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In this article, we have developed a simple mathematical model that captures the vital mechanisms of the hypothalamic-pituitary-adrenal (HPA) axis self-regulatory activities. For this, a system of three-component non-linear delay differential equations has been proposed and analysed to observe the ultradian and circadian variabilities of the hormone secretion of the HPA axis in normal subjects. Our analysis reveals that a feedback mechanism is sufficient to show the ultradian variability of the hormone secretion pattern but fails to show the circadian variability. A central nervous system-driven pulse generator coupled with the primary feedback mechanism can exhibit the ultradian as well as circadian variability in the hormone secretion of the HPA axis. The model can also predict different dynamics of the normal HPA axis following physiological changes (viz. adrenalectomy and hypophysectomy) and pathological changes (viz. infusion of different hormones).

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In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples. KEYWORDS ecological model, epidemic model, fractional order, global stability, local stability 6776

The coronavirus disease 2019 , which emerged from Wuhan, China, is now a pandemic, affecting across the globe. Government of different countries have developed and adopted various policies to contain this epidemic and the most common were the social distancing and lockdown. We proposed a SEIR epidemic model that accommodates the effects of lockdown and individual based precautionary measures and used it to estimate model parameters from the epidemic data up to 2nd April, 2020, freely available in GitHub repository for COVID-19, for nine developed and developing countries. We used the estimated parameters to predict the disease burden in these countries with special emphasis on India, Bangladesh and Pakistan. Our analysis revealed that the lockdown and recommended individual hygiene can slow down the outbreak but unable to eradicate the disease from the society. With the current human-to-human transmission rate and existing level of personal precautionary, the number of infected individuals in India will be increasing at least for the next 3 months and the peak will come in 5 months. We can, however, reduce the epidemic size and prolong the time to arrive epidemic peak by seriously following the measures suggested by the authorities. We need to wait for another one month to obtain more data and epidemiological parameters for giving a better prediction about the pandemic. It is to be mentioned that research community is working for drugs and/ or vaccines against COVID19 and the presence of such pharmaceutical interventions will significantly alter the results.

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