Salisu M. Garba

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.

Effect of cross-immunity on the transmission dynamics of two strains of dengue

A deterministic model for the transmission dynamics of two strains of dengue disease is presented. The model, consisting of mutually exclusive epidemiological compartments representing the human and vector dynamics, has a locally asymptotically stable, disease-free equilibrium whenever the maximum of the associated reproduction numbers of the two strains is less than unity. The model can have infinitely many co-existence equilibria if infection with one strain confers complete cross-immunity against the other strain and the associated reproduction number of each strain exceeds unity. On the other hand, if infection with one strain confers partial immunity against the other strain, disease elimination, competitive exclusion or co-existence of the two strains can occur. The effect of seasonality on dengue transmission dynamics is explored using numerical simulations, where it is shown that the oscillation pattern differs between the strains, depending on the degree of the cross-immunity between the strains.

Global Stability Analysis of SEIR Model with Holling Type II Incidence Function

A deterministic model for the transmission dynamics of a communicable disease is developed and rigorously analysed. The model, consisting of five mutually exclusive compartments representing the human dynamics, has a globally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number (ℛ 0), is less than unity; in such a case the endemic equilibrium does not exist. On the other hand, when the reproduction number is greater than unity, it is shown, using nonlinear Lyapunov function of Goh-Volterra type, in conjunction with the LaSalles invariance principle, that the unique endemic equilibrium of the model is globally asymptotically stable under certain conditions. Furthermore, the disease is shown to be uniformly persistent whenever ℛ 0 > 1.

Dynamics of Mycobacterium and bovine tuberculosis in a Human-Buffalo Population

A new model for the transmission dynamics of Mycobacterium tuberculosis and bovine tuberculosis in a community, consisting of humans and African buffalos, is presented. The buffalo-only component of the model exhibits the phenomenon of backward bifurcation, which arises due to the reinfection of exposed and recovered buffalos, when the associated reproduction number is less than unity. This model has a unique endemic equilibrium, which is globally asymptotically stable for a special case, when the reproduction number exceeds unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics of the two diseases in the Kruger National Park, show that the distribution of the associated reproduction number is less than unity (hence, the diseases would not persist in the community). Crucial parameters that influence the dynamics of the two diseases are also identified. Both the buffalo-only and the buffalo-human model exhibit the same qualitative dynamics with respect to the local and global asymptotic stability of their respective disease-free equilibrium, as well as with respect to the backward bifurcation phenomenon. Numerical simulations of the buffalo-human model show that the cumulative number of Mycobacterium tuberculosis cases in humans (buffalos) decreases with increasing number of bovine tuberculosis infections in humans (buffalo).

Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation

One-dimensional models are important for developing, demonstrating and testing new methods and approaches, which can be extended to more complex systems. We design for a linear delay differential equation a reliable numerical method, which consists of two time splits as follows: (a) It is an exact scheme at the early time evolution - ? ≤ t ≤ ? , where ? is the discrete value of the delay; (b) Thereafter, it is a nonstandard finite difference (NSFD) scheme obtained by suitable discretizations at the backtrack points. It is shown theoretically and computationally that the NSFD scheme is dynamically consistent with respect to the asymptotic stability of the trivial equilibrium solution of the continuous model. Extension of the NSFD to nonlinear epidemiological models and its good performance are tested on a numerical example.

Backward bifurcation analysis of epidemiological model with partial immunity

This paper presents a two stage SIS epidemiological model in animal population with bovine tuberculosis (BTB) in African buffalo as a guiding example. The proposed model is rigorously analyzed. The analysis reveals that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) coexists with a stable endemic equilibrium (EE) when the associated reproduction number ( R v ) is less than unity. It is shown under two special cases of the presented model, that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine. Numerical simulations of the model show that, the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough.

Mathematical analysis of West Nile virus model with discrete delays

Abstract The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number ( ℝ 0 ) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ℝ 0 > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δ b and Δ v are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δ b > 0 and Δ v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).

Dynamical behavior of an epidemiological model with a demographic Allee effect

As the Allee effect refers to small density or population size, it cannot be deduced whether or not the Allee mechanisms responsible for an Allee effect at low population density or size will affect the dynamics of a population at high density or size as well. We show using susceptible–exposed–infectious (SEI) model that such mechanisms combined with disease pathogenicity have a detrimental impact on the dynamics of a population at high population level. In fact, the eventual outcome could be an inevitable population crash to extinction. The tipping point marking the unanticipated population collapse at high population level is mathematically associated with a saddle–node bifurcation. The essential mechanism of this scenario is the simultaneous population size depression and the increase of the extinction threshold owing to disease virulence and the Allee effect. Using numerical continuation software MatCont another saddle–node bifurcation is detected, which results in the re-emergence of two non-trivial equilibria since highly pathogenic species cause their own extinction but not that of their host.

Assessing the Effectiveness of Ebola Control Measures in Kikwit, DRC 1995

We develop mathematical models to replicate the Ebola infection dynamics in Kikwit Districtof Congo DRC 1995. The models are used to assess various control measures undertaken during the outbreak and determine the type of control measures that were most effective in controlling the outbreak. Two models were designed for two distinct periods of the Ebola outbreak, that is the period before Ebola was declared an epidemic in Kikwit (January 1995 toApril 1995) and the period after Ebola was declared an outbreak in Kikwit (May 1995 to August1995. We also carry out cost benefit analysis for each intervention method. The aim of this study is to assess and suggest the best possible measures for effectively combating Ebola infection.

Two-stage Epidemic Model of Bovine Tuberculosis in African Buffalo

We present a two stage SIS epidemic model in animal population with bovine tuberculosis in African buffalo as a guiding example. The proposed model is rigorously analyzed. The analysis reveals that the model may exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium. Two special cases when this phenomenon of backward bifurcation does not arise are highlighted. Further, it is shown via threshold analysis approach that a vaccine could have positive or negative impact. Numerical simulations of the model demonstrate that, the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough.