Stability analysis of epidemic models of Ebola hemorrhagic fever with non-linear transmission

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), 17–27]; Doungmo Goufo et al. [E. F. Doungmo Goufo, R. Maritz, J. Munganga, Adv. Diff. Equ., 2014 (2014), 9 pages]; Pooseh et al. [S. Pooseh, H. S. Rodrigues, D. F. M. Torres, In: Numerical Analysis and Applied Mathe- matics, ICNAAM, American Institute of Physics, Melville, (2011), 739–742]), 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 bounded- ness 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.