Comparison between two tritrophic food chain models with multiple delays and anti-predation effect

Exploring the predator–prey linkage in food chain system is the most familiar research work in population biology. Recently, some research experiments show that predator–prey interaction not only governed by direct hunting but also influenced by some indirect effect such as fear effect (felt by prey) that may change the physiological behavior of prey. Based upon this fact, we consider a tritrophic food chain model incorporating with anti-predation response (fear effect) and multiple time delays for biomass conversion from prey to middle predator and middle to top predator. We analyze the resulting delay differential equations and explore how the anti-predation response level affects the population dynamics. We also investigate the effect of delay parameters, for which the model system switches its stability through Hopf-bifurcation. We compare all of our results between two different food chain models consisting of two different functional responses. Some numerical simulations are performed to validate the effectiveness of the derived theoretical results.

Modelling the role of optimal social distancing on disease prevalence of COVID-19 epidemic

COVID-19 first spread from Wuhan, China in December 2019 but it has already created one of the greatest pandemic situations ever witnessed. According to the current reports, a situation has arisen when people need to understand the importance of social distancing and take enough precautionary measures more seriously. Maintaining social distancing and proper hygiene, staying at isolation or adopting the self-quarantine strategy are some common habits which people should adopt to avoid from being infected. And the growing information regarding COVID-19, its symptoms and prevention strategies help the people to take proper precautions. In this present study, we have considered a SAIRS epidemiological model on COVID-19 transmission where people in the susceptible environment move into asymptotically exposed class after coming contact with asymptotically exposed, symptomatically infected and even hospitalised people. The numerical study indicates that if more people from asymptotically exposed class move into quarantine class to prevent further virus transmission, then the infected population decreases significantly. The disease outbreak can be controlled only if a large proportion of individuals become immune, either by natural immunity or by a proper vaccine. But for COVID-19, we have to wait until a proper vaccine is developed and hence natural immunity and taking proper precautionary measures is very important to avoid from being infected. In the latter part, a corresponding optimal control problem has been set up by implementing control strategies to reduce the cost and count of overall infected individuals. Numerical figures show that the control strategy, which denotes the social distancing to reduce disease transmission, works with a higher intensity almost after one month of implementation and then decreases in the last few days. Further, the control strategy denoting the awareness of susceptible population regarding precautionary measures first increases up to one month after implementation and then slowly decreases with time. Therefore, implementing control policies may help to reduce the disease transmission at this current pandemic situation as these controls reduce the overall infected population and increase the recovered population.

Epidemic model of COVID-19 outbreak by inducing behavioural response in population

COVID-19 has spread around the world since December 2019, creating one of the greatest pandemics ever witnessed. According to the current reports, this is a situation when people need to be more careful and take the precaution measures more seriously, unless the condition may become even worse. Maintaining social distances and proper hygiene, staying at isolation or adopting the self-quarantine method are some of the common practices that people should use to avoid the infection. And the growing information regarding COVID-19 and its symptoms help the people to take proper precautions. In this present study, we consider an SEIRS epidemiological model on COVID-19 transmission which accounts for the effect of an individual's behavioural response due to the information regarding proper precautions. Our results indicate that if people respond to the growing information regarding awareness at a higher rate and start to take the protective measures, then the infected population decreases significantly. The disease fatality can be controlled only if a large proportion of individuals become immune, either by natural immunity or by a proper vaccine. In order to apply the latter option, we need to wait until a safe and proper vaccine is developed and it is a time-taking process. Hence, in the latter part of the work, an optimal control problem is considered by implementing control strategies to reduce the disease burden. Numerical figures show that the control denoting behavioural response works with higher intensity immediately after implementation and then gradually decreases with time. Further, the control policy denoting hospitalisation of infected individuals works with its maximum intensity for quite a long time period following a sudden decrease. As, the implementation of the control strategies reduce the infected population and increase the recovered population, so, it may help to reduce the disease transmission at this current epidemic situation.

Interplay between reproduction and age selective harvesting: A case study of Hilsa (Tenualosa ilisha) fish at Sundarban estuary of northern Bay of Bengal, India

New offspring due to sexual reproduction is not an instantaneous process from its birth, it takes time to be sexually mature. On the other hand, harvesting of commercially profitable fish population before the perfect size or weight is reached is not only a commercial loss but also risks the extinction of the population. Now, we discuss the issue of Ganges-Brahmaputra-Meghna basin, northern Bay of Bengal for the age-selective harvesting of Hilsa shad (Tenualosa ilisha) which lays eggs after its sexual maturation. Harvesting of hilsa before its sexual maturation risks its extinction and due to lamer body weight, it is not a commercially profitable policy. This is a reality of Sundarban estuary for hilsa fish harvesting, therefore, biologically and economically both India and Bangladesh are facing several problems. Empirical data of Frasergunje Fishing Harbor shows a clear picture as the supporting document of this mathematical problem.

Analysis of a predator–prey model with herd behavior and disease in prey incorporating prey refuge

In this work, we have introduced an eco-epidemiological model of an infected predator–prey system. Incorporation of prey refuge gives that a fraction of the infected prey is available to the predator for consumption. Moreover, to make the model more realistic to the environment, we have introduced strong Allee effect in the susceptible population. Boundedness and positivity of the solution have been established. Local stability conditions of the equilibrium points have been found with the help of Routh–Hurwitz criterion and it has been observed that if a prey population is infected with a lethal disease, then both the prey (susceptible and infected) and predator cannot survive simultaneously in the system for any parametric values. The disease transmission rate and the attack rate on the susceptible have an important role to control the system dynamics. For different values of these two key parameters, we have got only healthy or disease-free or predation-free or a fluctuating disease-free or even a fluctuating predator-free system with some certain parametric conditions.

Parameter uncertainty in biomathematical model described by one-prey two-predator system with mutualism

In this paper, a three-species system consisting of two predators which are in mutualism with each other and preying on the same single prey is considered. Also, the prey and first predator are harvested under optimal conditions. The values of the biological parameters depend on the collection of data from the experts as well as on the nature of the environment in which prey–predator system are considered. So the biological parameters are not precise in reality. This paper presents a different approach to study the prey–predator model with imprecise biological parameters. All the possible equilibrium points are identified and the local as well as global stability criteria under impreciseness are discussed. The possibility of existence of bionomic equilibrium is discussed. The optimal harvesting policy is studied using Pontryagin’s maximum principle. Numerical examples are provided to support the proposed approach.

Analysis of a hand–foot–mouth disease model

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease (HFMD) infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number ([Formula: see text]) of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls (two treatment controls with quarantine control). We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows the reliability of our model from the practical point of view.

Stability and Bionomic Analysis of Fuzzy Prey-Predator Harvesting Model in Presence of Toxicity: A Dynamic Approach

This paper deals with a prey-predator model in which both the species are infected by some toxicants which are released by some other species or source with fuzzy biological parameters. The application of fuzzy differential equation in the modeling of prey-predator populations with the effect of toxicants is presented. The dynamical behavior and harvesting of the fuzzy exploited system are studied by using the utility function method. Sufficient conditions for the local stability of the positive equilibrium are obtained by analyzing the characteristic equation. Furthermore, the possibility of the existence of bionomic equilibrium is studied under imprecise biological parameters. The study of the presence of toxic substance and harvesting in the modeling system can have significant impact on the existence of both the species, which is in line with reality. Numerical simulation results are presented to validate the theoretical analysis.

Stability analysis and optimal control of an epidemic model with vaccination

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R0) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0when R0< 1. When R0> 1 endemic equilibrium E1exists and the system becomes locally asymptotically stable at E1under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey-predator harvesting model in different form, here we consider a simple prey-predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin's maximum principle. Numerical examples are presented to support the proposed model.

Stability analysis of an eco-epidemiological model incorporating a prey refuge

The present paper deals with the problem of a predator-prey model incorporating a prey refuge with disease in the prey-population. We assume the predator population will prefer only infected population for their diet as those are more vulnerable. Dynamical behaviours such as boundedness, permanence, local and global stabilities are addressed. We have also studied the effect of discrete time delay on the model. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.