Disease Basic Reproduction Numbers Research Articles

Role of Alternative Food in Controlling Chaotic Dynamics in an Eco-Epidemiological Model with Strong Allee Effects in Prey Populations

This paper explores a predator–prey system with disease in the predator population, focusing on the impact of alternative food sources. Investigating the eco-epidemiological systems with strong Allee effects in prey populations, the study analyzes local stability, introduces ecological and disease basic reproduction numbers, and observes the community structure. Extensive numerical simulations reveal varied global behaviors, including stable focus, limit cycles, period-doubling, and chaos in response to changes in the infection levels. The research emphasizes the role of alternative food in mitigating chaotic dynamics, noting that increased availability promotes stability, while decreased availability leads to a shift from chaos to a stable focus. Overall, the study underscores the significance of incorporating alternative food sources in conservation efforts for ecosystems with predator populations experiencing strong Allee effects, offering insights into the complex dynamics of eco-epidemiological systems and their implications for biodiversity conservation and disease management.

Alternative Strategies for the Estimation of a Disease's Basic Reproduction Number: A Model-Agnostic Study.

This work presents a model-agnostic evaluation of four different models that estimate a disease’s basic reproduction number. The evaluation presented is twofold: first, the theory behind each of the models is reviewed and compared; then, each model is tested with eight impartial simulations. All scenarios were constructed in an experimental framework that allows each model to fulfill its assumptions and hence, obtain unbiased results for each case. Among these models is the one proposed by Thompson et al. (Epidemics 29:100356, 2019), i.e., a Bayesian estimation method well established in epidemiological practice. The other three models include a novel state-space method and two simulation-based approaches based on a Poisson infection process. The advantages and flaws of each model are discussed from both theoretical and practical standpoints. Finally, we present the evolution of Covid-19 outbreak in Colombia as a case study for computing the basic reproduction number with each one of the reviewed methods.

Analysis of a Model for Coronavirus Spread

The spread of epidemics has always threatened humanity. In the present circumstance of the Coronavirus pandemic, a mathematical model is considered. It is formulated via a compartmental dynamical system. Its equilibria are investigated for local stability. Global stability is established for the disease-free point. The allowed steady states are an unlikely symptomatic-infected-free point, which must still be considered endemic due to the presence of asymptomatic individuals; and the disease-free and the full endemic equilibria. A transcritical bifurcation is shown to exist among them, preventing bistability. The disease basic reproduction number is calculated. Simulations show that contact restrictive measures are able to delay the epidemic’s outbreak, if taken at a very early stage. However, if lifted too early, they could become ineffective. In particular, an intermittent lock-down policy could be implemented, with the advantage of spreading the epidemics over a longer timespan, thereby reducing the sudden burden on hospitals.

Dengue Cases in Colombia: Mathematical Forecasts for 2018-2022.

INTRODUCTION Dengue is a disease caused by any one of five virus serotypes and transmitted to humans by the Aedes aegypti mosquito. Climate change and health conditions have combined to make dengue a global public health problem. The situation is especially serious in Colombia, where by week 36 of 2018, dengue incidence was 96 cases per 100,000 population, with a total of 111 deaths. Different mathematical and statistical models have been proposed to understand the dynamics of transmission and consequently to apply control strategies to reduce the number of dengue cases. OBJECTIVE Forecast the number of dengue cases expected in Colombia from 2018 through 2022 with the stochastic Auto-Regressive Integrated Moving Average (ARIMA) model and use the results to adjust the parameters of an ordinary differential equations model in order to determine the disease's basic reproduction number in the year presenting the highest number of dengue cases. METHODS An ecological time series study was conducted to forecast dengue incidence in Colombia from 2018 through 2022. The data were compiled from Colombia's National Health Institute series on dengue cases reported by epidemiological week from 2009 to 2017. The stochastic ARIMA time series model was applied. Forecasts were then analyzed, and the year with the highest number of predicted cases was used to adjust the parameters of an ordinary differential equations model (ODE) through nonlinear least squares regression to calculate the vectorial capacity of the transmitting mosquito. RESULTS Forecasts of the total number of dengue cases per year in Colombia for the following five years were: 32,411 (2018); 88,221 (2019); 56,392 (2020); 47,940 (2021); and 77,344 (2022). The highest number of cases was forecast for 2019. Values for the parameters affecting dengue transmission that year (by the year's four quarters), such as recovery rate (0.0992, 0.0838, 0.1177, and 0.1535, respectively), vectorial capacity of the transmitting mosquito (0.1720, 0.1705, 0.1204, and 0.2147, respectively) and the basic dengue reproduction number (1.73, 2.03, 1.02, and 1.40, respectively) were estimated, indicating that most cases would occur in the second quarter and, since the basic reproduction number values were >1, the disease would persist in the country throughout the entire year. CONCLUSIONS ARIMA model forecasts for 2018 through 2022 predicted the highest incidence of dengue cases in Colombia would occur in 2019. Comparison of ARIMA model forecasts and the ODE model allowed projections of possible variations in dengue cases reported, and the basic reproduction number predicted that dengue would persist throughout 2019. KEYWORDS Arboviruses, climate, dengue, models, theoretical, basic reproduction number, prognosis, Colombia.

Modelling the role of opportunistic diseases in coinfection

In this paper, we formulate a model for evaluating the effects of an opportunistic disease affecting only those individuals already infected by a primary disease. The opportunistic disease act on a faster time scale and it is represented by an SIS epidemic model with frequency-dependent transmission. The primary disease is governed by an SIS epidemic model with density-dependent transmission, and we consider two different recovery cases. The first one assumes a constant recovery rate whereas the second one takes into account limited treatment resources by means of a saturating treatment rate. No demographics is included in these models.Our results indicate that misunderstanding the role of the opportunistic disease may lead to wrong estimates of the overall potential amount of infected individuals. In the case of constant recovery rate, an expression measuring this discrepancy is derived, as well as conditions on the opportunistic disease imposing a coinfection endemic state on a primary disease otherwise tending to disappear. The case of saturating treatment rate adds the phenomenon of backward bifurcation, which fosters the presence of endemic coinfection and greater levels of infected individuals. Nevertheless, there are specific situations where increasing the opportunistic disease basic reproduction number helps to eradicate both diseases.

External Source of Infection and Nutritional Efficiency Control Chaos in a Predator–Prey Model with Disease in the Predator

This paper deals with an eco-epidemiological approach with disease circulating through the predator species. Disease circulation in the predator species can be possible by contact as well as by external sources. Here, we try to discuss the role of external source of infection along with nutritional value on system dynamics. To establish our findings, we have worked out the local and global stability analysis of the equilibrium points with Hopf bifurcation analysis associated with interior equilibrium point. The ecological consequence by ecological basic reproduction number as well as the disease basic reproduction number or basic reproductive ratio are obtained and we have analyzed the community structure of the particular system with the help of ecological and disease basic reproduction numbers. Further we pay attention to the chaotic dynamics which is produced by disease circulating in predator species by contact. Our numerical simulations reveal that eco-epidemiological system without external source of infection induced chaotic dynamics for increasing force of infection due to contact, whereas in the presence of external source of infection, it exhibits stable solution. It is also observed that nutritional value can prevent chaotic dynamics. We conclude that chaotic dynamics can be controlled by the external source of infection as well as nutritional value. We apply basic tools of nonlinear dynamics such as Poincare section and maximum Lyapunov exponent to investigate chaotic behavior of the system.

A cannibalistic eco-epidemiological model with disease in predator population

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.

Complex dynamics and its stabilization in an eco-epidemiological model with alternative food

The present paper deals with a predator–prey system with disease in the predator population and we have also studied the effect of alternative food on the system dynamics. We have analyzed the local stability of the biological feasible equilibria and introduced the concept of the ecological and the disease basic reproduction numbers. We have observed the community structure of our model system. We have performed extensive numerical simulation to observe the global behavior of our system. We have observed stable focus, limit cycle, period-doubling and chaos for variation of the force of infection. It is observed that when the infection level increases, the system enters into chaos from stable focus. We have also observed the role of alternative food on the chaotic dynamics. When the alternative food increases, the system remains chaotic dynamics but when the alternative food decreases, the system enters into a stable focus from chaos.

Proper Predation and Disease Transmission in Predator Population Stabilize Predator–Prey Oscillations

Parasites and pathogens has great impact upon the predator–prey system. Disease in ecological system can not be ignored and disease in prey species is well-studied. But disease in predator population paid little attention to the literature. Present study explores a predator–prey model with disease circulating in predator species only. The ecological as well as disease basic reproduction numbers are derived from local stability of different equilibrium points. We have also explained the biological significance of these numbers and studied the community structure of model system by these numbers. The conditions for Hopf bifurcation and persistence of the system has been analyzed under some conditions. Our numerical results reveal that disease introduction in predator species stabilizes the predator–prey species. We have also observed that proper predation stabilizes the predator–prey oscillations.

A study of harvesting in a predator–prey model with disease in both populations

Disease control by managers is a crucial response to emerging epidemics, and in the context of global change, emerging risks associated with parasites, invasive species, and infectious diseases are an important issue especially for developing countries. Our objective is to provide a mathematical framework to study the response of a predator–prey model to a disease in both populations and harvesting of prey species. We have worked out the conditions for local stability of the equilibrium points as well as persistence of the system. We have derived the ecological and disease basic reproduction numbers. These enable us to determine the community structure of the system. Harvesting may play a crucial role in a host–parasite system, and reasonable harvesting can remove parasite burden from the host. Our numerical results reveal that the reasonable harvesting prevents the oscillations of the species. We conclude that harvesting can be an effective strategy for controlling the spread of disease. Copyright © 2016 John Wiley & Sons, Ltd.

A Study of Chaotic Dynamics and Its Possible Control in a Predator-Prey Model with Disease in the Predator

The effect of parasites and pathogens in the prey population received a lot of attention, but disease in the predator population has been studied comparatively little in literature. In this study, we have considered a predator-prey model with disease in the predator population. Here, we have analyzed the local stability of the model system around the equilibria. We have derived the ecological as well as the disease basic reproduction numbers and analyzed the community structure of the model system by these numbers. Our numerical results reveal that disease introduction in the predator population produces chaotic dynamics. We have observed stable focus, limit cycles, period doubling, and chaos for variation of the force of infection in the predator population. It is also observed that half-saturation constants are responsible for occurrence and control of chaos. It is found that chaos may be prevented by increasing the predation efficiency of the infected predator.

Disease-Induced Chaotic Oscillations and its Possible Control in a Predator–Prey System with Disease in Predator

The effect of parasites and pathogens in prey populations received a lot of attention but disease in predator population has been studied comparatively little in literature. In the present paper we consider a predator–prey model with disease in predator population. We analyze the local stability of model system around the equilibria. We derive the ecological as well as disease basic reproduction numbers and study the community structure of model system by these numbers. We work out the conditions of Hopf bifurcation and persistence of system. Our numerical results reveals that system shows chaotic dynamics for increasing the infection in predator. It is also observe that half saturation constant is responsible for occurrence and control of chaos.

A Model Based Theoretical Study on Cannibalistic Prey–Predator System with Disease in Both Populations

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.

Cannibalism may control disease in predator population: result drawn from a model based study

In the present study, we propose and analyze a predator–prey system with disease in the predator population. To understand the role of cannibalism, we modify the model considering predator population is of cannibalistic type. Local and global stability around the biologically feasible equilibria are studied. The conditions for the persistence of the system are worked out. We also analyze and compare the community structure of the model systems with the help of ecological and disease basic reproduction numbers. Finally, through numerical simulation, we observe that inclusion of cannibalism in predator population may control the disease transmission in the susceptible predator population. Copyright © 2014 John Wiley & Sons, Ltd.

Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control.

We derive and study a deterministic compartmental model for malaria transmission with varying human and mosquito populations. Our model considers disease-related deaths, asymptomatic immune humans who are also infectious, as well as mosquito demography, reproduction and feeding habits. Analysis of the model reveals the existence of a backward bifurcation and persistent limit cycles whose period and size is determined by two threshold parameters: the vectorial basic reproduction number Rm, and the disease basic reproduction number R0, whose size can be reduced by reducing Rm. We conclude that malaria dynamics are indeed oscillatory when the methodology of explicitly incorporating the mosquito's demography, feeding and reproductive patterns is considered in modeling the mosquito population dynamics. A sensitivity analysis reveals important control parameters that can affect the magnitudes of Rm and R0, threshold quantities to be taken into consideration when designing control strategies. Both Rm and the intrinsic period of oscillation are shown to be highly sensitive to the mosquito's birth constant λm and the mosquito's feeding success probability pw. Control of λm can be achieved by spraying, eliminating breeding sites or moving them away from human habitats, while pw can be controlled via the use of mosquito repellant and insecticide-treated bed-nets. The disease threshold parameter R0 is shown to be highly sensitive to pw, and the intrinsic period of oscillation is also sensitive to the rate at which reproducing mosquitoes return to breeding sites. A global sensitivity and uncertainty analysis reveals that the ability of the mosquito to reproduce and uncertainties in the estimations of the rates at which exposed humans become infectious and infectious humans recover from malaria are critical in generating uncertainties in the disease classes.

Global dynamics of a delayed eco-epidemiological model with Holling type-III functional response

In this paper, an eco-epidemiological model with Holling type-III functional response and a time delay representing the gestation period of the predators is investigated. In the model, it is assumed that the predator population suffers a transmissible disease. The disease basic reproduction number is obtained. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the endemic-coexistence equilibrium are established, respectively. By using the persistence theory on infinite dimensional systems, it is proved that if the disease basic reproduction number is greater than unity, the system is permanent. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the endemic-coexistence equilibrium, the disease-free equilibrium and the predator-extinction equilibrium of the system, respectively. Copyright © 2013 John Wiley & Sons, Ltd.

Allee effects in a predator--prey system with a saturated recovery function and harvesting

In this paper, we will consider the Allee effects onpredator--prey system with a saturated recovery function andharvesting. Local stability analysis of biologically feasibleequilibrium points is worked out with help of ecological as wellas disease basic reproduction numbers. We proved that theequilibrium $P_0=(0,0)$ of the predator--prey system is (i) asaddle point in weak Allee effects (WAE) and (ii) asymptoticallystable in strong Allee effects (SAE). We proved that theequilibrium $P_1=(\beta,0)$ of the system is a saddle point if$R_0(1)<1$ and unstable if $R_0(1)>1$ in SAE case. Also we provedthat the equilibrium $P_2=(1,0)$ of the system is a saddle pointif $R_0(1)>1$ and asymptotically stability if $R_0(1)<1$ in SAEcase. It is shown that the coexistence equilibria is notasymptotically stable. Numerical simulations are carried out fora hypothetical set of parameter values to substantiate ouranalytical findings.

ROLE OF HARVESTING IN CONTROLLING CHAOTIC DYNAMICS IN THE PREDATOR–PREY MODEL WITH DISEASE IN THE PREDATOR

Predator–prey model with harvesting is well studied. The role of disease in such system has a great importance and cannot be ignored. In this study we have considered a predator–prey model with disease circulating in the predator population only and we have also considered harvesting in the prey and in the susceptible predator. We have studied the local stability, Hopf bifurcation of the model system around the equilibria. We have derived the ecological and the disease basic reproduction numbers and we have observed its importance in the community structure of the model system and in controlling disease propagation in the predator population. We have paid attention to chaotic dynamics for increasing the force of infection in the predator. Chaotic population dynamics can exhibit irregular fluctuations and violent oscillations with extremely small or large population abundances. In this study main objective is to show the role of harvesting in controlling chaotic dynamics. It is observed that reasonable harvesting on the prey and the susceptible predator prevents chaotic dynamics.

A Mathematical Study of a Predator-Prey Dynamics with Disease in Predator

We consider a predator-prey model where parasitic infection is spread in only predator population. We work out the local stability analysis of equilibrium point by the help of basic reproduction numbers. We also analyze the community structure of model system by the help of ecological as well as disease basic reproduction numbers. We derive Hopf bifurcation condition and permanence and impermanence of model system. We perform a numerical experiment and observe that parasitic infection in predator population stabilizes predator-prey oscillations.

A predator–prey mathematical model with both the populations affected by diseases

A prey–predator model with disease in both the population is proposed and analyzed. Local stability analysis of biologically feasible equilibrium points is worked out with help of ecological as well as disease basic reproduction numbers. We also explain the community composition of ecosystem by basic reproduction numbers. It is shown that the system is globally stable around the interior equilibrium point under certain parametric conditions. Moreover, the system is persistent under some feasible parametric conditions. Our analysis reveals that force of infection and predation rate are the key parameters on the dynamics of the system. Extensive numerical experiments with hypothetical set of parameters are carried out to substantiate our analytical findings. The oscillatory coexistence of the species which is very common in nature is observed for predator disease free system. The introduction of disease in the predator population enhances the extinction of predator population. We also observe that the system will be disease free and all population will persist through proper predational strategies.