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In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin’s maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases.

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The porphyritic granite body (~ 600 Ma) around Mylliem (25°32'N : 91°52'E), east Khasi hills district, Meghalaya occurs as a distinct intrusive body into the host rock of low-grade meta-sediments belonging to Proterozoic Shillong Group. The porphyritic granite body manifests prominent lath-shaped feldspar phenocrysts giving rise to dominant porphyritic texture. Further, this porphyritic granite body is characterized by primary foliation (defined by parallely arranged feldspar laths), tongues and appophyses and xenoliths of older metamorphics. Petrographically, the Mylliem porphyritic granite is grey leucocratic, coarse grained, phanerocrystalline with modal variants ranging from granite to granodiorite tending to tonalite. Simple statistical studies based on correlation coefficient values involving modal variables of the Mylliem porphyritic granite bodies appear significant in majority of the cases indicating magmatic crystallization. Use of relevant thermometric method indicates temperature of equilibration of the body in the range of 369°C to 507°C. The crystallization of the Mylliem porphyritic granite pluton initiated at an average lithostatic pressure in the tune of ~11 Kb followed by dominant P H 2 O controlled milieu. Spatially projected mineralogical parameters suggest a general trend of the magmatic cooling of the body from margin inward with sudden influx of volatile matters that occasionally offsets mineralogical trends.

During the COVID-19 pandemic, one of the major concerns was a medical emergency in human society. Therefore it was necessary to control or restrict the disease spreading among populations in any fruitful way at that time. To frame out a proper policy for controlling COVID-19 spreading with limited medical facilities, here we propose an SEQAIHR model having saturated treatment. We check biological feasibility of model solutions and compute the basic reproduction number ( ). Moreover, the model exhibits transcritical, backward bifurcation and forward bifurcation with hysteresis with respect to different parameters under some restrictions. Further to validate the model, we fit it with real COVID-19 infected data of Hong Kong from 19th December, 2021 to 3rd April, 2022 and estimate model parameters. Applying sensitivity analysis, we find out the most sensitive parameters that have an effect on . We estimate using actual initial growth data of COVID-19 and calculate effective reproduction number for same period. Finally, an optimal control problem has been proposed considering effective vaccination and saturated treatment for hospitalized class to decrease density of the infected class and to minimize implemented cost.