Abstract:
Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included,
which happens to be fractional (that derivative is called the derivative). We proved that the model is well-de ned and 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 the
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. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold
behavior. The results obtained here may be signi cant for the ght and prevention
against Ebola haemorrhagic fever that has so far exterminated hundreds of families and
is still a ecting many people in West-Africa and other parts of the world.
The full comprehension and handling of the phenomenon of shattering, sometime happening
during the process of polymer chain degradation [129, 142], remains unsolved
when using the traditional evolution equations describing the degradation. This traditional
model has been proved to be very hard to handle as it involves evolution of
two intertwined quantities. Moreover, the explicit form of its solution is, in general,
impossible to obtain. We explore the possibility of generalizing evolution equation modeling
the polymer chain degradation and analyze the model with the conventional time
derivative with a new parameter. We consider the general case where the breakup rate
depends on the size of the chain breaking up. In the process, the alternative version of
Sumudu integral transform is used to provide an explicit form of the general solution
representing the evolution of polymer sizes distribution. In particular, we show that
this evolution exhibits existence of complex periodic properties due to the presence of
cosine and sine functions governing the solutions. Numerical simulations are performed
for some particular cases and prove that the system describing the polymer chain degradation
contains complex and simple harmonic poles whose e ects are given by these
functions or a combination of them. This result may be crucial in the ongoing research
to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and
their variants naturally govern solutions to most of generalized fractional evolution models
such as kinetic, di usion or relaxation equations. The question is to say whether or
not this is always true! Whence, three generalized evolution equations with an additional
fractional parameter are solved analytically with conventional techniques. These
are processes related to stationary state system, relaxation and di usion. In the analysis,
we exploit the Sumudu transform to show that investigation on the stationary
state system leads to results of invariability. However, unlike other models, the generalized
di usion and relaxation models are proven not to be governed by Mittag-Le er
functions or any of their variants, but rather by a parameterized exponential function,
new in the literature, more accurate and easier to handle. Graphical representations
are performed and also show how that parameter, called ; can be used to control the
stationarity of such generalized models.
