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Measles is a higher contagious disease that can spread in a community population depending on the number of people (children) susceptible or infected and also depending on their movement in the community. In this paper we present a fractional SEIR metapopulation system modeling the spread of measles. We restrict ourselves to the dynamics between four distinct cities (patches). We prove that the fractional metapopulation model is well posed (nonnegative solutions) and we provide the condition for the stability of the disease-free equilibrium. Numerical simulations show that infection will be proportional to the size of population in each city, but the disease will die out. This is an expected result since it is well known for measles (Bartlett (1957)) that, in communities which generate insufficient new hosts, the disease will die out.

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This paper seeks to unveil the niche of delay differential equation in harmonizing low HIV viral haul and thereby articulating the adopted model, to delve into structured treatment interruptions. Therefore, an ordinary differential equation is schemed to consist of three components such as untainted CD4+ T-cells, tainted CD4+ T-cells (HIV) and CTL. A discrete time delay is ushered to the formulated model in order to account for vital components, such as intracellular delay and HIV latency which were missing in previous works but have been advocated for future research. It was divested that when the reproductive number was less than unity, the disease free equilibrium of the model was asymptotically stable. Hence the adopted model with or without the delay component articulates less production of virions as per the

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate (Brockmann et al. [D. Brockmann, L. Hufnagel, Phys. Review Lett., 98 (2007) , compared to models with the conventional derivative. In this paper, an Ebola epidemic model with non linear transmission is analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional. We proved that the model is well-defined, well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Finally, numerical simulations are provided for particular expressions of the non-linear transmission (with parameters κ = 0.01, κ = 1 and p = 2). The obtained simulations are in concordance with the usual threshold behavior. The results obtained here are significant for the fight and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still infecting many people in West-Africa.

Until now, all the investigations on fractional or generalized Navier-Stokes equations have been done under some restrictions on the different values that can take the fractional order derivative parameterβ. In this paper, we analyze the existence and stability of nonsingular solutions to fractional Navier-Stokes equations of type (ut+u·∇u+∇p-Re-1(-∇)βu=f  in  Ω×(0,T]) defined below. In the case whereβ=2, we show that the stability of the (quadratic) convergence, when exploiting Newton’s method, can only be ensured when the first guessU0is sufficiently near the solutionU. We provide interesting well-posedness and existence results for the fractional model in two other cases, namely, when1/2<β<1andβ≥1/2+(3/4).

The continuous fission equation with derivative of fractional orderα, describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalizedG-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

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