In bio-social models, cooperative behavior has evolved as an adaptive strategy, playing multi-functional roles. One of such roles in populations is to increase the success of the survival and reproduction of individuals and their families or social groups. Moreover, collective decision-making in cooperative behavior is an aspect that is used to study the dynamic behavior of individuals within a social group. In this paper, we have focused on population dynamics by considering a predator–prey model as our main exemplification, where the generalist predator has adopted a cooperative hunting strategy while consuming their prey. In particular, we have analyzed the dynamic nature of the system when a nonlocal term is introduced in cooperation. First, the Turing instability condition has been studied for the local model around the coexisting steady-state, followed by the Turing and non-Turing patterns in the presence of the nonlocal interaction term. This work is also concerned with the existence of travelling wave solutions for predator–prey interaction with the nonlocal cooperative hunting strategy. Such solutions are reported for local as well as for nonlocal models. We have characterized the invading speed of the predator with the help of the minimal wave speed of travelling wave solutions connecting the predator-free state to the co-existence state. The travelling waves are found to be non-monotonic in this system. The formation of wave trains has been demonstrated for an extended range of nonlocal interactions. Finally, the importance of psychological effects in shaping the dynamics of nonlocal collective behavior is demonstrated with several representative examples.

There are many positive and negative factors present in the predator–prey interaction which affect the net growth of the species. Fear of predation is one such factor that creates psychological stress in a prey species, which causes a negative impact on their overall growth. This work considers a predator–prey model where the prey species faces a reduction in their growth out of fear, and the predator has an alternative food source that helps the prey to hide in a safer place. As an extension into the nonlocal spatio-temporal model, a nonlocal term is considered in the prey growth to incorporate a fear-effect range around their spatial location. Linear stability analysis helps to analyse the temporal model and produces a wide range of interesting results, including the presence of a certain amount of fear or even prey refuge, which helps in population coexistence. Furthermore, the numerical simulations of the local and nonlocal spatio-temporal models show different types of spatial–temporal patterns, such as Turing and non-Turing patterns. Nevertheless, an increase in fear level reduces the range of the Turing domain for the local model, whereas the opposite happens when the range of nonlocal interaction is increased.

Understanding how emotions and psychological states influence both individual and collective actions is critical for expressing the real complexity of biosocial and ecological systems. Recent breakthroughs in mathematical modeling have created new opportunities for systematically integrating these emotion-specific elements into dynamic frameworks ranging from human health to animal ecology and socio-technical systems. This review builds on mathematical modeling approaches by bringing together insights from neuroscience, psychology, epidemiology, ecology, and artificial intelligence to investigate how psychological effects such as fear, stress, and perception, as well as memory, motivation, and adaptation, can be integrated into modeling efforts. This article begins by examining the influence of psychological factors on brain networks, mental illness, and chronic physical diseases (CPDs), followed by a comparative discussion of model structures in human and animal psychology. It then turns to ecological systems, focusing on predator–prey interactions, and investigates how behavioral responses such as prey refuge, inducible defense, cooperative hunting, group behavior, etc., modulate population dynamics. Further sections investigate psychological impacts in epidemiological models, in which risk perception and fear-driven behavior greatly affect disease spread. This review article also covers newly developing uses in artificial intelligence, economics, and decision-making, where psychological realism improves model accuracy. Through combining these several strands, this paper argues for a more subtle, emotionally conscious way to replicate intricate adaptive systems. In fact, this study emphasizes the need to include emotion and cognition in quantitative models to improve their descriptive and predictive ability in many biosocial and environmental contexts.

The proposed mathematical model explores the intricate dynamics of a predator-prey system involving prey infection and cooperative hunting of predators. The model incorporates habitat complexity, emphasizing its influence on ecological interactions. The well-posedness of the system has rigorously been examined in a temporal setting and also conducted stability analysis. The bifurcation analysis reveals the existence of several local bifurcations on the system, namely transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation. Furthermore, these investigations delineate the two-dimensional bifurcations including Bogdanov-Takens and cusp bifurcations for different parametric combinations. With suitable choices of parameter values, the proposed model exhibits diverse dynamic phenomena, including bistable and tri-stable behavior. Latin hypercube sampling is utilized to conduct uncertainty analysis on input parameters, aiming to observe their effects on population dynamics. Subsequently, Kendall’s tau and Spearman’s rank correlation coefficients are also computed to investigate the impact of these uncertainties on the population. In the later part, a spatio-temporal system is proposed with two-dimensional diffusion terms to obtain the conditions for Turing instability. Numerical simulations have been conducted to observe the emergence of spatial patterns and the impact of predator cooperation in these patterns. The study provides valuable insights into the dynamics of complex ecological systems, emphasizing the interplay of spatial and temporal factors in shaping population dynamics and predator-prey interactions.

A compartmental SIRIS epidemiological system containing two separate susceptible compartments (depending on immunity power) has been assessed in the present work involving governmental action, public reaction and social behavioral dynamics. In addition, the impact of environmental perturbations as well as time-dependent control techniques have been investigated. Present study analyzes that a few previously infectious individuals become susceptible to infection again after they have recovered, some infected persons build immunity after infection and some previously diseased populations become contaminated again after having recovered. More particularly, this study demonstrates the significance of social and governmental interventions on disease dynamics, along with the relevance of nonlinear dynamical modeling of epidemiological systems. As indicated by numerical simulation, the activities of government, social behavior act an essential role in preventing a pandemic scenario and if government takes action at incipient phases during an outbreak, the system becomes infection-free much sooner. Sensitivity analysis is used to assess how changes in different parameters of a model affect the spread of a disease. In this case, Latin hypercube sampling is used to perform both uncertainty and sensitivity analyses on input parameters. This sampling method helps to observe how these parameters impact the reproduction number of the disease. After that, Kendall’s tau and Spearman’s rank correlation coefficients are calculated to delve deeper into how these uncertainties affect the dynamics of the disease. Moreover, it is remarkable that random variations might inhibit the propagation of ailment, that can contribute in emergence of beneficial control strategies to govern the dynamics of disease. In this model, the policies implemented by government and pharmaceutical therapy are regarded as most adequate control pair, and it is determined that simultaneous execution of control mechanisms considerably diminishes the ailment burden.

This work considers a stage-structured predator–prey harvesting system with anti-predator behavior. It discusses the impact of the selective harvesting of a species above a specific age or size by incorporating time delays in the harvesting terms. Analytical and numerical simulation of the proposed system suggest that the coexistence steady state is stable at low and moderate values of the delay parameter associated with the maturation of the predator. However, there is an interval of maturation delay where the system exhibits oscillatory behavior. A very long maturation time can cause the extinction of the predator species. The effect of anti-predator behavior on the proposed system is analyzed. Moreover, the time delays associated with harvesting of prey and adult predator, while crossing some threshold values, drive the system toward instability. There also have some interesting results based on the harvesting rate.

In an environment, the food chains are balanced by the prey-predator interactions. When a predator species is provided with more than one prey population, it avails the option of prey switching between prey species according to their availability. So, prey switching of predators mainly helps to increase the overall growth rate of a predator species. In this work, we have proposed a two prey-one predator system where the predator population adopts switching behavior between two prey species at the time of consumption. Both the prey population exhibit a strong Allee effect and the predator population is considered to be a generalist one. The proposed system is biologically well-defined as the system variables are positive and do not increase abruptly with time. The local stability analysis reveals that all the predator-free equilibria are saddle points whereas the prey-free equilibrium is always stable. The intrinsic growth rates of prey, the strong Allee parameters, and the prey refuge parameters are chosen to be the controlling parameters here. The numerical simulation reveals that in absence of one prey, the other prey refuge parameter can change the system dynamics by forming a stable or unstable limit cycle. Moreover, a situation of bi-stability, tri-stability, or even multi-stability of equilibrium points occurs in this system. As in presence of the switching effect, the predator chooses prey according to their abundance, so, increasing refuge in one prey population decreases the count of the second prey population. It is also observed that the count of predator population reaches a comparatively higher value even if they get one prey population at its fullest quantity and only a portion of other prey species. So, in the scarcity of one prey species, switching to the other prey is beneficial for the growth of the predator population.

This study presents a comprehensive model of predator-prey interactions within a toxic environment, with a particular focus on the effect of toxicant compounds on the development of populations. By incorporating environmental disturbances, the dynamics of the model are investigated to enhance the system's authenticity. Analytical explanations have been provided for the deterministic system solutions, including positivity, uniform boundedness and persistence. The deterministic portion of the investigation entails a comprehensive examination of occurrence and stability criteria pertaining to every possible equlibria. The bifurcation studies conducted on the system exhibit the appearance of local bifurcations, including transcritical, saddle-node and Hopf bifurcations. Moreover, these evaluations establish the parametric region in which Bautin, Bogdanov-Takens and cusp bifurcation occur. Under a relevant selection of parametric values, the suggested system has the capacity to manifest a wide range of dynamic phenomena, such as bi-stable behavior, emergence of limit cycles, and presence of homoclinic loops. Furthermore, in a stochastic environment, the use of Lyapunov functions explains the existence of a global positive solution. It has additionally been argued that the proposed system exhibits ultimate stochastic boundedness. Subsequently, specific and adequate criteria demonstrate the eradication of both species as well as the long-Term survival of prey communities. We have also investigated the impact of the exogenous input rate of toxic substances and the coefficient of toxic substances in both species on the behavior of the whole system, both in deterministic and stochastic scenarios. Theoretical findings have been confirmed by various numerical investigations.

In this paper, a stage-structured predator–prey model is considered in a toxic environment, emphasizing on the affect of the toxic substances on growth of the species in presence of time delays. The positivity and boundedness of solutions have been discussed. The analytical and numerical simulations of the proposed model show that the coexistence equilibrium point is stable for low and moderate values of the delay parameter associated with the predator's maturity. Even though there is an interval of the maturation time in which the system exhibits oscillatory behaviour, a longer maturity duration may lead to the extinction of the predator species. The impact of toxicity on the proposed system is also investigated. The stability of the system repeatedly switches as the time delay, related to the changes of toxicity incurred in the adult predator, varies, and finally becomes unstable for a significant value of it. Besides that, the delay related to the prey's toxicity has also led some exciting results.

In the context of foraging behaviour, a species can be classified as a generalist or a specialist based on the breadth of their diet. Specialist species have a restricted diet and occupy a much narrower niche, whereas generalist species consume a wide range of resources and thrive in a variety of habitats. In this article, we propose an ecological model with two types of prey, with different fertility rates and nutritional levels, devoured by their respective specialist and the generalist predators. Further, it is assumed that the hunting process of generalist predator follows the switching mechanism. The growth of generalist predator is also influenced by external food sources and intra-specific competition. Our analyses reveal that the only species that may suffer extinction possibility are the specialist predators. The specialists relying on higher reproducing prey may face the danger of extinction, but this is not the case for those specialist predators that consume nourishing prey. Coexistence of all species is achievable if (i) specialists are sufficiently efficient in comparison to the number of available prey and (ii) the expansion of generalist predator is reduced due to shortage of external food sources. For lower hunting efficiency of both the specialist predators, the coexistence of all specialists with the generalist is expected to be unachievable in nature. In this case, only the specialist who consumes more reproductive and nutritious prey may cohabit with the generalist. Our findings may provide possibilities for empirical research on individual specialization.

COVID-19 has turned into one of the greatest pandemics ever witnessed within a very short period. The governments of almost every country have announced to maintain physical distancing and to use precautionary measures to prevent the high disease transmission. Here, a compartmental epidemiological model of COVID-19 transmissions has been formulated. People of the susceptible class move to the asymptotically exposed class by coming close to asymptotically exposed people, symptomatically infected people, quarantined people, and hospitalized people. The analysis reveals that when most people from the symptomatically infected class move to quarantine, then even the higher probability of virus transmission hardly makes any impact on the growth of the infected population and the count of the infected people starts to reduce. In the case of coronavirus, there are no existing vaccines which means maintaining physical distancing and hygiene are the only way-outs to avoid the infection. In the optimal control problem, social distancing is considered as one of the important control interventions to mitigate the disease prevalence. Moving the symptomatically infected population to quarantine or hospitals is taken as the other two control strategies. The trajectory profiles of the asymptomatically exposed class show that a lesser number of people become infected in the presence of the control interventions. In conclusion, it can be stated that the simultaneous use of all control interventions reduces the virus transmission in the present pandemic situation and also decrease the count of the infected population in the environment. Epidemic in complex networks (Adapted with permission from the Health and Art (HEART), Universal Scientific Education and Research Network (USERN); Painting by Ummugulsun Topcu).The code of this chapter is 01001101 01101111 01100100 01100101 01101100. Epidemic in complex networks (Adapted with permission from the Health and Art (HEART), Universal Scientific Education and Research Network (USERN); Painting by Ummugulsun Topcu).

Degradation of habitat is a direct outcome of anthropogenic activities, which includes urbanization, mining, the emission of industrial waste, and many others. Many living organisms experience severe surviving challenges as a result of habitat degradation. Here, we have studied the impact of habitat destruction caused by human activities on the dynamics of a prey-predator interaction with prey refuge. Our analyses reveal that a higher rate of habitat destruction than the habitat regeneration rate is always detrimental to the survival of predators. Predator species may still be threatened with extinction even if the rate of habitat degradation is slightly lower than the habitat recovery rate. So, in order to maintain biodiversity, we must appropriately step up our efforts to slow down the rate of habitat degradation as well as accelerate the habitat restoration. Further, our investigation suggests that in order to achieve cohabitation, we should effectively control the habitat deterioration caused by human activity, rather than artificially introducing or eliminating the hiding places of prey species.

The authors of Integrated Scienceof Global EpidemicsEpidemics, global were asked how they would see the futureFuture of their field 30 years later. This Chapter presents the authors’ views on how futureFuture epidemics are in 2050, along with thoughts about how to prepare for future epidemics. Thunderhead

An eco-epidemiological predator-prey model with Holling type-II functional response is proposed in this work. In the presence of disease, the prey population has been divided into two subpopulations: susceptible and infected prey. The predator can access the full healthy prey population for hunting but a predator is provided with a fraction of the infected prey as infected prey refuge term is incorporated here. Also, a strong Allee effect in susceptible population is introduced to make the model more realistic. Boundedness and positivity of the system strengthen that the proposed model is well-posed. The strong Allee threshold and the infected refuge parameter have been taken as the key parameters to control the system dynamics. The numerical simulation gives that regulating the refuge parameter can turn an oscillating state into a stable coexistence state. Also, the system changes its dynamics from two interior equilibrium points to no interior point when this refuge parameter crosses the saddle-node bifurcation threshold. Besides, the strong Allee threshold can also change the dynamics of a system from oscillating state to steady-state through Hopf bifurcation

The present model is dealt with prey-predator interactions in two different patches where only prey species are allowed to disperse among the patches. Each of these two patches has different predator population but the predator in Patch-2 only is affected with a disease. The proposed model is biologically welldefined. Also, the feasibility of the equilibrium points and corresponding stability conditions are analysed. It is found that the disease among predator, even in one patch, makes an important role to control the whole system dynamics as it starts to oscillates by regulating the disease transmission rate. Moreover, the disease transmission rate has a stabilizing as well as destabilizing effect on the system dynamics. From the results, it is observed that a high dispersal rate decreases the count of infected predator in a patch in presence of prey dispersal. There is another interesting result: it is observed that the prey dispersal cannot destabilize the coexistence state, i.e., the system which is stable in absence of dispersal remains stable when the prey species disperse between two patches.

The first COVID-19 case was reported at Wuhan in China at the end of December 2019 but till today the virus has caused millions of deaths worldwide. Governments of each country, observing the severity, took non-pharmaceutical interventions from the very beginning to break the chain of higher transmission. Fortunately, vaccines are available now in most countries and people are asked to take recommended vaccines as precautionary measures. In this work, an epidemiological model on COVID-19 is proposed where people from the susceptible and asymptomatically infected phase move to the vaccinated class after a full two-dose vaccination. The overall analysis says that the disease transmission rate from symptomatically infected people is most sensitive on the disease prevalence. Moreover, better disease control can be achieved by vaccination of the susceptible class. In the later part of the work, a corresponding optimal control problem is considered where maintaining social distancing and vaccination procedure change with time. The result says that even in absence of social distancing, only the vaccination to people can significantly reduce the overall infected population. From the analysis, it is observed that maintaining physical distancing and taking vaccines at an early stage decreases the infection level significantly in the environment by reducing the probability of becoming infected.

In the present article, global characteristics of a generalized SIRS (susceptible–infected–recovered–susceptible) epidemic model have been investigated incorporating government policy, public response and social behavioral reaction. The effects of environmental fluctuations and time-dependent control strategies on the disease dynamics have also been analyzed. In the case of deterministic model, it is shown that the disease invades in this system when the basic reproduction number (R0) is greater than 1, whereas the dynamics of the stochastic model can be controlled by its associated basic reproduction number R̃s. Specifically, this work emphasizes the importance of nonlinear dynamic analysis of epidemic modeling, as well as the significant impact of social and government actions on disease dynamics. Numerical figure depicts that the governmental action plays a crucial role to control an epidemic situation, and the system turns out to be disease-free sooner if the government takes action at an early stage during a disease outbreak. Furthermore, one of the most key developments is that random fluctuations can prevent disease outbreaks, which can lead to the development of useful control techniques to restrict disease dynamics. The governmental actions and the clinical treatment are considered to be the effective control pair in this model, and it can be observed that the simultaneous implementation of the control strategies significantly reduces the disease burden.

Dengue is a mosquito-borne viral disease where the species Aedes aegypti acts as the vector. In this work, we have proposed a biologically well-posed epidemic model to study the dynamics of dengue strain. It is observed that people infected with severe dengue are more infectious than people with mild symptoms in terms of disease transmission. Also, continuous removal of mosquitoes decreases the chances of becoming infected with Dengue significantly. In the later part, an optimal control problem is formulated where a lesser number of people become infected due to time-dependent people's awareness regarding taking precaution and mosquito removal measures. The numerical simulation shows that the transmission of the virus among mosquitoes becomes lowest when both the control policies are implemented. So, from the overall analysis, it is observed that maintaining effective non-pharmaceutical precautions at an early stage and removing mosquitoes from the system at certain time intervals decrease the infection level significantly in the environment by reducing the probability of becoming infected.

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.

Prey switching strategy is adopted by a predator when they are provided with more than one prey and predator prefers to consume one prey over others. Though switching may occur due to various reasons such as scarcity of preferable prey or risk in hunting the abundant prey. In this work, we have proposed a prey-predator system with a particular type of switching functional response where a predator feeds on two types of prey but it switches from one prey to another when a particular prey population becomes lower. The ratio of consumption becomes significantly higher in the presence of prey switching for an increasing ratio of prey population which satisfies Murdoch's condition [15]. The analysis reveals that two prey species can coexist as a stable state in absence of predator but a single prey-predator situation cannot be a steady state. Moreover, all the population can coexist only under certain restrictions. We get bistability for a certain range of predation rate for first prey population. Moreover, varying the mortality rate of the predator, an oscillating system can be obtained through Hopf bifurcation. Also, the predation rate for the first prey can turn a steady-state into an oscillating system. Except for Hopf bifurcation, some other local bifurcations also have been studied here. The figures in the numerical simulation have depicted that, if there is a lesser number of one prey present in a system, then with time, switching to the other prey, in fact, increases the predator population significantly.

In this work, we have introduced a tritrophic food-chain model where consumer hunt for prey with Holling type-III functional response. The birth rate of the prey population has been reduced due to the fear of predation, i.e., a fear effect is considered in the prey population. Moreover, a fraction of the prey is available to the consumer for consumption and this has been done by incorporation of prey refuge term. The predation between consumer and predator follows Beddington-DeAngelis response. Boundedness and positivity of the system prove that the proposed model is well-posed. Also, there are some parametric restrictions under which the system is permanent. Routh-Hurwitz criterion shows the local stability conditions of the equilibrium points and on the other hand Lyapunov LaSalle theorem guarantees that the locally stable equilibrium points are globally stable. Also, Matlab validates the analytical results with the help of diagrams. The occurrence of transcritical bifurcations have been shown and conditions for the existence of a limit cycle in the system through Hopf bifurcation also have been stated. Both the analytical and numerical results suggest that a certain amount of fear can make the system steady. It is also noted that the prey refuge has both stabilizing and destabilizing effect on the system.

In this work a predator-prey model is proposed where prey species shows anti-predator behaviour to save themselves from predator’s attack. Moreover, a strong Allee effect has been introduced in the prey population to make the model more realistic to the environment. Both generalist and specialist predators have been taken to observe the system dynamics minutely. The predator population decreases with increasing value of ‘rate coefficient of anti-predator behaviour’ when generalist predator is considered but in presence of the specialist predator, the predator population first increases up to a threshold value and then decreases. It means a very small rate of anti-predation does not affect the predator’s growth very much but if prey attacks the predator at a larger rate, then the predator population decreases. Also, the existence of one dimensional and two-dimensional bifurcations have been observed by making different parameters as bifurcating parameters around the steady states.

Fear of predation plays an important role in the growth of a prey species in a prey-predator system. In this work, a two-species model is formulated where the prey species move in a herd to protect themselves and so it acts as a defense strategy. The birth rate of the prey here is affected due to fear of being attacked by predators and so, is considered as a decreasing function. Moreover, there is another fear term in the death rate of the prey population to emphasize the fact that the prey may die out of fear of predator too. But, in this model, the function characterizing the fear effect in the death of prey is assumed in such a way that it is increased only up to a certain level. The results show that the system performs oscillating behavior when the fear coefficient implemented in the birth of prey is considered in a small amount but it changes its dynamics through Hopf bifurcation and becomes stable for a higher value of the coefficient. Regulating the fear terms ultimately makes an impact on the growth of the predator population as the predator is taken as a specialist predator here. The increasing value of the fear terms (either implemented in birth or death of prey) decrease the count of the predator population with time. Also, the fear implemented in the birth rate of prey makes a higher impact on the growth of the predator population than in the case of the fear-induced death rate.

We have considered a compartmental epidemiological model with infectious disease to observe the influence of environmental stress on disease transmission. The proposed model is well-defined as the population at each compartment remains positive and bounded with time. Dynamical behaviour of the model is observed by the stability and bifurcation analysis at the equilibrium points. Also, numerical simulation supports the theoretical proofs and the result shows that the system undergoes a forward bifurcation around the disease-free equilibrium. Our results indicate that with the increase of environmental pollution, the overall infected population increases. Also, the disease transmission rate among the susceptible and stressed population from asymptomatically infected individuals plays a crucial role to make a system endemic. A corresponding optimal control problem has also been proposed to control the disease prevalence as well as to minimize the cost by choosing the vaccination policy before being infected and treatment policy to the infected as control variables. Numerical figures indicate that the vaccination provided to susceptible needs some time to reduce the disease transmission but the vaccination provided to stressed individuals works immediately after implementation. The treatment policy for symptomatically infected individuals works with a higher rate at an earlier stage but the intensity decreases with time. Simultaneous implementation of all control interventions is more useful to reduce the size of overall infective individuals and also to minimize the economic burden. Hence, this literature clearly expresses the impact of environmental pollution (specifically the influence of environmental stress) on the disease transmission in the population.

A predator-prey model is proposed in this work where the prey population is infected by a disease. Here, healthy prey species show defence mechanism while they are attacked by the predator. Moreover as the infected prey are already physically weak, so, predator apply cooperative hunting strategy while consume infected prey to get more food. It helps the predator population to grow with a higher rate. But calculation gives that if they start to hunt the infected prey with a larger cooperative hunting rate, then ultimately predator population decrease with time. Boundedness and positivity of the system variables show that the proposed model system is well-posed. Routh-Hurwitz criterion provides the local stability conditions of the equilibrium points. Also, the system becomes permanent under certain parametric restrictions. The numerical results, verified using MATLAB, support the analytical findings. Numerical simulations give that the parameter denoting cooperative hunting rate can change the system dynamics and we can get oscillating behaviour by regulating this parameter. Moreover transcritical and saddle-node bifurcations occur by regulating the death rate of predator around the critical points. Occurrence of Bogdanov-Takens, generalized Hopf and Cusp bifurcations have also been observed here.

In this work, a model is proposed to analyze the effect of wild plant species on biologically-based technologies for pest control. It is assumed that the pest species have a second food source (wild host plants) except crops. Analytical results prove that the model is well-posed as the system variables are non-negative and uniformly bounded. The permanence of the system has been verified. Equilibrium points and corresponding stability analysis have also been performed. Numerical figures have supported the fact that the interior steady state if it exists, remains stable for any transmission rate. Henceforth biological control has a stabilizing effect. Furthermore, the results prove that biological control is beneficial not only for wild plants but for crops too.

In this work, we have introduced an ecological model of a prey–predator system. It is assumed that the prey species grows logistically, but the total number of predator is constant in the time interval. Positivity and boundedness of the solution ensure that the proposed model is well-posed. Local stability conditions of the equilibrium points have been analysed by the Routh–Hurwitz criterion. The persistence of the system has also been shown under a parametric restriction. Numerical analysis has indicated that both axial and interior steady states can exist only for moderate consumption rate (searching efficiency). But if this rate becomes high (or low), then only the prey-free equilibrium (or one of the interior equilibriums) exists as a steady state. Further, the equilibrium points can change their stability through transcritical and saddle-node bifurcations by varying the consumption rate of the predator. Analytical results provide an interesting phenomenon about this model: the system can never show any oscillating behaviour for any parametric values, i.e. no limit cycle can occur through Hopf bifurcation around an equilibrium point. The axial equilibrium becomes stable from an unstable situation when the consumption rate becomes high and the interior state which is stable remains stable as time goes by.

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.

The synthetic drugs that are becoming increasingly popular in drugmarkets are some of the most destructive drugs which have made headlines causing serious social and health care issues in the past few years. In this work, a synthetic drug transmission model with general contact rate and Holling Type-II functional responses among susceptible and drug addicts (both psychological and physiological) is proposed. Sensitivity analysis provides that controlling the contact rate among the drug addicts and the susceptible is better than the treatment. Further, an optimal control problem has been formulated to minimize the cost and drug addiction by choosing the counselling treatment (including awareness programmes) as a control variable. Numerical analysis has indicated that if the control policy is implemented, then it will be economically viable for long term. In the proposed model system, it is observed how counselling can prevent the psychologically addicted persons from taking more drugs.

This work aims to study a population-dispersal dynamics for predator–prey interactions in a two patch environment with strong Allee effect among prey species in both patches. It is assumed that the prey species are movable and their dispersal between patches is directed from lower fitness to the higher fitness patch (exhibiting balanced dispersal). Existence and stability criterion of the interior equilibrium point of the system is analyzed in presence as well as in absence of dispersal speed. It has been observed that Allee threshold takes an important role to destabilize the system while the prey individuals evolve in their own patches independently. Moreover dispersal cannot destabilize populations at the interior equilibrium, i.e., if a predator–prey equilibrium without dispersal is in stable state then this situation cannot be destabilized when prey species move between two patches. Numerical simulations using MATLAB validate the analytical results. The occurrence of transcritical as well as Hopf bifurcation has also been reported.

We have considered a toxin-dependent dynamical model to study the effects of an environmental toxin on the spread of infectious diseases in the population. The well-posedness of the model is discussed by showing the positivity and boundedness of the state variables. Model analysis is performed as well as the stability and bifurcation of equilibrium points are also established. Numerical simulation is performed to verify the theoretical results and it shows that the system possesses a transcritical bifurcation around the disease-free equilibrium. Our analytical results indicate that there exists a threshold value of the environmental toxin, i.e., if the environmental toxin amount is lower than the threshold, the system has a stable disease-free equilibrium and if the environmental toxin amount is higher than the threshold value, then the system has a unique endemic equilibrium. Numerical simulation also shows that the environmental toxin plays a crucial role in spreading of infectious diseases. An optimal control problem has been formulated in the later part to minimize the cost and disease fatality by choosing the treatment and the depuration of the toxin as control variables. Numerical analysis indicates that if only control via depuration is used, then it will be more economical for the time period whereas treatment works well but with a lesser intensity. Moreover, simultaneous use of both the control interventions is more useful for the system dynamics and it reduces the number of infective individuals and also minimizes the economic cost generated from disease burden. This study gives a clear view of the impact of toxins on the spread of infectious diseases in the population and it helps the disease control agencies and governments to take suitable precautions to control the disease.

Epidemic outbreak is one of the primary issue that induce behavioural changes in healthy individuals to avoid contracting infection. We here propose a compartmental model for the transmission of HIV/AIDS including treatment and Pre-exposure prophylaxis (PrEP). It accounts for the effect of individual's behavioural response due to information of PrEP also. Taking PrEP by uninfected individuals actually can prevent the acquisition of HIV infection. Model analysis has been performed as well as the local and global stability of equilibrium points is established. Further, an optimal control problem has been formulated to minimize the cost and disease fatality by choosing the treatment and the effect of information regarding PrEP as control variables. Numerical analysis gives that if only control via information regarding PrEP is used, then it will be economical for early phase of the time period whereas treatment works well for long term control. Moreover, simultaneous use of both the control interventions is more useful than any single applied control policy and it reduces the number of infective individuals and also minimizes the economic cost generated from disease burden and controls. In the model it can be observed how uptake of PrEP can effect in the population by reducing the number of HIV infections. Moreover the combined effect of both the control policies is found to be more economical during the entire epidemic period whereas the implementation of a single policy is comparatively found less economical viable.

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.

Here, we have proposed a predator-prey model with Michaelis-Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh-Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson-Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.