Nonlinear Incidence Research Articles

A Lyapunov functional for vaccination model on the dynamics of cholera epidemic with non-linear incidence of infection

A new deterministic susceptible-exposed-infectious-vaccinated-removed-pathogen (SEIVRB) cholera epidemic model with combined mass action incidence and saturated incidence rates is proposed and analyzed. A threshold level of vaccine coverage necessary for controlling or eradicating cholera has been determined and analyzed using the next generation matrix approach. It is shown that the higher values of vaccine coverage that are lower than the threshold value significantly reduces the number of infected individuals whenever basic reproduction number is less than unity, and the cholera would persist in the populations whenever the model basic reproduction number exceeds unity. The global stability for cholera free equilibrium state and cholera endemic equilibrium state of the model system is investigated using Lyapunov functional approach and Lasalle invariance principle, which are found to be globally asymptotically stable at both equilibrium states. Numerical simulations and graphical illustrations is presented to support the analytical results found in the study.

DYNAMICAL ANALYSIS OF TB MODEL WITH NONLINEAR INCIDENCE RATE AND OPTIMAL CONTROL

We propose and analyze a TB transmission model with nonlinear incidence rate, immunization and medical treatment. First, the existences and stabilities of the equilibrium are studied. The results indicate the basic reproduction number [Formula: see text] is the threshold of disease extinction and persistence. The disease-free equilibrium is globally asymptotically stable when [Formula: see text], and the disease will gradually disappear. The unique positive equilibrium is local stability and the disease is uniformly persistent when [Formula: see text]. Second, optimal control is added to the original model because of limited resources. Finally, the stability of the equilibrium and the theoretical results of optimal control are verified by numerical simulations, and the sensitivity of the parameters is analyzed by the PRCC method.

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate.

The current study deals with the stochastic reaction-diffusion epidemic model numerically with two proposed schemes. Such models have many applications in the disease dynamics of wildlife, human life, and others. During the last decade, it is observed that the epidemic models cannot predict the accurate behavior of infectious diseases. The empirical data gained about the spread of the disease shows non-deterministic behavior. It is a strong challenge for researchers to consider stochastic epidemic models. The effect of the stochastic process is analyzed. So, the SIR epidemic model is considered under the influence of the stochastic process. The time noise term is taken as the stochastic source. The coefficient of the stochastic term is a Borel function, and it is used to control the random behavior in the solutions. The proposed stochastic backward Euler scheme and the proposed stochastic implicit finite difference scheme (IFDS) are used for the numerical solution of the underlying model. Both schemes are consistent in the mean square sense. The stability of the schemes is proven with Von-Neumann criteria and schemes are unconditionally stable. The proposed stochastic backward Euler scheme converges toward a disease-free equilibrium and does not converge toward an endemic equilibrium but also possesses negative behavior. The proposed stochastic IFD scheme converges toward disease-free equilibrium and endemic equilibrium. This scheme also preserves positivity. The graphical behavior of the stochastic SIR model is much similar to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The efficacy of the proposed scheme is shown in the graphical behavior of the test problem for the various values of the parameters.

Traveling waves for a diffusive virus infection model with humoral immunity, cell‐to‐cell transmission, and nonlinear incidence

This paper is concerned with the existence and nonexistence of traveling waves of a virus infection model with humoral immunity, cell‐to‐cell transmission, and general nonlinear incidence. Our results show that the existence of traveling wave solutions is determined not only by the basic reproduction number of virus infection and the antibody response reproduction number but also by the critical wave speed . More precisely, we obtain the existence of traveling wave solution connecting infection‐free equilibrium and antibody‐free infection equilibrium for , and and connecting infection‐free equilibrium and antibody‐present infection equilibrium for , and . The existence of traveling wave solution connecting infection‐free equilibrium and antibody‐present infection equilibrium is discussed. Some numerical simulations are carried out to illustrate our analytical results.

Threshold Analysis of a Stochastic SIRS Epidemic Model with Logistic Birth and Nonlinear Incidence

The paper mainly investigates a stochastic SIRS epidemic model with Logistic birth and nonlinear incidence. We obtain a new threshold value (R0m) through the Stratonovich stochastic differential equation, different from the usual basic reproduction number. If R0m<1, the disease-free equilibrium of the illness is globally asymptotically stable in probability one. If R0m>1, the disease is permanent in the mean with probability one and has an endemic stationary distribution. Numerical simulations are given to illustrate the theoretical results. Interestingly, we discovered that random fluctuations can suppress outbreaks and control the disease.

Dynamics of a stochastic epidemic-like rumor propagation model with generalized nonlinear incidence and time delay

Rumors are abnormal children of public opinion, they usually arise with environmental crisis and have serious negative functions. Hence, how to control rumor propagation has become an urgent topic for the government. Since the communication between people in the real world is inevitably interfered by random factors, and time delay exists in the process of rumor propagation not only influence rumor-forwarding individuals by rumor-hesitant individuals but also in government decision-making. Based on these premises, we propose a stochastic delayed rumor propagation model with generalized incidence function. By setting the suitable Lyapunov functions, we show that the asymptotic behavior of stochastic rumor spreading model is constrained by some conditions. Finally, our analytical results are supported by numerical simulations. The results show that (1) the increase of noise intensity in social environment will reduce rumor propagation; (2) time delay has an impact on rumor propagation to a certain extent. Extending the thinking time may reduce the number of rumor-forwarding individuals and even lead to the disappearance of rumors; (3) improve scientific literacy and accelerate rumor refutation (i.e. reduce the value of β), strengthen rumor suppression (i.e. increase the value of γ) can effectively control rumor transmission.

Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate

In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population.

Dynamical analysis of a heroin–cocaine epidemic model with nonlinear incidence and spatial heterogeneity

In this paper, we investigated a new heroin–cocaine epidemic model which incorporates spatial heterogeneity and nonlinear incidence rate. The main project of this paper is to explore the threshold dynamics in terms of the basic reproduction number , which was defined by applying the next-generation operator. The threshold type results shown that if , then the drug-free steady state is globally asymptotically stable. If , then heroin–cocaine spread is uniformly persistent. Furthermore, the globally asymptotic stability of the drug-free steady state has been established for the critical case of by analysing the local asymptotic stability and global attractivity.

Stability analysis and optimal control in an epidemic model on directed complex networks with nonlinear incidence

In this paper, stability and optimal control for SIS epidemic systems with birth and death in directed complex networks are investigated. A class of epidemic systems are constructed based on SIS models combining with two kinds of topological characters as in-degree and out-degree in directed networks, in which the nonlinear incidence rate is introduced that relevant to network’s topology containing the psychological effect coefficient. The disease-free and the endemic equilibria are derived and the epidemic threshold for our systems is obtained. Then, the local stability about the disease-free and the endemic equilibria are discussed via the corresponding Jacobian matrices and the characteristic equation. In addition, global asymptotical stability about two kinds of equilibria are also considered based on the Lyapunov function and stable analysis method. From the view of Pontryagin Minimum Principle, the optimal control issue about the system is also considered, and the corresponding optimal control systems and optimal control tactics are obtained. Several numerical examples are provided to demonstrate the effectiveness of our main method. Meanwhile, by means of the Forward–Backward Sweep method, optimal control systems are solved in the aspect of numeric and the control aim under the derived strategies is achieved.

Stability and bifurcations of an SIR model with a nonlinear incidence rate

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium , an endemic equilibrium , and the basic reproduction number of the model are obtained. If is locally asymptotically stable and if is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near approaches and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations.

Relative prevalence-based dispersal in an epidemic patch model.

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

The stability analysis of a co-circulation model for COVID-19, dengue, and zika with nonlinear incidence rates and vaccination strategies

This paper aims to study the impacts of COVID-19 and dengue vaccinations on the dynamics of zika transmission by developing a vaccination model with the incorporation of saturated incidence rates. Analyses are performed to assess the qualitative behavior of the model. Carrying out bifurcation analysis of the model, it was concluded that co-infection, super-infection and also re-infection with same or different disease could trigger backward bifurcation. Employing well-formulated Lyapunov functions, the model’s equilibria are shown to be globally stable for a certain scenario. Moreover, global sensitivity analyses are performed out to assess the impact of dominant parameters that drive each disease’s dynamics and its co-infection. Model fitting is performed on the actual data for the state of Amazonas in Brazil. The fittings reveal that our model behaves very well with the data. The significance of saturated incidence rates on the dynamics of three diseases is also highlighted. Based on the numerical investigation of the model, it was observed that increased vaccination efforts against COVID-19 and dengue could positively impact zika dynamics and the co-spread of triple infections.

Hopf bifurcation in a networked delay SIR epidemic model

A time delay networked Susceptible–Infectious–Recovered (SIR) epidemic model with a nonlinear incidence rate is considered on a graph of Laplacian diffusion. The model introduces population mobility through the graph network. Several stability theorems are proved at all possible different equilibrium points of the model. Further, Hopf bifurcation analysis for the endemic equilibrium is investigated. Numerical results are presented to support the theoretical findings.

Analysis of an optimal control problem for a spatio‐temporal SIR model with nonlinear density dependent diffusion terms

SummaryThis paper focuses on the study of an optimal control problem for a new spatio‐temporal SIR epidemic model with nonlinear density dependent diffusion terms and a class of nonlinear incidence functions. We consider two types of control variables, vaccination for the susceptible and treatment for the infected. For fixed controls, by means of Schauder fixed point theorem, we prove that the proposed model admits a weak biologically feasible solution, the uniqueness of the latter is also investigated. Furthermore, using the state and adjoint problems, first order necessary optimal conditions are obtained. Finally, numerical simulations are carried out for particular diffusion terms incorporating the heard mentality of individuals, when it comes to the spatial movement, and for particular incidence functions, as well as by varying the parameters of the objective functional, to illustrate the possible optimal control strategies and their effect on the studied population.

Study of Turing patterns in a SI reaction-diffusion propagation system based on network and non-network environments

The study of rumor propagation dynamics is of great significance to reduce false news and ensure the authenticity of news information. In this paper, a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied. First, through stability analysis, we obtain the conditions for the existence and local stability of the positive equilibrium point. By selecting suitable variable as the control parameter, the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained. Then, using the above theorem and multi-scale standard analysis, the expression of amplitude equation around Turing bifurcation point is obtained. By analyzing the amplitude equation, different types of Turing pattern are divided such as uniform steady-state mode, hexagonal mode, stripe mode and mixed structure mode. Further, in the numerical simulation part, by observing different patterns corresponding to different values of control variable, the correctness of the theory is verified. Finally, the effects of different network structures on patterns are investigated. The results show that there are significant differences in the distribution of users on different network structures.

The Dynamics of a Delayed Ecoepidemiological Model with Nonlinear Incidence Rate

In this paper, the general framework for calculating the stability of equilibria, Hopf bifurcation of a delayed prey-predator system with an SI type of disease in the prey population, is investigated. The impact of the incubation period delay on disease transmission utilizing a nonlinear incidence rate was taken into account. For the purpose of explaining the predation process, a modified Holling type II functional response was used. First, the existence, uniform boundedness, and positivity of the solutions of the considered model system, along with the behavior of equilibria and the existence of Hopf bifurcation, are studied. The critical values of the delay parameter for which stability switches and the nature of the Hopf bifurcation by using normal form theory and center manifold theorem are identified. Additionally, using numerical simulations and a hypothetical dataset, various dynamic characteristics are discovered, including stability switches, chaos, and Hopf bifurcation scenarios.

Global Dynamics of a Two-Strain Disease Model with Amplification, Nonlinear Incidence and Treatment

In recent years, antibiotic resistance to the most effective treatments has emerged and spread. This has led to a decline in the efficacy of antibiotics used to treat patients, with drug-resistant strain experiencing much higher failure rates and serious side effects. In this study, we considered two-strains (e.g., drug-susceptible and drug-resistant) susceptible-infected-recovery disease model with amplification, nonlinear incidence and treatment. We assumed amplification develops mainly through the choice of naturally happening mutations in the presence of inappropriate treatment. We performed a rigorous analytical analysis of the model properties and solutions to predict late-time behavior of the disease dynamics and find that the model contains four equilibrium points: disease-free equilibrium, monoexistence endemic equilibrium 1 concerning drug-susceptible strain, monoexistence endemic equilibrium 2 concerning drug-resistant strain and coexistence equilibrium regarding to drug-susceptible as well as drug-resistant strains. Two basic reproduction numbers $${R}_{0\mathrm{s}}$$ and $$R_{{0{\text{m}}}}$$ are found, and we have presented that if both are less than one $$({\text{i}}.{\text{e}}.\max \left[ {R_{{0{\text{s}}}} ,{ }R_{{0{\text{m}}}} } \right] < 1)$$ , the disease fade-out, and if both greater than one $$({\text{i}}.{\text{e}}.\max \left[ {R_{{0{\text{s}}}} ,{ }R_{{0{\text{m}}}} } \right] > 1)$$ the epidemic situation occurs. Moreover, epidemics occur regarding to any strain when the basic reproduction number remains above the value 1 and disease fade-out with regard to any strain when the basic reproduction number remains below the value 1. In all equilibrium points, the global stability analysis was determined with the help of appropriate Lyapunov functions. In addition, we also found that the drug-resistant strain prevalence increases when the drug-susceptible strain is treated due to the poor-quality treatment (i.e., amplification). We also performed the sensitivity analysis through evaluation of Partial Rank Correlation Coefficients (PRCC) to identify the most important model parameters and found that transmission rate of both strains had the maximum influence on disease outbreak. To support those analytical results, numerical simulations of the model were performed using ODE45 MATLAB routine.

Stability and optimal control for delayed rumor-spreading model with nonlinear incidence over heterogeneous networks

On the multilingual online social networks of global information sharing, the wanton spread of rumors has an enormous negative impact on people’s lives. Thus, it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation. In this paper, considering the multilingual environment and intervention mechanism in the rumor-spreading process, an improved ignorants–spreaders-1–spreaders-2–removers (I2SR) rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks. Firstly, based on the mean-field equations corresponding to the model, the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium. Secondly, by applying Lyapunov stability theory and graph theory, the global stability of rumor-spreading equilibrium is analyzed in detail. In particular, aiming at the lowest control cost, the optimal control scheme is designed to optimize the intervention mechanism, and the optimal control conditions are derived using the Pontryagin’s minimum principle. Finally, some illustrative examples are provided to verify the effectiveness of the theoretical results. The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time, which provides guiding insights for public opinion managers to control rumors.

The global stability and optimal control of the COVID-19 epidemic model

This paper considers stability analysis of a Susceptible-Exposed-Infected-Recovered-Virus (SEIRV) model with nonlinear incidence rates and indicates the severity and weakness of control factors for disease transmission. The Lyapunov function using Volterra–Lyapunov matrices makes it possible to study the global stability of the endemic equilibrium point. An optimal control strategy is proposed to prevent the spread of coronavirus, in addition to governmental intervention. The objective is to minimize together with the quantity of infected and exposed individuals while minimizing the total costs of treatment. A numerical study of the model is also carried out to investigate the analytical results.

Threshold Dynamics and Probability Density Function of a Stochastic Avian Influenza Epidemic Model with Nonlinear Incidence Rate and Psychological Effect

Threshold Dynamics and Probability Density Function of a Stochastic Avian Influenza Epidemic Model with Nonlinear Incidence Rate and Psychological Effect