, numerical simulations are carried out to illustrate the analytical results. Based on the simulation results, it can be stated that the size of the rhino population increases in the presence of additional food and habitat complexity in spite of the poaching activities to a certain extent. The system shows complex dynamical behavior like Hopf bifurcation with respect to Poaching activity. Sensitivity analysis with respect to four important parameters viz., poaching effect, quality and quantity of the additional food and habitat complexity is also discussed. [ABSTRACT FROM AUTHOR]