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# Applied mathematical modelling with new parameters and applications to some real life problems

 dc.contributor.advisor Doungmo Goufo, Emile Franc dc.contributor.author Mugisha, Stella dc.date.accessioned 2018-10-26T10:19:49Z dc.date.available 2018-10-26T10:19:49Z dc.date.issued 2018-09 dc.identifier.citation Mugisha, Stella (2018) Applied mathematical modelling with new parameters and applications to some real life problems, University of South Africa, Pretoria, dc.identifier.uri http://hdl.handle.net/10500/24973 dc.description.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, en 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. dc.format.extent 1 online resource (112 leaves) : illustrations (some color), graphs (some color) en dc.language.iso en en dc.subject Conventional derivative with a new parameter en dc.subject Ebola epidemic model en dc.subject Non-linear incidence en dc.subject Existence en dc.subject Stability en dc.subject Depolymerization en dc.subject Replicated Fractional poles en dc.subject Simple and complex harmonic motion en dc.subject Shattering en dc.subject Generalized evolution models en dc.subject Exponential with a parameter en dc.subject Sumudu transform en dc.subject Mittag-Leffler functions en dc.subject.ddc 614.4015118 dc.subject.lcsh Epidemiology -- Mathematical models en dc.subject.lcsh Ebola virus disease -- Epidemiology en dc.title Applied mathematical modelling with new parameters and applications to some real life problems en dc.type Thesis en dc.description.department Mathematical Sciences en dc.description.degree Ph. D. (Applied Mathematics) en
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