Threshold Value R0 Research Articles

Interconnected Takagi-Sugeno system and fractional SIRS malware propagation model for stabilization of Wireless Sensor Networks

This paper is devoted to investigate the dynamical behaviors of malware attack on a familiar kind of complex heterogeneous networks, namely Wireless Sensor Network, and discuss an effective immunization treatment based on fractional interconnected Takagi-Sugeno (T-S) fuzzy systems. Our approach is based on the mathematical modeling to establish a controlled fractional network-based SIRS malware propagation model that better describes the attacking behavior of malicious objects on Wireless Sensor Network. After that, we point out some qualitative properties of the proposed network-based SIRS malware propagation model such as the existence of positively invariant set, backward bifurcation and asymptotic behavior. Especially, in order to study the model's stability, we evaluate an epidemiological threshold value R0, namely basic reproductive ratio, which ensures the existence of at least one endemic equilibrium P⁎ and the local asymptotic stability of malware-free equilibrium P0. As a consequence of theoretical result, the malware-free equilibrium P0 is unstable when R0>1 and hence, the rest of this paper is to address a stabilization problem for the proposed controlled network-based model and establish some sufficient conditions related to linear matrix inequalities and positive definite matrices. Finally, we illustrate the obtained theoretical results by a computational example.

Bifurcation analysis of a delayed stochastic HBV epidemic model: Cell-to-cell transmission

In the present study, we develop a model for hepatitis B virus (HBV) that integrates cell-to-cell transmission of the virus, CTL immunological response, delay effects, and the effect of fluctuating environmental noise. The inheritance randomness of the system shifting the system from a deterministic to another system consists of stochastic differential equations (SDEs). After model formulation, the dynamical aspects of the model are investigated by using the key threshold quantity R0. If the threshold value R0 is assumed less than unity, then it is proven that the solution trajectories are fluctuating around the disease-free state (DFS) of the associated deterministic model. The fluctuation of the trajectories around the DFS explains the phenomenon of extinction of the stochastic models. It is also proved that the trajectories of the SDE model oscillate around the endemic state of the underlying ODE system when R0s>1 explains the sure persistence theory of the proposed model. One notable finding of the present manuscript is the positive association between the fluctuating amplitude and the intensities of the white noises. This understanding can provide useful control tools for managing the worst dynamic(s) of the infection. The exact value of the parameter for time delay determines the stability of the model under consideration. By analyzing the time-delayed parameter, it was identified that the model is stable by keeping the value of the parameter below the crucial value that causes the Hopf bifurcation. This conclusion is supported by the numerical examples having epidemiological significance. Furthermore, the findings suggest that stochastic environmental disturbance can impact the spread of infectious illnesses. Notably, the high levels of noise can help to slow down the spread of epidemics in the population.

Modeling criminal careers of different levels of offence

We set up and analyse a mathematical model, the Serious Crime Model, which describes the interaction of mild and serious offenders and potential criminals. However we get more complete results for a simpler version of this model, the Mild Crime Model, with no serious offenders. For the full Serious Crime Model there are two key parameters R01 and R02 corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the Simpler Mild Crime Model there is just one such parameter R01. Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R01<1 in the backwards case. For backwards bifurcation there is another threshold value R0* such that the upper non-trivial equilibrium is unstable for R01<R0* and stable for R01>R0*. For forwards bifurcation there is a second additional threshold value R0** such that the stability of the unique non-trivial equilibrium switches from unstable to stable as R01 passes through R0**. At the end we return to the full Serious Crime Model and discuss the behaviour of this model. The results are meaningful and interesting because in all of the other epidemiological and sociological models of which we are aware, analogous thresholds to R0* and R0** do not exist. For forwards bifurcation the unique non-trivial equilibrium, and for backwards bifurcation with two subcritical endemic equilibria the higher non-trivial equilibrium, are also usually always locally asymptotically stable. So our models exhibit unusual and interesting behaviour.

The Influence of Nonlinear Cannibalism to Logistic Equation

A single species model with Holling II type cannibalism term is proposed and studied in this paper. Local and global stability property of the system are investigated. By applying the iterative method, we show that the system always admits the unique globally asymptotically stable positive equilibrium. A threshold value R0, which depends on the cannibalism rate and the transform rate, is obtained. Depending on R0 &gt; 1, R0 = 1 or R0 &lt; 1, the final density of the species will smaller or equal to or bigger than the system without cannibalism. Our study shows that if the cannibalism rate is too large, and transform rate is too small, then R0 &gt; 1 and cannibalism has negative effect on the final density of the species, which increase the extinction property of the species.

An SIS epidemic model in a patchy environment with pulse vaccination and quarantine

An SIS epidemic model in an m-patch environment with pulse vaccination and quarantine at two different fixed moments is formulated by impulsive differential equations in this paper. The sufficient conditions for the extinction and persistence of the disease are derived and the threshold value R0 is obtained by using the persistence theory of impulsive systems, the impulsive-type Floquet theory and the perturbation techniques. That is, the infection-free periodic solution is globally asymptotically stable if R0<1, and the impulsive system becomes uniformly persistent if R0>1. When m=2, two special cases are considered to illustrate joint impact of the pulse vaccination and quarantine and population mobility on the disease dynamics. Numerical simulations are given to show the effectiveness of the theoretical results.

The impact of time delay in the transmission of Japanese encephalitis without vaccination

In this manuscript, the influence of time delay in the transmission of Japanese encephalitis without vaccination model has been studied. The time delay is because of the existence of an incubation period during which the Japanese encephalitis virus reproduces enough in the mosquitoes with the goal that it tends to be transmitted by the mosquitoes to people. The motivation behind this manuscript is to assess the influence of the time delay it takes to infect susceptible human populations after interacting with infected mosquitoes. The steadystate and the threshold value R0 of the delay model were resolved. This value assists with setting up the circumstance that ensures the asymptotic stability of relating equilibrium points. Utilizing the delay as a bifurcation parameter, we built up the circumstance for the presence of a Hopf bifurcation. Moreover, we infer an express equation to decide the stability and direction of Hopf bifurcation at endemic equilibrium by using center manifold theory and normal structure strategy. It has been seen that delay plays a vital role in stability exchanging. Furthermore, the presence of Hopf bifurcation is affected by larger values of virus transmission rate from an infected mosquito to susceptible individuals and the natural mortality of humans in a model. Finally, to understand some analytical outcomes, the delay framework is simulated numerically.

A general non-local delay model on oncolytic virus therapy

The oncolytic virus is regarded as a novel, powerful, and biologically safe method of cancer treatment. A general delay differential system was driven by the age-dependent model better to understand the interaction between tumor cells and viruses. General continuous functions F(x,y) and G(x) depict the tumor proliferation rate and virus infection rate. The critical threshold value R0 was calculated that plays a determinant role in whether virus therapy occurs. The non-local delay term ∫t−τtβG(x(θ))v(θ)e−α(t−θ)dθ makes our model hard to analyze when using the traditional eigenvalue method. The method combining implicit function theorem and comparison theorem is used to overcome this problem. Furthermore, we support the fact that virotherapy can lead to tumor remission by using the fluctuation method. Lastly, Bayesian information criterion was adopted to select a better model when fitting the experimental data.

Dynamical analysis and effects of law enforcement in a social interaction model

Crime is a rising problem all over the world. Over the years, several researches have been conducted to coin suitable model systems and intervention strategies that would decrease delinquent behavior and promote prosocial development. Being dynamic and complex process, the spread of crime requires a system level approach. In this paper, we formulate and analyze a series of dynamical model systems of the dilation of crime incorporating law enforcement. The population models of social interactions have been constructed based on predator–prey interaction models with Holling type II response function. Two different types of populations (criminal minded and non-criminal minded) have been assumed in the given community/society. According to law policy for crime control, we scrutinize the dynamic behavior of the model system concerning law enforcement on the criminal minded population. Along with analytical expressions for the existence of different equilibrium points and their stability, we have also provided their geometrical interpretations using isocline analysis. The expressions obtained for the existence and stability of equilibrium points have been used to investigate the effects of coefficient of law enforcement and the logistic growth term on the prevalence of crime. It is observed that for a threshold value of law enforcement, a stable limit cycle exists. In particular, a threshold of law enforcement is determined beyond which the associated place/community could be made crime free. A threshold value R0 (similar to basic reproduction number in epidemiology) has also been introduced. It is shown that when R0<1, crime free equilibrium is stable. The elimination and persistence of crime have been discussed via two parameters (law enforcement and half saturation constant) bifurcation diagram. Theoretical results have been supported via numerical simulations followed by discussion and societal impacts of the present study.

Estimates of Intrinsic Growth Rates and Basic Reproduction Number (R0) for the First Historical Zika Outbreak in Salta, Argentina

Scopes and Objectives: After entering South America in May 2015 through northeast Brazil, the Zika virus spread to Argentina between April and June 2016 and reached Salta province the following year. We analyzed some aspects of the first historical outbreak of Zika that occurred in Salta, Argentina, in the year 2017. We obtained elementary estimates, such as the intrinsic growth rate of the cases accumulated in the first weeks of the outbreak, using expressions that relate it to the basic reproduction number thereof. Study design: Retrospective-descriptive studies and relational analysis.&#x0D; Place and Duration of Study: Department of Mathematics, Faculty of Exact Sciences and Faculty of Engineering. National University of Salta, Argentina, from September 2019 to June 2020.&#x0D; Methodology: Descriptive and relational analysis. Estimates of parameters and Simulation tests were also carried out in order to qualitatively describe the first Zika historical outbreak in Salta. &#x0D; Results: Our study revealed that the Zika virus in the province of Salta mainly affects the localities of the departments of Or´an, General San Mart´ın, and Rivadavia, with infection forces α2017 ≈ 0.42 week-1 (SD 0.05) and α2017 ≈ 0.32 week-1 (SD 0.02) with the refined exponential model. On the other hand, we obtained estimates of the basic reproduction number R0 ≈ 1:105 95% CI[1:104 - 1:106] and R0 ≈ 1:111 95% CI[1:110 - 1:112]. &#x0D; Conclusion: Both the values of the estimates of the infection forces and R0 would seem to indicate that the first outbreak of Zika in Salta was of relatively low intensity and of short duration, coinciding with patterns generally present emerging diseases. We found practically no differences with the estimates provided by the two expressions of basic reproduction number used. Although the estimates slightly exceed the threshold value R0 = 1, with respect to otherestimates, we consider them quite reasonable for the first historical outbreak occurred in Salta, since it was short-lived and of little intensity.

Backward bifurcation and stability analysis of a network-based SIS epidemic model with saturated treatment function

In this paper, we present a study on a network-based SIS epidemic model with a saturated treatment function to characterize the saturation phenomenon of limited medical resources. In this model, we first obtain a threshold value R0, which is the threshold condition for the stability of the disease-free equilibrium. We show that a backward bifurcation occurs under certain conditions. More precisely, the saturated treatment function leads to a such bifurcation. In this case, R0<1 is not sufficient to eradicate the disease from the population. Furthermore, we also study the stability of the endemic equilibrium and the corresponding stability condition is given. Numerical experiments are conducted and their results validate the theoretical results.

Bifurcation analysis of a network-based SIR epidemic model with saturated treatment function.

In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R0, which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R0=1. More specifically, we derived a condition that determines the direction of bifurcation at R0=1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R0<1. Therefore, in this case, R0<1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R0=1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state.

Analysis of an age-structured multi-group heroin epidemic model

This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method.

Stability and traveling waves of a vaccination model with nonlinear incidence

In this paper, we propose a spatial vaccination model with nonlinear incidence. First, we consider the well-posedness of solutions of the model. Second, in the case of the bounded spatial habitat Ω⊆Rn, we investigate the global stability of the model. More precisely, it is shown that, if the threshold value R0≤1, then the disease-free equilibrium E0 is globally asymptotically stable; if R0>1, then there exists a unique disease equilibrium E∗ which is globally asymptotically stable. Third, in the case of the unbounded spatial habitat Ω=Rn, we study the existence of traveling wave solutions of the model. Here we show that when the threshold value R0>1, then there exists c∗>0 such that there exist positive traveling wave solutions of the model connecting the two equilibria E0 and E∗ with speed c>c∗. And when R0>1, there is not such a traveling wave solution with speed c<c∗. Numerical simulations are performed to illustrate our analytic results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R0.

A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity

A class of SIR epidemic model with generalized nonlinear incidence rate is presented in this paper. Temporary immunity and stochastic perturbation are also considered. The existence and uniqueness of the global positive solution is achieved. Sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established. Moreover, the threshold behavior is discussed, and the threshold value R0 is obtained. We show that if R0<1, the disease eventually becomes extinct with probability one, whereas if R0>1, then the system remains permanent in the mean.

Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate.

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold R0 which completely governs the disease dynamics: when R0 < 1, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when R0 > 1, the disease is permanent. It is interesting that the threshold value R0 bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.

Traveling waves in a diffusive influenza epidemic model with vaccination

In this paper, a mathematical model of influenza disease with vaccination is formulated to incorporate a spatially homogeneous structure. The explicit formula of the basic reproduction number R0 for the model is obtained. It is shown that the existence and non-existence of the traveling wave solutions for the model are determined completely by the threshold value R0. Here, by introducing an auxiliary system and applying Schauder fixed point theorem, we show that the auxiliary system admits a nontrivial traveling wave solution. And then, by the limit arguments and Arzelà–Ascoli’s theorem, we establish the existence of the traveling wave solutions for the model if R0>1 and c > c*. Furthermore, by the two-sided Laplace transform, we show that the model has no nontrivial traveling wave solutions when R0>1 and c ∈ [0, c*). In addition, we further study the non-existence of the traveling wave solutions for the model when R0<1 and c ≥ 0.

Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks

In this paper, a fractional SIR model with birth and death rates on heterogeneous complex networks is proposed. Firstly, we obtain a threshold value R0 based on the existence of endemic equilibrium point E∗, which completely determines the dynamics of the model. Secondly, by using Lyapunov function and Kirchhoff’s matrix tree theorem, the globally asymptotical stability of the disease-free equilibrium point E0 and the endemic equilibrium point E∗ of the model are investigated. That is, when R0<1, the disease-free equilibrium point E0 is globally asymptotically stable and the disease always dies out; when R0>1, the disease-free equilibrium point E0 becomes unstable and in the meantime there exists a unique endemic equilibrium point E∗, which is globally asymptotically stable and the disease is uniformly persistent. Finally, the effects of various immunization schemes are studied and compared. Numerical simulations are given to demonstrate the main results.

Permanence and extinction in a nonautonomous discrete SIRVS epidemic model with vaccination

In this paper, by applying a nonstandard finite difference scheme, we formulate a discretized SIRVS epidemic model which takes into account vaccination. Under quite weak assumptions, the threshold value conditions on permanence and extinction of disease are established. Some new threshold values in product forms R0* and R1* are obtained. We show that the disease is permanent if R0*>1, and if R1*<1, then the disease is extinct. When the model degenerates into a periodic model, a sharp threshold value R0 is obtained for permanence versus extinction of disease. In order to illustrate our analytic analysis, some numerical simulations are also included in the end.

Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold valueR0&lt;1, then the susceptible pest eradication periodic solution is globally asymptotically stable and ifR0&gt;1, then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractiveT-periodic solution.

Traveling waves in a Kermack–Mckendrick epidemic model with diffusion and latent period

This paper is devoted to the study of a Kermack–Mckendrick epidemic model with diffusion and latent period. We first consider the well-posedness of solutions of the model. Furthermore, using the Schauder fixed point theorem and Laplace transform, we show that if the threshold value R0>1, then there exists c∗>0 such that for every c>c∗, the model admits a traveling wave solution, and if R0<1 and c≥0; or R0>1 and c∈(0,c∗), then the model admits no traveling wave solutions. Hence, the existence and non-existence of traveling wave solutions is determined completely by R0, and the constant c∗ is the minimum speed for the existence of traveling wave solutions of the model.