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In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundaryvalue problem D α u(t) = f (t, u(t)) with a RiemannLiouville fractional derivative via the different boundary-value problems u(0) = u(T), and the threepoint boundary condition u(0) = β 1 u(η) and u(T) = β 2 u(η), where T > 0, t ∈ I = [0, T], 0 < α < 1, 0 < η < T, 0 < β 1 < β 2 < 1.

Highlights COVID-19 is transmitted from asymptomatic individuals to susceptible individuals. COVID-19 is transmitted from symptomatic individuals to susceptible individuals. Since R0=1.6 is greater than 1, the COVID-19 will spread exponentially. If COVID-19 is not controlled, it is estimated that about 20 million people will become infected in the next three years.

By using the fractional Caputo-Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

Background. Protease inhibitor (PI)–based combination antiretroviral therapy (cART) is administered during pregnancy to prevent perinatal human immunodeficiency virus (HIV) transmission. However, PI use has been associated with adverse birth outcomes, including preterm delivery and small-for-gestational-age (SGA) births. The mechanisms underlying these outcomes are unknown. We hypothesized that PIs contribute to these adverse events by altering progesterone levels.Methods. PI effects on trophoblast progesterone production were assessed in vitro. A mouse pregnancy model was used to assess the impact of PI-based cART on pregnancy outcomes and progesterone levels in vivo. Progesterone levels were assessed in plasma specimens from 27 HIV-infected and 17 HIV-uninfected pregnant women.Results. PIs (ritonavir, lopinavir, and atazanavir) but not nucleoside reverse transcriptase inhibitors (NRTIs) or nonnucleoside reverse transcriptase inhibitors reduced trophoblast progesterone production in vitro. In pregnant mice, PI-based cART but not dual-NRTI therapy was associated with significantly lower progesterone levels that directly correlated with fetal weight. Progesterone supplementation resulted in a significant improvement in fetal weight. We observed lower progesterone levels and smaller infants in HIV-infected women receiving PI-based cART, compared with the control group. In HIV-infected women, progesterone levels correlated significantly with birth weight percentile.Conclusions. Our data suggest that PI use in pregnancy may lead to lower progesterone levels that could contribute to adverse birth outcomes.

We present a fractional-order model for the COVID-19 transmission with Caputo-Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the problem and give approximate solution in convergent series. We prove the existence of a unique solution and the stability of the iteration approach by using fixed point theory. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative.

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We provide a SEIR epidemic model for the spread of COVID-19 using the Caputo fractional derivative. The feasibility region of the system and equilibrium points are calculated and the stability of the equilibrium points is investigated. We prove the existence of a unique solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. To predict the transmission of COVID-19 in Iran and in the world, we provide a numerical simulation based on real data.