In this research work, we propose an unprecedented hybrid algorithm which involves the coupling of a new integral transform, namely, the Elzaki transform and the well‐known homotopy perturbation method called the Elzaki homotopy transformation perturbation method (EHTPM) to solve for the exact solution of three distinct types of Fisher's equation, namely, the Fisher's equation of two cases, the sixth‐order Fisher's equation and the nonlinear diffusion equation of the Fisher type. These equations are prominent in mathematical biology and highly applicable in genetic propagation, population dynamics, stochastic processes, combustion theory, as well as a prototype model for a spreading flame, and so on. The efficacy and authenticity of this method was established via convergence and error analysis, and shows that the solutions obtained from EHTPM are unique and convergent. The results of EHTPM when compared with the exact results, homotopy results, and results from the other existing literature via table of comparison, 3D plots, and the convergence plots validates that EHTPM is an efficient and reliable tool of providing exact solutions to a wider class of nonlinear partial differential equations in a simple and straightforward manner, with no discretization, linearization, computation of Adomian polynomials, and devoid of errors.

In this research work, we propose an algorithm which involves the coupling of a new integral transform called the Elzaki transform and the well-known homotopy perturbation method on the Burgers-Huxley equation which is a type of nonlinear advection-diffusion partial differential equation. The Burgers-Huxley equation which models reaction mechanisms, diffusion transports and nerve ion propagation which is applicable in traffic flows, acoustics, turbulence theory, hydrodynamics and generally mechanics is the fusion of the well-known Burgers equation and the Huxley equation. We proffer an analytical solution in the form of a Taylor multivariate series of displacement x and time t using the proposed Elzaki homotopy transformation perturbation method (EHTPM) to three cases of the Burgers-Huxley equation. This solution converges rapidly to a closed form which is the same as the exact solutions obtained using the normal analytical methods from the existing literatures. The exact results and that of our proposed EHTPM when compared via tables and 3D plots show an excellent agreement devoid of errors.

Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour.Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.

No abstract

In this research, an unrivalled hybrid scheme which involves the coupling of the new Elzaki integral transform (an improved version of Laplace transform) and a modified differential transform called the projected differential transform (PDTM) have been implemented to solve the generalized Burgers-Fisher's equation; which springs up due to the fusion of the Burgers' and the Fisher's equation; describing convective effects, diffusion transport or interaction between reaction mechanisms, traffic flows; and turbulence; consequently finding meaningful applicability in the applied sciences viz: gas dynamics, fluid dynamics, turbulence theory, reaction-diffusion theory, shock-wave formation, traffic flows, financial mathematics, and so on. Using the proposed Elzaki projected differential transform method (EPDTM), a generalized exact solution (Solitary solution) in form of a Taylor multivariate series has been obtained; of which the highly nonlinear terms and derivatives handled by PDTM have been decomposed without expansion, computation of Adomian or He's polynomials, discretization, restriction of parameters, and with less computational work whilst achieving a highly convergent results when compared to other existing analytical/exact methods in the literature, via comparison tables, 3D plots, convergence plots and fluid-like plots. Thus showing the distinction, novelty and huge advantage of the proposed method as an asymptotic alternative, in providing generalized or solitary wave solution to a wider class of differential equations.

No abstract

We have proposed an unprecedented deterministic model of Lassa Haemorrhagic fever (LHF) model with nonlinear force of LHF infection to capture the transmission dynamics and long term effects of the disease. The Qualitative analyses we have conveyed on this model using well established methods viz: Cauchy’s differential theorem, Birkhoff & Rota’s theorem verifies and reveals the well-posedness, and carrying capacity of the model respectively. We established that a LHF-free equilibrium termed the disease-free equilibrium (DFE) exists for this model and this equilibrium however from our stability analyses, tends to be stable when the basic reproduction number computed via next generation matrix method is less than unity (one); and unstable if otherwise. Furthermore, we have carried out a sensitivity analyses to check variation effects of the model parameters when increased or decreased using the normalized forward-sensitivity index; unraveling the most sensitive parameters which requires the attention of the healthcare workers as; the effective contact rates W, and the rodents’ recruitment rate e. After which numerical simulations of the model were carried out to verify our qualitative analyses (Stability and sensitivity analysis) and to study the dynamical behavior of the model; showing that the presence of saturation instantaneously causes the system to approach a DFE/LHF-Free equilibrium.From these qualitative analyses and numerical simulation results, we recommend early intervention and early treatment of Lassa haemorrhagic virus infection (LAHV) with Ribavirin on the infected, maximum hygiene practices and periodic evacuation of rodents in households in order to curb the recruitment of wild/rodents.

No abstract