Pulse vaccination in the periodic infection rate SIR epidemic model

A pulse vaccination SIR model with periodic infection rate $\beta (t)$ have been proposed and studied. The basic reproductive number $R_0$ is defined. The dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability. It has been shown that the infection-free periodic solution is globally stable provided $R_0 < 1$ and is unstable if $R_0>1$. Standard bifurcation theory have been used to show the existence of the positive periodic solution for the case of $R_0 \to1^+$. Finally, the numerical simulations have been performed to show the uniqueness and the global stability of the positive periodic solution of the system.Comment: 17pages and 3figures, submmission to Mathematical Bioscience

Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations

Effects of impulsive harvesting and an evolving domain in a diffusive logistic model

In order to understand how the combination of domain evolution and impulsive harvesting affect the dynamics of a population, we propose a diffusive logistic population model with impulsive harvesting on a periodically evolving domain. Initially the ecological reproduction index of the impulsive problem is introduced and given by an explicit formula, which depends on the domain evolution rate and the impulsive function. Then the threshold dynamics of the population under monotone or nonmonotone impulsive harvesting is established based on this index. Finally numerical simulations are carried out to illustrate our theoretical results, and reveal that a large domain evolution rate can improve the population survival, no matter which impulsive harvesting takes place. Contrary, impulsive harvesting has a negative effect on the population survival, and can even lead to the extinction of the population.Comment: 26 pages, 8 figure

Study of Mathematical Modeling for Plant Disease Transmission: A Systematic Literature Review during 2012-2022

Many models representing disease transmission have been constructed and analyzed mathematically. However, literature studies on the mathematical models for vector-borne disease are sparse, especially on the plant disease transmission model. This study aims to obtain information about the research conducted and find room for developing the model, including mathematical analysis, intervention used, and biological factors considered. We employ a Systematic Literature Review (SLR) to explore all of the studies on plant disease transmission modeling collected from four digital databases. First, the JabRef reference manager helps conduct the inclusion and exclusion processing. Then, we obtain 60 selected articles that passed the criterion. Next, the VOSviewer application is resulting a bibliometric analysis of the database containing chosen articles. Finally, we classify the model constructed based on the system used and elaborate on the intervention used. The results show that the existing researcher clusters are not linked to each other, and the models only consider usual interventions such as roguing and insecticide spraying. Hence, there is much room to build collaboration between the researcher and develop models for plant disease transmission by considering the other various intervention and biological factors in the model to improve further

Effects of additional food availability and pulse control on the dynamics of a Holling-(p+1) type pest-natural enemy model

In this paper, a novel pest-natural enemy model with additional food source and Holling-($p$+1) type functional response is put forward for plant pest management by considering multiple food sources for predators. The dynamical properties of the model are investigated, including existence and local asymptotic stability of equilibria, as well as the existence of limit cycles. The inhibition of natural enemy on pest dispersal and the impact of additional food sources on system dynamics are elucidated. In view of the fact that the inhibitory effect of the natural enemy on pest dispersal is slow and in general deviated from the expected target, an integrated pest management model is established by regularly releasing natural enemies and spraying insecticide to improve the control effect. The influence of the control period on the global stability and system persistence of the pest extinction periodic solution is discussed. It is shown that there exists a time threshold, and as long as the control period does not exceed that threshold, pests can be completely eliminated. When the control period exceeds that threshold, the system can bifurcate the supercritical coexistence periodic solution from the pest extinction one. To illustrate the main results and verify the effectiveness of the control method, numerical simulations are implemented in MATLAB programs. This study not only enriched the related content of population dynamics, but also provided certain reference for the management of plant pest

Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging

The population dynamics of a three-species ecological system with impulsive effect are investigated. Using the theories of impulsive equations and small-amplitude perturbation scales, the conditions for the system to be permanent when the number of predators released is less than some critical value can be obtained. Furthermore, because the predator in the system follows the predictions of optimal foraging theory, it follows that optimal foraging promotes species coexistence. In particular, the less beneficial prey can support the predator alone when the more beneficial prey goes extinct. Moreover, the influences of the impulsive effect and optimal foraging on inherent oscillations are studied using simulation, which reveals rich dynamic behaviors such as period-halving bifurcations, a chaotic band, a periodic window, and chaotic crises. In addition, the largest Lyapunov exponent and the power spectra of the strange attractor, which can help analyze the chaotic dynamic behavior of the model, are investigated. This information will be useful for studying the dynamic complexity of ecosystems

Finite-Time Stability Analysis and Control for a Class of Stochastic Singular Biological Economic Systems Based on T-S Fuzzy Model

This paper studies the problem of finite-time stability and control for a class of stochastic singular biological economic systems. It shows that such systems exhibit the distinct dynamic behavior when the economic profit is a variable rather than a constant. Firstly, the stochastic singular biological economic systems are established as fuzzy models based on T-S fuzzy control approach. These models are described by stochastic singular T-S fuzzy systems. Then, novel sufficient conditions of finite-time stability are obtained for the stochastic singular biological economic systems, and the state feedback controller is designed so that the population (state of the systems) can be driven to the bounded range by the management of the open resource. Finally, by using Matlab software, numerical examples are given to illustrate the effectiveness of the obtained results

Modelling of a seasonally perturbed competitive three species impulsive system

The population of biological species in the ecosystem is known sensitive to the periodic fluctuations of seasonal change, food resources and climatic conditions. Research in the ecological management discipline conventionally models the behavior of such dynamic systems through specific impulsive response functions, but the results of such research are applicable only when the environments conform exactly to the conditions as defined by the specific response functions that have been implemented for specific scenarios. This means that the application of previous work may be somewhat limited. Moreover, the intra and inter competitions among species have been seldom studied for modelling the prey-predator ecosystem. To fill in the gaps this paper models the delicate balance of two-prey and one-predator system by addressing three main areas of: ⅰ) instead of using the specific impulse response this work models the ecosystem through a more general response function; ⅱ) to include the effects due to the competition between species and ⅲ) the system is subjected to the influences of seasonal factors. The seasonal factor has been implemented here in terms of periodic functions to represent the growth rates of predators. The sufficient condition for the local and global asymptotic stability of the prey-free periodic solution and the permanence of the system have been subsequently obtained by using the Comparison techniques and the Floquet theorems. Finally, the correctness of developed theories is verified by numerical simulation, and the corresponding biological explanation is given.2017005,2017019: Shanxi Agricultural University of Science and Technology Innovation Fund Projects

Mathematical modeling of fall armyworm spodoptera frugiperda infestations in maize crops and its impact on final maize biomass

A Dissertation Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Mathematical and Computer Sciences and Engineering of the Nelson Mandela African Institution of Science and TechnologyFall armyworm (FAW-Spodoptera frugiperda), a highly destructive and fast spreading agricul tural pest native to North and South America, poses a real threat to global food security. It is estimated that intermittent FAW outbreaks could cause up to $US 13 billion per annum in crop losses throughout sub-Saharan Africa. Considering this projected loss it is imperative that various tools and techniques be utilized to infer on the various factors that affect FAW maize in teraction and in-turn affect the final maize biomass. Mathematical modeling has proved to be an important tool that is capable of shedding light on the FAW-maize interaction dynamics. In this study, three mathematical models were proposed to evaluate the impact of memory effects and controls, seasonality and Integrated Pest Management strategy (farming awareness and larvae predation) on FAW infestations in maize crops and on final maize biomass. Firstly, to evaluate the impact of memory effects and control, a new dynamical system for FAW-maize biomass interaction via Caputo fractional-order operator was proposed and analyzed. In the proposed model, four equilibrium points which revealed the existence of a threshold parameter defined by R0 were computed and analyzed. Further, it was observed that, R0, the average number of newborns produced by one individual female moth during its life span was an integral compo nent for stability of the aforementioned model equilibria. Secondly, to evaluate the implications of seasonality on FAW maize interaction and on the final maize biomass, a non-autonomous mathematical model was proposed and analyzed. The analysis revealed that the model solution was non-negative, unique, permanent and bounded admitting global asymptotic and continuous periodic function. Further, the model was extended into an optimal control problem with the aim of determining optimal pesticides and traditional methods that are capable of minimizing FAW egg and larvae populations at minimum cost. Results from the study demonstrated that a combination of pesticides use at low intensity with traditional methods at higher intensity could eradicate FAW in a maize field in a period less than half the life span of the crop in the field. Thirdly, to evaluate the impact of farming awareness campaigns and larvae predation, a fractional-order model that incorporated farming awareness campaigns and larvae predation was proposed and analysed. Overall, the study highlighted that, non-time dependent farming awareness campaigns should be close to 100% all the time to eradicate the FAW. However, when time-dependent farming awareness was implemented, it was observed that even less than 50% intensity level could lead to eradication of FAW. In all the proposed models, comprehen sive numerical simulations were carried out in MATLAB programming language to support the analytical findings. In a nutshell, the results of this study showed that mathematical models can be important tools to evaluate FAW and maize interaction dynamics