title: Stability, bifurcation and chaos analysis of vector-borne disease model with application to Rift Valley fever.
abstract: This paper investigates a RVF epidemic model by qualitative analysis and numerical simulations. Qualitative analysis have been used to explore the stability dynamics of the equilibrium points while visualization techniques such as bifurcation diagrams, Poincaré maps, maxima return maps and largest Lyapunov exponents are numerically computed to confirm further complexity of these dynamics induced by the seasonal forcing on the mosquitoes oviposition rates. The obtained results show that ordinary differential equation models with external forcing can have rich dynamic behaviour, ranging from bifurcation to strange attractors which may explain the observed fluctuations found in RVF empiric outbreak data, as well as the non deterministic nature of RVF inter-epidemic activities. Furthermore, the coexistence of the endemic equilibrium is subjected to existence of certain number of infected Aedes mosquitoes, suggesting that Aedes have potential to initiate RVF epidemics through transovarial transmission and to sustain low levels of the disease during post epidemic periods. Therefore we argue that locations that may serve as RVF virus reservoirs should be eliminated or kept under control to prevent multi-periodic outbreaks and consequent chains of infections. The epidemiological significance of this study is: (1) low levels of birth rate (in both Aedes and Culex) can trigger unpredictable outbreaks; (2) Aedes mosquitoes are more likely capable of inducing unpredictable behaviour compared to the Culex; (3) higher oviposition rates on mosquitoes do not in general imply manifestation of irregular behaviour on the dynamics of the disease. Finally, our model with external seasonal forcing on vector oviposition rates is able to mimic the linear increase in livestock seroprevalence during inter-epidemic period showing a constant exposure and presence of active transmission foci. This suggests that RVF outbreaks partly build upon RVF inter-epidemic activities. Therefore, active RVF surveillance in livestock is recommended.
authors.affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa; Modelling, International Center of Insect Physiology and Ecology, Nairobi, Kenya; Departmento de Matemática e Informática, Universidade Eduardo Mondlane, Maputo, Mozambique.
authors.forename: Sansao A
authors.initials: SA
authors.lastname: Pedro
authors.affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa.
authors.forename: Shirley
authors.initials: S
authors.lastname: Abelman
authors.affiliation: Modelling, International Center of Insect Physiology and Ecology, Nairobi, Kenya; Departement de Physique, Universite de Yaoundé I, Yaoundé, Cameroun.
authors.forename: Frank T
authors.initials: FT
authors.lastname: Ndjomatchoua
authors.affiliation: Human Health, International Center of Insect Physiology and Ecology, Nairobi, Kenya.
authors.forename: Rosemary
authors.initials: R
authors.lastname: Sang
authors.affiliation: Modelling, International Center of Insect Physiology and Ecology, Nairobi, Kenya.
authors.forename: Henri E Z
authors.initials: HE
authors.lastname: Tonnang
mesh_terms: D000465:Algorithms; D006801:Humans; D008962:Models, Theoretical; D012295:Rift Valley Fever / Q000453:epidemiology*; D012296:Rift Valley fever virus