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Matsuda and Abrams (Theor Popul Biol 45(1):76–91, 1994) initiated the exploration of self-extinction in species through evolution, focusing on the advantageous position of mutants near the extinction boundary in a prey-predator system with evolving foraging traits. Previous models lacked theoretical investigation into the long-term effects of harvesting. In our model, we introduce constant-effort prey and predator harvesting, along with individual logistic growth of predators. The model reveals two distinct evolutionary outcomes: (i) Evolutionary suicide, marked by a saddle-node bifurcation, where prey extinction results from the invasion of a lower forager mutant; and (ii) Evolutionary reversal, characterized by a subcritical Hopf bifurcation, leading to cyclic prey evolution. Employing an innovative approach based on Gröbner basis computation, we identify various bifurcation manifolds, including fold, transcritical, cusp, Hopf, and Bogdanov-Takens bifurcations. These contrasting scenarios emerge from variations in harvesting parameters while keeping other factors constant, rendering the model an intriguing subject of study. [ABSTRACT FROM AUTHOR]

The impacts of pandemic situation on the health and economy of any nation are indeed significant. Several paradigms are present, where the nations's economy has stumbled due to impact of infectious diseases. Hence, we formulate an econo-epidemics model to study economic impact of infectious disease. The structure is amalgamated with Solow growth functions to describe effect of epidemiological factors on economy. Numerous numerical experiments, viz. bifurcation analysis, production of capital growth profile, etc., is conducted to provide valuable insights into the potential outcomes and dynamics of the economy. Additionally, we consider the impact of demographic stochasticity on the economy of third-tier countries, Bangladesh and Tanzania, as these nations face unique challenges and vulnerabilities. Enumeration of probability of economic losses in upcoming years and stationary distributions on the capital growth data of those countries provide an outline of strengthening the economy after a period of depreciation caused by an infectious disease outbreak. Overall, the statistical experiments and analysis of capital data provide valuable information about the potential economic consequences of infectious diseases. This knowledge can guide the formulation of policies and schemes to mitigate economic disasters in the future and promote a resilient economy that is better prepared to handle pandemics. [ABSTRACT FROM AUTHOR]

The developing world has been facing a significant health issue due to cholera as an endemic communicable disease. Lusaka was Zambia's worst affected province, with 5414 reported cases of cholera during the outbreak from late October 2017 to May 12, 2018. To explore the epidemiological characteristics associated with the outbreak, we fitted weekly reported cholera cases with a compartmental disease model that incorporates two transmission routes, namely environment-to-human and human-to-human. Estimates of the basic reproduction number show that both transmission modes contributed almost equally during the first wave. In contrast, the environment-to-human transmission appears to be mostly dominating factor for the second wave. Our study finds that a massive abundance of environmental vibrio's with a huge reduction in water sanitation efficacy triggered the secondary wave. To estimate the expected time to extinction (ETE) of cholera, we formulate the stochastic version of our model and find that cholera can last up to 6.5–7 years in Lusaka if any further outbreak occurs at a later time. Results indicate that a considerable amount of attention is to be paid to sanitation and vaccination programs in order to reduce the severity of the disease and to eradicate cholera from the community in Lusaka. [ABSTRACT FROM AUTHOR]

Modeling and analysis of biological growth curves are an age-old study area in which much effort has been dedicated to developing new growth equations. Recent efforts focus on identifying the correct model from a large number of equations. The relative growth rate (RGR), developed by Fisher (1921), has largely been used in the statistical inference of biological growth curve models. It is convenient to express growth equations using RGR, where RGR can be expressed as functions of size or time. Even though RGR is model invariant, it has limitations when it comes to identifying actual growth patterns. By proposing interval-specific rate parameters (ISRPs), Pal et al. (2018) appeared to solve this problem. The ISRP is based on the mathematical structure of the growth equations. Therefore, it is not model invariant. The current effort is to develop a measure of growth that is model invariant like RGR and shares the advantages of ISRP. We propose a new measure of growth, which we call instantaneous maturity rate (IMR). IMR is model invariant, which allows it to distinguish growth patterns more clearly than RGR. IMR is also scale-invariant and can take several forms including increasing, decreasing, constant, sigmoidal, bell-shaped, and bathtub. A wide range of possible IMR shapes makes it possible to identify different growth curves. The estimation procedure of IMR under a stochastic setup has been developed. Statistical properties of empirical IMR estimators have also been investigated in detail. In addition to extensive simulation studies, real data sets have been analyzed to prove the utility of IMR. [ABSTRACT FROM AUTHOR]

In contrast to long standing view on predator–prey interactions that predators have only direct effect on prey by killing, recent field experimentation on terrestrial vertebrates showed that indirect effect of predators' fear may alter the behavioral changes on prey, including foraging and reproduction. Usually, prey perceive the signals from predators (chemical and/or vocal cues) and change their life-history and behavior to reduce the probability of being killed. Recently, Wang et al. (J Math Biol 73:1179–1204, 2016) proposed and analyzed a predator–prey model by considering the fear effect on prey population. They concluded that the model dynamics may exhibit both supercritical and subcritical Hopf bifurcation, while the classical predator–prey model exhibits only supercritical Hopf bifurcation. The cost of fear on prey may dramatically reduce foraging and reproduction, which may change the ecosystem stability. In the present investigation, we explore the possible applications of fear in prey due to predators' signals and error based Z-control mechanism by manipulating the abundance of predator population. Our results suggest that by manipulating or controlling the abundance of predator one can achieve a desired prey population density. We also observe that Z-control mechanism has the property to produce a stable steady-state or a stable limit cycle by excluding the bi-stability situation as observed by Wang et al. We perform extensive numerical simulations to illustrate our analytical findings. [ABSTRACT FROM AUTHOR]

Abrupt transitions leading to algal blooms are quite well known in aquatic ecosystems and have important implications for the environment. These ecosystem shifts have been largely attributed to nutrient dynamics and food web interactions. Contamination with heavy metals such as copper can modulate such ecological interactions which in turn may impact ecosystem functioning. Motivated by this, we explored the effect of copper enrichment on such regime shifts in planktonic systems. We integrated copper contamination to a minimal phytoplankton–zooplankton model which is known to demonstrate abrupt transitions between ecosystem states. Our results suggest that both the toxic and deficient concentration of copper in water bodies can lead to regime shift to an algal-dominated alternative stable state. Further, interaction with fish density can also lead to collapse of population cycles thus leading to algal domination in the intermediate copper ranges. Environmental stochasticity may result in state transition much prior to the tipping point and there is a significant loss in the bimodality on increasing intensity and redness of noise. Finally, the impending state shifts due to contamination cannot be predicted by the generic early warning indicators unless the transition is close enough. Overall the study provides fresh impetus to explore regime shifts in ecosystems under the influence of anthropogenic changes like chemical contamination. [ABSTRACT FROM AUTHOR]

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey–predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov's regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics. [ABSTRACT FROM AUTHOR]

Middle East Respiratory Syndrome Coronavirus (MERS-CoV) can cause mild to severe acute respiratory illness with a high mortality rate. As of January 2020, more than 2500 cases of MERS-CoV resulting in around 860 deaths were reported globally. In the absence of neither effective treatment nor a ready-to-use vaccine, control measures can be derived from mathematical models of disease epidemiology. In this manuscript, we propose and analyze a compartmental model of zoonotic MERS-CoV transmission with two co-circulating strains. The human population is considered with eight compartments while the zoonotic camel population consist of two compartments. The expression of basic reproduction numbers are obtained for both single strain and two strain version of the proposed model. We show that the disease-free equilibrium of the system with single stain is globally asymptotically stable under some parametric conditions. We also demonstrate that both models undergo backward bifurcation phenomenon, which in turn indicates that only keeping R 0 below unity may not ensure eradication. To the best of the authors knowledge, backward bifurcation was not shown in a MERS-CoV transmission model previously. Further, we perform normalized sensitivity analysis of important model parameters with respect to basic reproduction number of the proposed model. Furthermore, we perform optimal control analysis on different combination interventions with four components namely preventive measures such as use of masks, isolation of strain-1 infected people, strain-2 infected people and infected camels. Optimal control analysis suggests that combination of preventive measures and isolation of infected camels will eventually eradicate the disease from the community. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a predator–prey system under the combined effects of fear, migration and switching phenomena. This study is new in the sense that the dynamics of such system was studied either through fear or migration, or switching, but the combined effects of such three factors are yet to be explored. We observe oscillatory behavior of the system in the absence of fear, migration and switching, whereas the system shows stable dynamics if anyone of these three factors is introduced. After analyzing the behavior of system with fear, migration and switching, we find that the system does not possess periodic solution whenever the predator population experiences intraspecies competition, but in the absence of intraspecies competition among predator population, fear of predators destabilizes the system, whereas on increasing the migration rates, the system first undergoes subcritical Hopf-bifurcation and then supercritical Hopf-bifurcation settling the system to stable coexistence. Existence of multiple limit cycles is also observed. Our results show that switching behavior of predator population supports the survival of prey and predator populations. We extend our model by assuming the fear parameters as time dependent. We find that the nonautonomous system exhibits periodic solutions, whereas the corresponding autonomous system shows stable focus. Moreover, we observe that if the autonomous system undergoes a Hopf-bifurcation through limit cycle oscillations, the corresponding nonautonomous system shows higher periodic solutions. Almost periodic behavior of the system is also observed by setting the fear parameters as almost periodic functions of time. [ABSTRACT FROM AUTHOR]

Growth curve models serve as the mathematical framework for the quantitative studies of growth in many areas of applied science. The evolution of novel growth curves can be categorized in two notable directions, namely generalization and unification. In case of generalization, a modeler starts with a simple mathematical form to describe the behavior of the data and increases the complexity of the equation by incorporating more parameters to obtain a more flexible shape. The unification refers to the process of obtaining a compact representation of a large number of growth equations. An enormous number of growth equations are made available in the literature by means of the generalization of existing growth laws. However, the unification of growth equations has received relatively less attention from the researchers. Two significant unification functions are available in the literature, namely the Box–Cox transformation by Garcia (For Biometry Model Inf Sci 1:63–68, 2005) and generalized logarithmic and exponential functions by Martinez et al. (Phys A 387:5679–5687, 2008; Phys A 388:2922–2930, 2009). Existing unification approaches are found to have limited applications if the growth equation is characterized by the relative growth rate (RGR). RGR has immense practical value in biological growth curve analysis, which has been amplified by the construction of size and time covariate models, in which; RGR is represented either as a function of size or time or both. The present study offers a unification function for the RGR growth curves. The proposed function combines a broad class of the growth curves and possesses a greater generality than the existing unification functions. We also propose the notion of generalized RGR, which is capable of making interrelations among the unifying functions. Our proposed method is expected to enhance the generality of software and may aid in choosing an optimal model from a set of competitor growth equations. [ABSTRACT FROM AUTHOR]

The recent Zika virus outbreak has been spreading rapidly all over the world, expanding its traditionally geographical affected regions, making it a global public health hazard and endangering millions of people. One unique property of the Zika virus compared to most vector-borne diseases is the fact that the virus is transmitted both by mosquitoes and by direct sexual contact. In the present manuscript, we formulate and analyze five mathematical compartmental models of Zika transmission. We model both transmission routes (i.e., vector-borne and sexual transmission). In order to make the model more realistic, heterogeneity in the sexual transmission is modeled in several ways. We fitted the five different models to data, inferred the parameters and selected the most appropriated model, which describes the Zika outbreak in Columbia. For all the models, we estimate the reproduction numbers, namely direct (sexual) transmission, vector transmission and the basic reproduction number (R0). The analysis revealed that the sexual transmission contribution to R0 is highest [15.36% (95% CI 12.83-17.4)] for the model which stratifies each gender to high-risk and low-risk individuals in their sexual behavior. For this model, the estimated R0 is 1.89 (95% CI 1.21-2.13), the direct transmission reproduction number is 0.42 (95% CI 0.29-0.64), and the vector transmission reproduction number is 1.51 (95% CI 1.23-1.87). The sensitivity analysis demonstrated that the value of R0 depends on three controllable parameters: the biting rate, the sexual transmission rate and the average ratio of mosquito to human. [ABSTRACT FROM AUTHOR]

In this present article, we propose and analyze a cannibalistic predator-prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator-prey oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model on transmission of Japanese Encephalitis disease has been developed considering some control parameters and time-dependent environmental-carrying capacity. Here, the total vector population is divided into two subpopulations such as susceptible mosquito and infected mosquito. Here also, total reservoir population (i.e., the population in which the encephalitis virus grows) such as pig, horse, etc., has been considered which is divided into three subpopulations such as susceptible reservoir, infected reservoir and recovered reservoir. Total human population is also divided into three subpopulations such as susceptible human, infected human and recovered human. The dynamical behaviors of the system have been investigated. Here, the basic reproduction number associated with the system has been analyzed with respect to control parameters both theoretically and numerically. The biological feasible equilibria and their stability properties have been discussed and the existence condition of the disease has been illustrated numerically. For a certain set of parametric values, effectiveness of control parameters of our proposed model has been checked numerically. At last, Hopf bifurcations have been made numerically without considering control parameters for the case of constant environmental-carrying capacity of mosquito. [ABSTRACT FROM AUTHOR]

The present paper deals with an eco-epidemiological model consisting of susceptible prey, infected prey and predator. We assume that the recruitment of prey follows the saturating functional form due to habitat saturation. We make a general assumption of the non-restricted conversion rate of the predator population due to consumption of the infected prey population. We study the existence and stability criteria of the equilibrium points. Our results suggest that the predator population may be eliminated from the system due to the negative effect of infected prey; however, the negative impact can be buffered by the alternative food. Alternative food helps predator population to survive and makes the system disease free. The outcomes from the model are verified numerically by taking a set of biologically feasible parameter values. [ABSTRACT FROM AUTHOR]

The increasing occurrence of algal blooms and their negative ecological impacts have led to intensified monitoring activities. This needs the proper identification of the most responsible factor/factors for the bloom formation. However, in natural systems, algal blooms result from a combination of factors and from observation it is difficult to identify the most important one. In the present paper, using a mathematical model we compare the effects of three human induced factors (fertilizer input in agricultural field, eutrophication due to other sources than fertilizers, and overfishing) on the bloom dynamics and DO level. By applying a sophisticated sensitivity analysis technique, we found that the increasing use of fertilizers in agricultural field causes more rapid algal growth and decreases DO level much faster than eutrophication from other sources and overfishing. We also look at the mechanisms how fertilizer input rate affects the algal bloom dynamics and DO level. The model can be helpful for the policy makers in determining the influential factors responsible for the bloom formation. [ABSTRACT FROM AUTHOR]

In this manuscript, we propose and analyze a compartmental model of visceral leishmaniasis (VL). We model the human population with six compartments including asymptomatic, symptomatic and PKDL-infected, animal population as second host and sandfly population as the vector. Furthermore, the non-adult stage of the sandfly population is introduced in the system, which was not considered before in the literature. We show that the increase in the number of host of sandfly population generates a backward bifurcation. Thus, multiple hosts will cause disease persistence even if the basic reproduction number ( $$R_{0}$$ ) is below unity. We perform a sensitivity analysis of important model parameters with respect to some epidemiologically significant responses. We validate our model by calibrating it to weekly VL incidence data from South Sudan for the year 2013. We perform cost-effectiveness analysis on different interventions: treatment, non-adult control, adult control and their different layered combinations based on their implementation cost (in USD) and case reduction. We also use a global sensitivity analysis technique to understand the effect of important parameters of our model on the implementation cost of different controls. This cost-effectiveness study and cost-sensitivity analysis are relatively new in existing literature of this disease. [ABSTRACT FROM AUTHOR]

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed. [ABSTRACT FROM AUTHOR]

In this article, an eco-epidemiological model with strong Allee effect in prey population growth is presented by a system of delay differential equations. The time lag in terms of the delay parameter corresponds to the predator gestation period. We inspect elementary mathematical characteristic of the proposed model such as uniform persistence, stability and Hopf bifurcation at the interior equilibrium point of the system. We execute several numerical simulations to illustrate the proposed mathematical model and our analytical findings. We use basic tools of nonlinear dynamic analysis as first return maps, Poincare sections and Lyapunov exponents to identify chaotic behavior of the system. We observe that the system exhibits chaotic oscillation due to the increase of the delay parameter. Such chaotic behavior can be suppressed by the strength of Allee effect. [ABSTRACT FROM AUTHOR]

We propose and analyze a deterministic eco-epidemiological system where the susceptible pest exhibits a weak Allee effect due to mating limitation. We make the following assumptions: (i) Allee effect is built in the reproduction process of susceptible pest where infected pest has no contribution; and (ii) growth rate of natural predator is negative for consuming infected pest. We also assume that the functional response of predator for susceptible pest is linear, hyperbolic, or sigmoidal, whereas for infected pest is linear, as infected pest are weakened and easy to catch. We study the dynamics of the system around each of the ecological feasible equilibrium and observe some interesting features of the system dynamics. The reduction of disease eradication, and predator-pest coexistence are observed around the predator free and disease free equilibrium respectively. It is also observed that the interior equilibrium is always unstable and the result is true for any choice of functional responses. Our results suggest that introduction of predator or natural enemies with sigmoidal functional response is a good candidate to represent disease eradication. [ABSTRACT FROM AUTHOR]

In this paper, we propose a prey-predator system with disease in both the prey and predator populations. We also consider that the predator population is cannibalistic in nature and the disease spread in the predator population through cannibalism. Local stability of the system around the biologically feasible equilibria are studied. We also analyze the community structure of the system in terms of ecological and disease basic reproduction numbers. We observe that cannibalism in predator population stabilized the prey-predator oscillations. It is also observed that cannibalism can control the disease in the system. The existence of Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter, and also derive the conditions for the permanence of the system. Furthermore, applying the normal form theory and central manifold theorem, the direction of Hopf bifurcation, the stability and the period of bifurcating periodic solutions are derived. Finally extensive numerical experiments are performed to validate analytical findings. [ABSTRACT FROM AUTHOR]

A mathematical analysis of a virus-marine bacteria interaction model is proposed to study the role of virus infection on a bacterial community in the presence of a nutrient. Our model includes four state variables, namely, nutrient concentration, susceptible bacteria biomass, infected bacteria biomass and phage virus concentration. The phage virus infection induces viral lysis, which releases into the marine environment on average $$b$$ (viral replication factor) viruses per cell. The dynamical behavior of the system is studied by deriving the existence and bounded conditions and criteria for local and global stability of solutions. The local stability for the interior positive equilibrium is studied by using Routh-Hurwitz conditions. The general bifurcation analysis is shown. The qualitative behaviour of the system is verified through numerical simulation. It is observed that inclusion of nutrients in the system changes the nature of stability of different equilibria. [ABSTRACT FROM AUTHOR]

Here we consider a reaction diffusion system of three plankton populations, a zooplankton feeding on two phytoplankton populations, in two different settings. Firstly, the two phytoplanktons are both non-toxic and both enhance the growth of the grazing zooplankton. Secondly, we assume that one of the phytoplankton releases toxin and thereby inhibits the growth of the zooplankton. Our analytic and numerical study shows that the spatiotemporal distribution of the plankton species is uniform when both phytoplankton populations are non-toxic. However, in the presence of toxin-producing phytoplankton, the biomass distribution of all the plankton populations becomes inhomogeneous. [ABSTRACT FROM AUTHOR]

In the recent past there have been several attempts to obtain the equilibrium distribution of multiple populations and their moments in the context of some biological or ecological processes (e.g., Matis and Kiffe in Biometrics 52:155, 1996; Matis and Kiffe in Environ Ecol Stat 9:237, 2002; Renshaw in J Math Appl Med Biol, 15:1, 1998). In particular, the method of cumulant truncation (Matis and Kiffe in Biometrics 52:155, 1996) is a pioneering work in this field. However it requires solving a large number of cumulant functions even in the case of two simultaneous differential equations. Besides the solutions are approximate and depend on the precision of the software. Renshaw (Math Biosci 168:57, 2000) provided a nice extension of the univariate truncated saddle point procedure to multivariate scenarios. But this approach involves a multivariate Newton-Raphson type iterative algorithm whose performance and convergence are critically dependent on the choice of the initial values. In the present paper we propose a new and simple approach to obtain the equilibrium distribution of populations and their conditional moments in a system of differential equations of any dimension. Our proposed method, which is a natural extension of the classical variational matrix approach, has several advantages which are discussed in detail in the paper; among other things it includes the derivation of additional conditions which can be interpreted as environmental surrogates. [ABSTRACT FROM AUTHOR]

We explore the mutual dependencies and interactions among different groups of species of the plankton population, based on an analysis of the long-term field observations carried out by our group in the North–West coast of the Bay of Bengal. The plankton community is structured into three groups of species, namely, non-toxic phytoplankton (NTP), toxic phytoplankton (TPP) and zooplankton. To find the pair-wise dependencies among the three groups of plankton, Pearson and partial correlation coefficients are calculated. To explore the simultaneous interaction among all the three groups, a time series analysis is performed. Following an Expectation Maximization (E-M) algorithm, those data points which are missing due to irregularities in sampling are estimated, and with the completed data set a Vector Auto-Regressive (VAR) model is analyzed. The overall analysis demonstrates that toxin-producing phytoplankton play two distinct roles: the inhibition on consumption of toxic substances reduces the abundance of zooplankton, and the toxic materials released by TPP significantly compensate for the competitive disadvantages among phytoplankton species. Our study suggests that the presence of TPP might be a possible cause for the generation of a complex interaction among the large number of phytoplankton and zooplankton species that might be responsible for the prolonged coexistence of the plankton species in a fluctuating biomass. [ABSTRACT FROM AUTHOR]