International Journal of Biomathematics

In this paper, a delayed mosquito population suppression model, where the number of sexually active sterile mosquitoes released is regarded as a given nonnegative function, and the birth process is density dependent by considering larvae progression and the intra-specific competition within the larvae, is developed and studied. A threshold value $r^{\ast }$ for the releases of sterile mosquitoes is determined, and it is proved that the origin is globally asymptotically stable if the number of sterile mosquitoes released is above the threshold value $r^{\ast }$. Besides, the case when the number of sterile mosquitoes released stays at a constant level $r$ is also considered. In the special case, it is also proved that the origin is globally asymptotically stable if and only if $r>r^{\ast }$ and that the model exhibits other complicated dynamics such as bi-stability and semi-stability when $r\le r^{\ast }$. Numerical examples are also provided to illustrate our main theoretical results.

To describe the interaction between viral infection and immune response, we establish a mathematical model with intracellular delay and investigate an optimal control problem to examine the effect of antiviral therapy. Dynamic analysis of the proposed model for the stability of equilibria and Hopf bifurcation is conducted. Theoretical results reveal that the dynamical properties are determined by both the immune-inactivated reproduction number and the immune-activated reproduction number. With the aim of minimizing the infected cells and viral load with consideration for the treatment costs, the optimal solution is discussed analytically. Numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences.

Due to human-caused deforestation, global warming, and other environmental factors, habitat fragmentation became widespread. This fragmentation has a variety of detrimental repercussions for many species as well as humans, especially in the agricultural economy. It causes insect outbreaks, the expansion of alien species, and disrupts biological management by rendering the habitat unsuitable for natural enemies in agriculture. Providing natural enemies with additional food is one method to improve the ecosystem and support them. In this study, we assumed that the ecosystem is separated into two patches and that predators can easily migrate from one patch to the next, while prey stays inside its patch’s territory. We looked at the impact of offering more food to predators in a patchy environment using dynamical systems theory. The permanence, stability, and various bifurcations that occur in the system are studied using a rigorous mathematical analysis. The study looks at how predator’s access to other food sources affects pest management. By adjusting the provided food’s characteristic qualities, such as (nutritional) quality and quantity, one can limit and manage the pest in one or both patches, as well as eliminate predators from the ecosystem. This research reveals that providing predators with additional food (of specified quality and quantity) can help in controlling chaotic behavior in the system. The findings of the study are supported by numerical simulations.

Phytoplanktons are drifting plants in an aquatic system. They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis. Zooplanktons are drifting animals found inside the aquatic bodies. For stable aquatic ecosystem, the growth of both Zooplankton and Phytoplankton should be in steady state but in previous eras, there has been a universal explosion in destructive Plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So, in this paper, a discretized two-dimensional Phytoplankton–Zooplankton model is investigated. The results for the existence and uniqueness, and conditions for local stability with topological classifications of the equilibrium solutions are determined. It is also exhibited that at trivial and semitrivial equilibrium solutions, discrete model does not undergo flip bifurcation, but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution under certain conditions. Further, state feedback method is deployed to control the chaos in the under consideration system. The extensive numerical simulations are provided to demonstrate theoretical results.

The energy-intensive industrial economy results in frequent Fog–Haze weather which increases the risk of influenza spread and brings new challenges for the prevention and control of influenza. Therefore, the study of spread mechanics and controlling strategies of influenza based on Fog–Haze will be scientifically meaningful. Considering that when the concentration of Fog–Haze is low, Fog–Haze contributes to the number of infectious individuals and Fog–Haze suppresses the transmission of the influenza virus when the concentration of Fog–Haze is high, we establish the Fog–Haze dynamics model. Then we prove the global existence and boundedness of the solution, and the global asymptotic stability of the solution is given by constructing a suitable Lyapunov functional. Under the Fog–Haze weather, we study the influenza virus transmission model that incorporates the incidence rate to reflect the Fog–Haze-dependent saturation effect and investigate the effect of Fog–Haze pollution on the transmission of influenza. We show that the basic reproduction number $R_0$ determines the global dynamics of the system: if $R_0<1$, the disease-free equilibrium is global asymptotically stable; the unique endemic equilibrium is global asymptotically stable if $R_0>1$. Simulations are carried out to validate the theoretical results. Our study provides further understanding of the dynamics of Fog–Haze and the effect of Fog–Haze pollution on the transmission of influenza.

The differential equations having delays take paramount interest in the research community due to their fundamental role to interpret and analyze the mathematical models arising in biological studies. This study deals with the exploitation of knack of artificial intelligence-based computing paradigm for numerical treatment of the functional delay differential systems that portray the dynamics of the nonlinear influenza-A epidemic model (IA-EM) by implementation of neural network backpropagation with Levenberg–Marquardt scheme (NNBLMS). The nonlinear IA-EM represented four classes of the population dynamics including susceptible, exposed, infectious and recovered individuals. The referenced datasets for NNBLMS are assembled by employing the Adams method for sufficient large number of scenarios of nonlinear IA-EM through the variation in the infection, turnover, disease associated death and recovery rates. The arbitrary selection of training, testing as well as validation samples of dataset are utilizing by designed NNBLMS to calculate the approximate numerical solutions of the nonlinear IA-EM develop a good agreement with the reference results. The proficiency, reliability and accuracy of the designed NNBLMS are further substantiated via exhaustive simulations-based outcomes in terms of mean square error, regression index and error histogram studies.

This paper is devoted to investigating the selection mechanism of the minimal wave speed for traveling waves to an epidemic model. The determinacy of linear and nonlinear selections is further discussed by the upper–lower solutions and comparison principle. A threshold is defined by the eigenvalue problem of the linearized system. We show that the nonlinear determinacy is obtained as long as there exists a lower solution with a faster decay and a speed parameter that is larger than the threshold. When the speed parameter equals to the threshold, if there exists an upper solution satisfying proper limit behavior, then the linear selection is realized. For a special function of infection rate, we obtain a threshold parameter that determines the linear and nonlinear selections.

In this study, the Z-type control method is applied to an intraguild crop-pest-natural enemy model, assuming that the natural enemy can predate on both crop and pest populations. For this purpose, the indirect Z-type controller is considered in the natural enemy population. After providing the design function for the crop-pest-natural enemy model with Z-control, we find the analytical expression of the update parameter. The findings indicate that the uncontrolled system can produce chaos through period-doubling bifurcation due to crop over-consumption by the pest population. We draw a Poincaré map to confirm the occurrence of chaos and compute the maximum Lyapunov exponent. As the observations further indicate that the pest population can be controlled by using an indirect Z-control mechanism in the natural enemy population, we postulate that, if natural enemy abundance can be governed by the update parameter, any desired pest population abundance can be achieved through the proposed Z-type controller, thus controlling the pest. To verify these assertions, extensive numerical simulations are performed to explore the potential for practical application of the proposed Z-type controller.

This paper is concerned with a Lotka–Volterra cooperation–reaction–diffusion–advection model in open advective environments. It is found that there are two critical advection rates, which classify the dynamic behavior of this system into three different scenarios, namely, (i) both species go extinct; (ii) one species survives in the long run, the other goes extinct and (iii) both species can persistently survive. The theoretical results provide some interesting highlights in ecological protection in streams and rivers.

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic’s future course when the public healthcare facilities, specifically the number of hospital beds, are limited. The feasibility and stability of the obtained equilibria are analyzed, and the basic reproduction number ($R_0$) is obtained. We show that the system exhibits transcritical bifurcation. To show the existence of Bogdanov–Takens bifurcation, we have derived the normal form. We have also discussed a generalized Hopf (or Bautin) bifurcation at which the first Lyapunov coefficient evanescences. To show the existence of saddle-node bifurcation, we used Sotomayor’s theorem. Furthermore, we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure. An optimal control setting is studied analytically using optimal control theory, and numerical simulations of the optimal regimen are presented as well.