scholarly journals Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step

Processes ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 2050
Author(s):  
Mohammed Hanaki ◽  
Jérôme Harmand ◽  
Zoubida Mghazli ◽  
Alain Rapaport ◽  
Tewfik Sari ◽  
...  
Keyword(s):  
Equilibrium Points ◽  
Growth Function ◽  
Mathematical Study ◽  
Reaction Step ◽  
Solid Matter ◽  
Limiting Step ◽  
Step Model ◽  
Existence And Stability ◽  
Degradation Step ◽  
Substrate Concentrations

A two-step model of the anaerobic digestion process is mathematically and numerically studied. The focus of the paper is put on the hydrolysis and methanogenesis phases when applied to the digestion of waste with a high content of solid matter: existence and stability properties of the equilibrium points are investigated. The hydrolysis step is considered a limiting step in this process using the Contois growth function for the bacteria responsible for the first degradation step. The methanogenesis step being inhibited by the product of the first reaction (which is also the substrate for the second one), and the Haldane growth rate is used for the second reaction step. The operating diagrams with respect to the dilution rate and the input substrate concentrations are established and discussed.

Bifurcation Analysis of a Prey–Predator Model with Beddington–DeAngelis Type Functional Response and Allee Effect in Prey

2020 ◽  
Vol 30 (16) ◽  
pp. 2050238
Author(s):  
Koushik Garain ◽  
Partha Sarathi Mandal
Keyword(s):  
Functional Response ◽  
Death Rate ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Growth Function ◽  
Global Dynamics ◽  
Density Dependent ◽  
Existence And Stability ◽  
Predator Model ◽  
The Impact

The article aims to study a prey–predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and Beddington–DeAngelis type functional response. We notice the changes in the existence and stability of the equilibrium points due to the Allee effect. To investigate the complete global dynamics of the Allee model, we present here a two-parametric bifurcation diagram which describes the effect of density dependent death rate parameter of predator on dynamical changes of the system. We have also analyzed all possible local and global bifurcations that the system could go through, namely transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Finally, the impact of the Allee effect in the considered system is investigated by comparing the dynamics of both the systems with and without Allee effect.

Assessing the impact of agrochemicals on schistosomiasis transmission: A mathematical study

2021 ◽  
pp. 2150049
Author(s):  
Liming Cai ◽  
Peixia Yue ◽  
Mini Ghosh ◽  
Xuezhi Li
Keyword(s):  
Mathematical Model ◽  
Sensitivity Analysis ◽  
Numerical Simulations ◽  
Disease Transmission ◽  
Agricultural Pollution ◽  
Parasitic Disease ◽  
Mathematical Study ◽  
Proposed Model ◽  
Existence And Stability ◽  
The Impact

Schistosomiasis is a snail-borne parasitic disease, which is affecting almost 240 million people worldwide. The number of humans affected by schistosomiasis is continuously increasing with the rise in the use of agrochemicals. In this paper, a mathematical model is formulated and analyzed to assess the effect of agrochemicals on the transmission of schistosomiasis. The proposed model incorporates the effects of fertilizers, herbicides and insecticides on susceptible snails and snail predators along with schistosomiasis disease transmission. The existence and stability of the equilibria in the model are discussed. Sensitivity analysis is performed to identify the key parameters of the proposed model, which contributes most in the transmission of this disease. Numerical simulations are also performed to assess the impact of fertilizers, herbicides and insecticides on schistosomiasis outbreaks. Our study reveals that the agricultural pollution can enhance the transmission intensity of schistosomiasis, and in order to prevent the outbreak of schistosomiasis, the use of pesticides should be controlled.

THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS

2004 ◽  
Vol 12 (04) ◽  
pp. 399-417 ◽  
Author(s):  
M. KGOSIMORE ◽  
E. M. LUNGU
Keyword(s):  
Equilibrium Point ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Treatment Programs ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
Disease Free ◽  
Hiv Aids

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α) are satisfied.

scholarly journals Investigating the Role of Environmental Transmission in COVID-19 Dynamics: A Mathematical Model Based Study

2020 ◽  
Author(s):  
Ibrahim M. ELmojtaba ◽  
Fatma Al-Musalhi ◽  
Asma Al-Ghassani ◽  
Nasser Al-Salti
Keyword(s):  
Mathematical Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
Environmental Transmission ◽  
Estimated Parameters ◽  
Existence And Stability

Abstract A mathematical model with environmental transmission has been proposed and analyzed to investigate its role in the transmission dynamics of the ongoing COVID-19 outbreak. Two expressions for the basic reproduction number R0 have been analytically derived using the next generation matrix method. The two expressions composed of a combination of two terms related to human to human and environment to human transmissions. The value of R0 has been calculated using estimated parameters corresponding to two datasets. Sensitivity analysis of the reproduction number to the corresponding model parameters has been carried out. Existence and stability analysis of disease free and endemic equilibrium points have been presented in relation with the obtained expressions of R0. Numerical simulations to demonstrate the effect of some model parameters related to environmental transmission on the disease transmission dynamics have been carried out and the results have been demonstrated graphically.

scholarly journals The Analysis of SEIRS-SEI Epidemic Models on Malaria with Regard to Human Recovery Rate

2017 ◽  
Vol 6 (3) ◽  
pp. 132-140
Author(s):  
Resmawan Resmawan ◽  
Paian Sianturi ◽  
Endar Hasafah Nugrahani
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Recovery Rate ◽  
Human Population ◽  
Equilibrium Points ◽  
Epidemic Models ◽  
Existence And Stability ◽  
Simulation Results ◽  
Model Formation ◽  
The Given

This article discusses SEIRS-SEI epidemic models on malaria with regard to human recovery rate. SEIRS-SEI in this model is an abbreviation of the population class used in the model, ie Susceptible, Exposed, Infected, and Recovered populations in humans and Susceptible, Exposed, and Infected populations in mosquito. These epidemic models belong to mathematical models which clarify a phenomenon of epidemic transmission of malaria by observing the human recovery rate after being infected and susceptible. Human population falls into four classes, namely susceptible humans, exposed humans, infected humans, and recovered humans. Meanwhile, mosquito population serving as vectors of the disease is divided into three classes, including susceptible mosquitoes, exposed mosquitoes, and infected mosquitoes. Such models are termed SEIRS-SEI epidemic models. Analytical discussion covers model formation, existence and stability of equilibrium points, as well as numerical simulation to find out the influence of human recovery rate on population dynamics of both species. The results show that the fixed point without disease ( ) is stable in condition  and unstable in condition . The simulation results show that the given treatment has an influence on the dynamics of the human population and mosquitoes. If the human recovery rate from the infected state becomes susceptible to increased, then the number of infected populations of both species will decrease. As a result, the disease will not spread and within a certain time will disappear from the population.

scholarly journals An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 36
Author(s):  
Santiago Alonso-Quesada ◽  
Manuel De la Sen ◽  
Raúl Nistal
Keyword(s):  
Control System ◽  
Equilibrium Point ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Design Parameters ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Existence And Stability ◽  
Disease Free

This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value Rc, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>Rc, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and Rc. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.

scholarly journals Considering Quarantine in the SIRA Malware Propagation Model

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
José Roberto C. Piqueira ◽  
Cristiane M. Batistela
Keyword(s):  
Data Science ◽  
Network Reliability ◽  
Equilibrium Points ◽  
Propagation Model ◽  
Data Networks ◽  
Preventive Actions ◽  
Malware Propagation ◽  
Existence And Stability ◽  
Block Based ◽  
Numerical Approaches

As the beginning of the 21st century was marked by a strong development in data science and, consequently, in computer networks, models for designing preventive actions against intruding, data stealing, and destruction became mandatory. Following this line, several types of epidemiological models have been developed and improved, considering different operational approaches. The development of the research line using traditional SIR(Susceptible, Infected, Removed) model for data networks started in the 1990s. In 2005, an epidemiological compartmental model containing antidotal nodes, SIRA (Susceptible, Infected, Removed, Antidotal), was introduced to study how the antivirus policies affect the network reliability. The idea here is to study the consequence of quarantine actions in a network by modifying the SIRA model, introducing quarantine nodes generating the SIQRA (Susceptible, Infected, Quarantine, Removed, Antidotal) model. Analytical and numerical approaches result in parameter conditions for the existence and stability of disease-free and endemic equilibrium points for two different cases: saturation and nonsaturation of the quarantine population block. Based on these results, operational actions can be planned to improve the network reliability.

scholarly journals Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xinxin Liu ◽  
Qingdao Huang
Keyword(s):  
Prey Species ◽  
Equilibrium Points ◽  
Maximum Sustainable Yield ◽  
Predator Prey ◽  
Prey System ◽  
Existence And Stability ◽  
Predator Model ◽  
Model Equations ◽  
The Given

AbstractA new way to study the harvested predator–prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system.

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