Filippov Model Research Articles

A joint-threshold Filippov model describing the effect of intermittent androgen-deprivation therapy in controlling prostate cancer

Intermittent androgen-deprivation therapy (IADT) can be beneficial to delay the occurrence of treatment resistance and cancer relapse compared to the standard continuous therapy. To study the effect of IADT in controlling prostate cancer, we developed a Filippov prostate cancer model with a joint threshold function: therapy is implemented once the total population of androgen-dependent cells (AC-Ds) and androgen-independent cells (AC-Is) is greater than the threshold value ET, and it is suspended once the population is less than ET. As the parameters vary, our model undergoes a series of sliding bifurcations, including boundary node, focus, saddle, saddle-node and tangency bifurcations. We also obtained the coexistence of one, two or three real equilibria and the bistability of two equilibria. Our results demonstrate that the population of AC-Is can be contained at a predetermined level if the initial population of AC-Is is less than this level, and we choose a suitable threshold value.

Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy

A nonsmooth Filippov avian-only influenza model with threshold strategies of culling susceptible and/or infected birds under different conditions is proposed to control the spread of avian influenza. The time delay representing the incubation period of avian influenza is incorporated into our model to make it more realistic, which is different from the traditional Filippov model. The stability of various types of equilibria and the existence of Hopf bifurcation are researched. Moreover, the existence of the sliding mode and its dynamics are investigated by Filippov convexity method. The theoretical analysis and numerical simulations indicate that, according to the value of thresholds and time delay, all solutions eventually converge to the regular equilibrium, pseudoequilibrium or stable periodic solution. We also indicate that time delay has a great influence on the sliding mode. Due to the many factors involved, and our main purpose is to study the impact of time delay on system stability, we have only conducted a simple analysis of the impact of time delay on the sliding mode. Furthermore, the boundary bifurcation switching stable regular equilibrium or stable limit cycle to a stable pseudoequilibrium can be exhibited by numerical simulations. Finally, with the increase of time delay, the global bifurcations from grazing bifurcation to buckling bifurcation and then to cross bifurcation are obtained. Our results indicate that although Filippov control strategies can effectively control the number of infected birds in many cases, the existence of time delay may challenge influenza control by the emergence of buckling bifurcation and cross bifurcation.

Stability analysis of Filippov prey–predator model with fear effect and prey refuge

Mathematical ecosystems play a crucial role in our comprehension and conservation of ecology. Within these ecosystems, prey exhibits protective instincts that compel refuging behaviors to avoid predation risk. When the ratio of prey to predators falls below a threshold, prey seeks refuge. However, when prey is abundant relative to predators, these protective instincts are overridden as prey ventures out to forage. Therefore, this study develops a Filippov prey–predator model with fear effect on prey and switching of prey refuge behavior based on the ratio of prey to predators. Analytical and numerical approaches are used to address the dynamic behaviors, bifurcation sets, existence, and stability of various equilibria in this model. Additionally, the regions of sliding and crossing segments are analyzed. The bifurcation sets of pseudo-equilibrium and local and global sliding bifurcations are investigated. The numerical simulations are conducted to investigate the interplay between fear factor and other relevant parameters within the Filippov model, such as the threshold ratio and prey refuge. These investigations shed light on the influence of them in the model. The results indicate that increasing the fear factor results in a decrease in both prey and predator densities, thereby changing the behavior of the dynamics from a limit cycle oscillation to a stable state and vice versa. Notably, despite these population changes, neither species faces complete extinction.

Qualitative analysis of a Filippov wild-sterile mosquito population model with immigration.

Effectively combating mosquito-borne diseases necessitates innovative strategies beyond traditional methods like insecticide spraying and bed nets. Among these strategies, the sterile insect technique (SIT) emerges as a promising approach. Previous studies have utilized ordinary differential equations to simulate the release of sterile mosquitoes, aiming to reduce or eradicate wild mosquito populations. However, these models assume immediate release, leading to escalated costs. Inspired by this, we propose a non-smooth Filippov model that examines the interaction between wild and sterile mosquitoes. In our model, the release of sterile mosquitoes occurs when the population density of wild mosquitoes surpasses a specified threshold. We incorporate a density-dependent birth rate for wild mosquitoes and consider the impact of immigration. This paper unveils the complex dynamics exhibited by the proposed model, encompassing local sliding bifurcation and the presence of bistability, which entails the coexistence of regular equilibria and pseudo-equilibria, as crucial model parameters, including the threshold value, are varied. Moreover, the system exhibits hysteresis phenomena when manipulating the rate of sterile mosquito release. The existence of three types of limit cycles in the Filippov system is ruled out. Our main findings indicate that reducing the threshold value to an appropriate level can enhance the effectiveness of controlling wild insects. This highlights the economic benefits of employing SIT with a threshold policy control to impede the spread of disease-carrying insects while bolstering economic outcomes.

Threshold control strategy for a Filippov model with group defense of pests and a constant-rate release of natural enemies.

In this paper, we establish an integrated pest management Filippov model with group defense of pests and a constant rate release of natural enemies. First, the dynamics of the subsystems in the Filippov system are analyzed. Second, the dynamics of the sliding mode system and the types of equilibria of the Filippov system are discussed. Then the complex dynamics of the Filippov system are investigated by using numerical analysis when there is a globally asymptotically stable limit cycle and a globally asymptotically stable equilibrium in two subsystems, respectively. Furthermore, we analyze the existence region of a sliding mode and pseudo equilibrium, as well as the complex dynamics of the Filippov system, such as boundary equilibrium bifurcation, the grazing bifurcation, the buckling bifurcation and the crossing bifurcation. These complex sliding bifurcations reveal that the selection of key parameters can control the population density no more than the economic threshold, so as to prevent the outbreak of pests.

Sliding dynamics and bifurcations of a human influenza system under logistic source and broken line control strategy.

This paper proposes a non-smooth human influenza model with logistic source to describe the impact on media coverage and quarantine of susceptible populations of the human influenza transmission process. First, we choose two thresholds $ I_{T} $ and $ S_{T} $ as a broken line control strategy: Once the number of infected people exceeds $ I_{T} $, the media influence comes into play, and when the number of susceptible individuals is greater than $ S_{T} $, the control by quarantine of susceptible individuals is open. Furthermore, by choosing different thresholds $ I_{T} $ and $ S_{T} $ and using Filippov theory, we study the dynamic behavior of the Filippov model with respect to all possible equilibria. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or bistability endemic equilibria under some conditions. The regular/virtulal equilibrium bifurcations are also given. Lastly, numerical simulation results show that choosing appropriate threshold values can prevent the outbreak of influenza, which implies media coverage and quarantine of susceptible individuals can effectively restrain the transmission of influenza. The non-smooth system with logistic source can provide some new insights for the prevention and control of human influenza.

Complex dynamics in a Filippov pest control model with group defense

In this paper, an integrated pest management Filippov model with group defense behavior is established, which takes the population density of pests as the control index of integrated pest management. First, under the condition that both subsystems have a globally asymptotically stable equilibrium, the dynamics of the established model are systematically analyzed, including the sliding mode dynamics, the existence and global stability of the real, virtual and pseudo equilibrium, as well as boundary-node and boundary-focus bifurcation. Next, we study the complex dynamics in the Filippov model when an unstable node (focus) or a stable limit cycle occurs in the subsystem by using numerical simulations. The results show that although there are no closed orbits in subsystem, a stable periodic solution may exist for Filippov system after switching perturbation. Finally, we conclude that the group defense behavior of pest makes it harder to control.

Rich dynamics of a Filippov avian-only influenza model with a nonsmooth separation line

Depopulation of birds has been authenticated to be an effective measure in controlling avian influenza transmission. In this work, we establish a Filippov avian-only model incorporating a threshold policy control. We choose the index—the maximum between the infected threshold level I_{T} and the product of the number of susceptible birds S and a ratio threshold value ξ—to decide on whether to trigger the control measures or not, which then leads to a discontinuous separation line and two pieces of sliding-mode domains. Meanwhile, one more sliding-mode domain gives birth to more complex dynamics. We investigate the global dynamical behavior of the Filippov model, including the real and/or virtual equilibria and the two sliding modes and their dynamics. The solutions will eventually stabilize at the real endemic equilibrium of the subsystem or the pseudoequilibria on the two sliding modes due to different threshold values. Therefore an effective and efficient threshold policy is essential to control the influenza by driving the number of infected birds below a certain level or at a previously given level.

Joint impacts of media, vaccination and treatment on an epidemic Filippov model with application to COVID-19

A non-smooth SIR Filippov system is proposed to investigate the impacts of three control strategies (media coverage, vaccination and treatment) on the spread of an infectious disease. We synthetically consider both the number of infected population and its changing rate as the switching condition to implement the curing measures. By using the properties of the Lambert W function, we convert the proposed switching condition to a threshold value related to the susceptible population. The classical epidemic model involving media coverage, linear functions describing injecting vaccine and treatment strategies is examined when the susceptible population exceeds the threshold value. In addition, we consider another SIR model accompanied with the vaccination and treatment strategies represented by saturation functions when the susceptible population is smaller than the threshold value. The dynamics of these two subsystems and the sliding domain are discussed in detail. Four types of local sliding bifurcation are investigated, including boundary focus, boundary node, boundary saddle and boundary saddle-node bifurcations. In the meantime, the global bifurcation involving the appearance of limit cycles is examined, including touching bifurcation, homoclinic bifurcation to the pseudo-saddle and crossing bifurcation. Furthermore, the influence of some key parameters related to the three treatment strategies is explored. We also validate our model by the epidemic data sets of A/H1N1 and COVID-19, which can be employed to reveal the effects of media report and existing strategy related to the control of emerging infectious diseases on the variations of confirmed cases.

A two-thresholds policy for a Filippov model in combating influenza

This work designs a two-thresholds policy for a Filippov model in combating influenza, so as to estimate when and whether to take control strategies, including the media coverage, antiviral treatment of infected individuals and vaccination of susceptible population. By introducing two tolerance thresholds [Formula: see text] and [Formula: see text] of susceptible and infected individuals, the two-thresholds policy is designed as: a vaccination program is implemented when the number of susceptible individuals is above [Formula: see text]; an antiviral treatment strategy is taken and the mass media begins to report information about influenza when the infection number is larger than [Formula: see text]; no control strategies are required in other cases. Furthermore, the global dynamics of the model are analyzed by varying these two thresholds, including the existence and dynamics of sliding mode, and the existence and global stability of equilibrium. It is shown that the model solutions ultimately converge to a pseudoequilibrium or a pseudoattractor on the switching surface, or a real equilibrium. The obtained results indicate that, by choosing susceptible and infected thresholds properly, the infection number can be remained below or at an acceptable level.

Role of triaxiality parameter γ in deformed nuclei

The well known, one-parameter Davydov–Filippov model (DFM) or its variants have been used often to study the inter band [Formula: see text]-g B(E2) ratios for the triaxially deformed nuclei. Here, its application for the axially deformed nuclei with [Formula: see text] less than [Formula: see text] is illustrated, which has not received attention so far. Instead of the usual practice of studying the gamma-ground B(E2) ratios, we focus on the absolute B(E2) values in DFM. It offers a wholly new role of the [Formula: see text]-variable in explaining the spectral features of the axially deformed nuclei. Further, it illustrates the intimate relationship of the [Formula: see text] variable and the quadrupole deformation variable [Formula: see text]. Additionally, a unified view of the various differing patterns of the variation of the [Formula: see text]-g B(E2) ratios versus [Formula: see text] is presented.

Threshold Strategy for Nonsmooth Filippov Stage-Structured Pest Growth Models

In order to control pests and eventually maintain the number of pests below the economic threshold, in this paper, based on the nonsmooth dynamical system, a two-stage-structured pest control Filippov model is proposed. We take the total number of juvenile and adult pest population as the control index to determine whether or not to implement chemical control strategies. The sliding-mode domain and conditions for the existence of regular and virtual equilibria, pseudoequilibrium, boundary equilibria, and tangent points are given. Further, the sufficient condition of the locally asymptotic stability of pseudoequilibrium is obtained. By numerical simulations, the local bifurcations of the equilibria are discussed. Our results show that the total number of pest populations can be successfully controlled below the economic threshold by taking suitable threshold policy.

Using non-smooth models to determine thresholds for microbial pest management.

Releasing infectious pests could successfully control and eventually maintain the number of pests below a threshold level. To address this from a mathematical point of view, two non-smooth microbial pest-management models with threshold policy are proposed and investigated in the present paper. First, we establish an impulsive model with state-dependent control to describe the cultural control strategies, including releasing infectious pests and spraying chemical pesticide. We examine the existence and stability of an order-1 periodic solution, the existence of order-k periodic solutions and chaotic phenomena of this model by analyzing the properties of the Poincaré map. Secondly, we establish and analyze a Filippov model. By examining the sliding dynamics, we investigate the global stability of both the pseudo-equilibria and regular equilibria. The findings suggest that we can choose appropriate threshold levels and control intensity to maintain the number of pests at or below the economic threshold. The modelling and control outcomes presented here extend the results for the system with impulsive interventions at fixed moments.

A Filippov model describing the effects of media coverage and quarantine on the spread of human influenza

Mass-media reports on an epidemic or pandemic have the potential to modify human behaviour and affect social attitudes. Here we construct a Filippov model to evaluate the effects of media coverage and quarantine on the transmission dynamics of influenza. We first choose a piecewise smooth incidence rate to represent media reports being triggered once the number of infected individuals exceeds a certain critical level Ic1. Further, if the number of infected cases increases and exceeds another larger threshold value Ic2 (>Ic1), we consider that the incidence rate tends to a saturation level due to the protection measures taken by individuals; meanwhile, we begin to quarantine susceptible individuals when the number of susceptible individuals is larger than a threshold value Sc. Then, for each susceptible threshold value Sc, the global properties of the Filippov model with regard to the existence and stability of all possible equilibria and sliding-mode dynamics are examined, as we vary the infected threshold values Ic1 and Ic2. We show generically that the Filippov system stabilizes at either the endemic equilibrium of the subsystem or the pseudoequilibrium on the switching surface or the endemic equilibrium Ec=(Sc,Ic2), depending on the choice of the threshold values. The findings suggest that proper combinations of infected and susceptible threshold values can maintain the number of infected individuals either below a certain threshold level or at a previously given level.

Dynamics of a Filippov epidemic model with limited hospital beds.

A Filippov epidemic model is proposed to explore the impact of capacity and limited resources of public health system on the control of epidemic diseases. The number of infected cases is chosen as an index to represent a threshold policy, that is, the capacity dependent treatment policy is implemented when the case number exceeds a critical level, and constant treatment rate is adopted otherwise. The proposed Filippov model exhibits various local sliding bifurcations, including boundary focus or node bifurcation, boundary saddle bifurcation and boundary saddle-node bifurcation, and global sliding bifurcations, including grazing bifurcation and sliding homoclinic bifurcation to pseudo-saddle. The impact of some key parameters including the threshold level on disease control is examined by numerical analysis. Our results suggest that strengthening the basic medical conditions, i.e. increasing the minimum treatment ratio, or enlarging the input of medical resources, i.e. increasing HBPR (i.e. hospital bed-population ratio) as well as the possibility and level of maximum treatment ratio, can help to contain the case number at a relatively low level when the basic reproduction number $R₋0>1$. If $R₋0<1$, implementing these strategies can help in eradicating the disease although the disease cannot always be eradicated due to the occurring of backward bifurcation in the system.

An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza.

Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level [Formula: see text], we derive the phase portrait for the dynamical system as we vary the infected threshold level [Formula: see text], focusing on the existence of endemic states; the endemic states are represented by real equilibria, pseudoequilibria and pseudo-attractors. We show generically that all solutions of this model will approach one of the endemic states. Our results suggest that the spread of avian influenza in bird populations is tolerable if the trajectories converge to the equilibrium point that lies in the region below the threshold level [Formula: see text] or if they converge to one of the pseudoequilibria or a pseudo-attractor on the surface of discontinuity. However, we have to cull birds whenever the solution of this model converges to an equilibrium point that lies in the region above the threshold level [Formula: see text] in order to control the outbreak. Hence a good threshold policy is required to combat bird flu successfully and to prevent overkilling birds.

Transition probabilities in negative parity bands of the 119I nucleus

Lifetimes in four negative-parity bands of 119I were measured using DSAM and RDM. 119I nuclei were produced in the 109Ag(13C,3n) reaction, γγ coincidences were collected using the NORDBALL array. RDDSA — a new method of RDM analysis — is described. This method allowed for the self-calibration of stopping power. From 31 measured lifetimes, 39 values of B(E2) were established. Calculations in the frame of the Core Quasi Particle Coupling (CQPC) model were focused on the problem of susceptibility of the nucleus to γ-deformation. It was established that nonaxial quadrupole deformation of 119I plays on important role. The Wilets–Jean model of a γ-soft nucleus describes the 119I nucleus in a more consistent way then the Davydov–Filippov model of a γ-rigid nucleus.

Low-lying collective bands in the even tungsten isotopes 180–186W

The γ-rigid model has been used to study the energy levels and the electromagnetic properties of 180, 182, 184, 186W. In this model, the intrinsic wave functions are obtained using the pairing plus the quadrupole–quadrupole interaction Hamiltonian of Baranger and Kumar. Good angular momentum states are projected approximately from such a triaxially symmetric intrinsic wave function. This model assumes the nucleus to be γ rigid but soft in the β degrees of freedom. The asymmetry parameter γ for a given nucleus is extracted using the experimental energies of the first 2+ and second 2+ states within the framework of the Davydov–Filippov model. The symmetry parameter β for each J state is determined from the minimization of the projected energy. The calculated energy levels of the ground and the 7 band, the B(E2) values, the electromagnetic moments, the E2 and E4 matrix elements, and the B(E2) ratios agree quite well with experimental results.

ON THE AXIAL SYMMETRY OF THE NILSSON MODEL NUCLEUS

The shape of the oscillator potential necessary to give the minimum energy for a systsm of single fermions in an anisotropic oscillator with Nilsson spin-orbit coupling is calculated. The number of particles is limited to 50 or less and is even. The preferred distortion parameters ( beta and gamma ) are plotted vs. number of particles, and energy maps (energy vs. beta ) are given for various amounts of spin-orbit coupling. Implications of the calculations, e.g., that the Bohr strong coupling approximation and the Davydov- Filippov model cannot easily be justified, are discussed. (D.L.C.)