International Journal of Biomathematics

The introduction of predators like red fox and dingoes in the 1930s took a big toll on the quokka population. This paper is mainly concerned with designing conservation policies for quokka population (an endangered species). For this purpose, we designed a reaction–diffusion tri-tropic food chain model consisting quokka and its two predators, red fox and dingoes. We have shown the global existence, non-negativity and uniform boundedness for the designed spatiotemporal model. Existence and stability analysis of the equilibrium points for the model is done. We applied the prevalent idea of basic reproduction number with its origin from epidemiology to the food chain model, to deduce a condition for extinction and persistence of predator population. Natural systems exhibit an amazing diversity of structures in both living and non-living systems. We found that in the presence of diffusion, the model has the potential of exhibiting Turing instability generating beautiful patterns. Numerical results reveal that quokka tends to avoid places resided by dingoes. Our research aims at finding a solution for the current quokka extinction problem by showing the effect of presence of alternative food for dingoes and prohibiting the external factor causing death of quokka population.

In this paper, we investigate the dynamics behavior of stochastic hybrid switching staged progression model for the transmission of HIV. The method used in this paper is Lyapunov functions analysis. We establish sufficient conditions for extinction of disease. Especially, we conclude that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, some examples and their simulations are provided to illustrate the theoretical results.

In December 2019, human cases of novel coronavirus infection were detected in Wuhan, China which have been named as COVID-19 by the World Health Organization (WHO). Since COVID-19 was first detected in China, the virus has reached more than 120 countries and was declared a global pandemic on March 11, 2020 by the WHO. In this paper, we have highlighted the influence of temperature on the spread of COVID-19. For this, the dynamic transmission of COVID-19 is modeled taking into account the influence of the temperature on the persistence of coronavirus in the environment. We also took into account the impact of proportion of people who respect the barrier measures published by the WHO on the scale of the COVID-19 pandemic. Taking into account the influence of the temperature on the persistence of the virus in the environment, the dynamic transmission has been described by a system of ordinary differential equations (ODEs). First, we analyzed the solutions of system in the case where the impact of the temperature on the virus is neglected. Second, we carried out the mathematical analysis of the solutions of the system in the non-autonomous case. Mathematical analyzes have enabled us to establish that the temperature and proportion of persons who respect the barrier rules can affect the spread of COVID-19. Some numerical simulations have been proposed to illustrate the behavior of the pandemic in some countries.

Aortic dissection (AD) is the most common catastrophic disease reported at cardiovascular emergency in hospitals. Herein, a tear in the tunica intima results into separation of layers of aortic wall leading to rupture and torrential bleed. Hypoxia and oxidative stress are associated with AD. The release of hypoxia inducible factor (HIF)-$1\alpha$ from the initial flap lesion in the tunica intima is the basis for aneurysmal prone factors. We framed a boundary value problem (BVP) to evaluate homeostatic saturation for oxygen dynamics using steady-state analysis. We prove uniqueness and existence of the solution of the BVP for gas exchange at capillary–tissue interface as a normal physiological function. Failure of homeostatic mechanism establishes hypoxia, a new quasi-steady-state in AD. We model permeation of two-layer fluid comprised of blood and HIF-$1\alpha$ through tunica media as a generalized $(3+1)$-dimensional nonlinear evolution equation and solve it using Lie group of transformations method. We note that the two-layer fluid permeates the tunica media as solitary wave including solitons such as bright soliton, dark soliton, peregrine soliton, topological soliton, kink soliton, breather soliton and multi-soliton complex. Also, we introduce the main result and discuss the implications of soliton solution, using graphic interpretation, to describe the early stage of progression of AD.

In this paper, a class of non-autonomous stochastic Nicholson’s blowflies systems with patch structure and time delays is formulated and studied. By constructing suitable Lyapunov functions and using the stochastic technical, the pth moment boundedness and almost sure growth bounds are discussed, which reveal that solutions of the system do not exceed the time value $t$ and the sample Lyapunov exponent is no more than zero. Then, the system is proved to be exponentially stable (or extinct) if the production rate is less than the mortality rate, which provides an effective reference for the population control. Moreover, taking into account the specific form of time-varying coefficients, related results for several classical stochastic Nicholson’s blowflies systems are studied, and they show the significant improvement of this paper. Finally, numerical simulations for several specific examples are carried out to illustrate our theoretical conclusions.

We formulate an influenza model with treatment and vaccination. Both time invariant and time-dependent uncertainty analyses and sensitivity analysis of the model parameter values are carried out to understand the dependence of the reproduction numbers and model state variables on their components. Results show that the relationship between treatment and epidemic size is nonlinear and that there exists a critical threshold treatment rate under which treatment is beneficial. Sensitivity analysis suggests that the most significant parameters are those related to infection transmission, infectiousness, duration of infectiousness and waning immunity. Further, there are important instances when the relationship between some parameters and model outputs changes behavior from negatively to positively correlated or vice versa because all sensitivity indices, except ${\beta }_S$ are functions of other parameters and thus will change with the change in parameter values. For example, treatment helps to lower the epidemic size, but may then become a “source” of infection likely due to resistance de novo. This knowledge is critical for proper public health planning and guidance of control strategies.

Bio-convection is an important phenomenon which is described by hydrodynamic instability and pattern in suspension of biased swimming microorganisms. This hydrodynamics instability arises due to the coupling force between the motion of the micoorganisms and fluid flow. It becomes more significant when nanoparticles are immersed in the base fluid with non-Newtonian rheology. This study presents the bio-convection for a viscoelastic Casson nanofluid flow over a stretching sheet. The Cattaneo–Christov double diffusion, induced magnetic field, thermal radiation, heat generation, viscous dissipation and chemical reaction are taken into account. The boundary condition is enriched with the suction / injection and melting phenomena at the surface. Highly coupled nonlinear governing equations are simplified into a system of coupled ordinary differential equation by using proper similarity transformation. The spectral quasi-linearization method (SQLM) is used to solve the transformed governing equations numerically. Good agreement is observed with the numerical data investigated in the previous outstanding works. It is observed that the density of the motile microorganisms depends on Peclet number and bio-convective Lewis number. Bio-convection Rayleigh number increases the possibility of bio-convection in the system which results in the enhancement of temperature. It is also examined that temperature and concentration profiles increase with the Eckert number and thermophoresis parameter.

It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

In this paper, a new delayed within host model for dengue fever with immune response and Beddington–DeAngelis incidence is investigated. The basic reproduction number is computed. In addition, a detailed analysis on the local and global dynamics of the model is conducted. Finally, sensitivity analysis is carried out on basic reproduction number and numerical simulations are given to elucidate our theoretical results.

In this paper, we propose a multi-scale waterborne disease model and are concerned with a heterogenous process of waning vaccine-induced immunity. A completely nested rule has been adopted to link the within- and between-host systems. We prove the existence, positivity and asymptotical smoothness of the between-host system. We derive the basic reproduction numbers associated with the two-scale system in explicit forms, which completely determine the behavior of each system. Uncertainty analysis reveals the trade-offs of the kinetics of the within-host system and the transmission of the between-host system. Numerical simulations suggest that the vaccine waning process plays a significant role in the estimation of the prevalence at population level. Furthermore, the environmental heterogeneity complicates the transmission patterns at the population level.

This paper reports novel theoretical and analytical results for a perturbed version of a SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô–Lévy stochastic differential equations in order to simulate sudden and unexpected external phenomena. By using some new and ameliorated mathematical approaches, we study the long-run characteristics of the perturbed delayed model. Within this scope, we give sufficient conditions for two interesting asymptotic properties: extinction and persistence of the epidemic. One of the most interesting results is that the dynamics of the stochastic model are closely related to the intensities of white noises and Lévy jumps, which can give us a good insight into the evolution of the epidemic in some unexpected situations. Our work complements the results of some previous investigations and provides a new approach to predict and analyze the dynamic behavior of epidemics with distributed delay. For illustrative purposes, numerical examples are presented for checking the theoretical study.