Infection-free Periodic Solution Research Articles

Dynamical analysis of the pest control model with multiple delays and impulsive effects

In recent years, the suppression of disease population in pest control has gained popularity and remains a pressing issue in agricultural production. Additionally, the mathematical modeling method is often exploited to study the dynamics of different infectious diseases. A new pest control model with multiple delays and impulsive effects is developed and investigated. Firstly, the global attractivity of infection-free periodic solution of the system is proved by applying the theory of impulsive differential equation and comparative theorem. Secondly, the persistence of the infection-free periodic solution of the proposed model is also proved. Finally, the theoretical analysis results are validated by numerical simulations and some suggestions on how to better control the pests are presented.

A periodic reaction-diffusion model of hospital infection with crowding effects

In this paper, we propose a periodic reaction-diffusion model of hospital infection with crowding effects. We introduce the basic reproduction number R0 for this model and show that the infection-free periodic solution is globally asymptotically stable if R0≤1, while the system admits a globally asymptotically stable positive periodic solution if R0>1. We also obtain the asymptotic behavior of the basic reproduction number when the diffusion rates go to infinity and zero, respectively. Further, we numerically study the effects of diffusion rates, the outflow rate by patient crowding and other parameters on R0 with and without seasonality, and found that the neglect of seasonality does overestimate the infection risk and the disease may be well controlled by avoiding overcrowding of patients rather than increasing the mobility of individuals or bacteria.

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.

A diffusive virus model with a fixed intracellular delay and combined drug treatments.

The method of administration of an effective drug treatment to eradicate viruses within a host is an important issue in studying viral dynamics. Overuse of a drug can lead to deleterious side effects, and overestimating the efficacy of a drug can result in failure to treat infection. In this study, we proposed a reaction-diffusion within-host virus model which incorporated information regarding spatial heterogeneity, drug treatment, and the intracellular delay to produce productively infected cells and viruses. We also calculated the basic reproduction number [Formula: see text] under the assumptions of spatial heterogeneity. We have shown that the infection-free periodic solution is globally asymptotically stable when [Formula: see text], whereas when [Formula: see text] there is an infected periodic solution and the infection is uniformly persistent. By conducting numerical simulations, we also revealed the effects of various parameters on the value of [Formula: see text]. First, we showed that the dependence of [Formula: see text] on the intracellular delay could be monotone or non-monotone, depending on the death rate of infected cells in the immature stage. Second, we found that the mobility of infected cells or virions could facilitate the virus clearance. Third, we found that the spatial fragmentation of the virus environment enhanced viral infection. Finally, we showed that the combination of spatial heterogeneity and different sets of diffusion rates resulted in complicated viral dynamic outcomes.

Time periodic reaction–diffusion equations for modeling 2-LTR dynamics in HIV-infected patients

Considering multiple effects of spatial factors (spatial heterogeneity and mobility) and temporal heterogeneity (the periodic administration of the drug) in HIV infection and treatment, we formulate a time periodic reaction–diffusion equation model to study the HIV viral dynamics. The basic reproduction number R0 is derived and the global threshold dynamics is explored in terms of R0. When R0<1, the infection-free periodic solution is globally attractive. When R0>1, the model system is uniformly persistent. The global stability analysis is also performed in the critical case of R0=1 with spatial heterogeneity but without time periodicity. The global asymptotic stability of the infection steady state for the spatial homogeneous model is discussed. Furthermore, numerical simulations reveal that (i) the value of R0 depends on spatial heterogeneity, which indicates that different human individuals or different organs in one individual may have different infection states even if the drug efficacy is the same; and (ii) the 2-LTR level at the steady state may increase and the steady-state viral load may decrease dramatically if raltegravir is administered from the beginning of thetreatment.

On a delayed epidemic model with non-instantaneous impulses

We introduce a non-instantaneous pulse vaccination model. Non-instantaneous impulsive nonlinear differential equations provide an adequate biomathematical model of some medical problems. In this paper we study some basic properties such as the attractiveness of the infection-free periodic solution and the permanence of some sub-population for a vaccine model where a constant fraction of the susceptible population is vaccinated in some periodic way. Our model is a system of nonlinear differential equations with impulses.

Complex dynamics of a time periodic nonlocal and time-delayed model of reaction–diffusion equations for modeling CD4[formula omitted] T cells decline

Pyroptosis, an intensively inflammatory form of programmed cell death triggered during abortive HIV infection, is associated with the release of inflammatory cytokines. The inflammatory cytokines can attract more CD4 + T cells to be infected. Based on the new biological perspective, a time periodic reaction–diffusion equation model with spatial heterogeneity and latent period is developed to investigate whether or not pyroptosis can explain CD4+ T cells decline during HIV infection. Threshold dynamics is explored in terms of the basic reproduction number ℛ0. It is shown that the infection-free periodic solution is globally attractive if ℛ0<1, while there exists positive infection periodic solution and the virus is uniformly persistent if ℛ0>1, which might be a new finding for viral infection dynamical models. Theoretical analyses and simulations for the spatially homogeneous model demonstrate rich dynamics, including the occurrence of Turing instability, which may result in Hopf bifurcation and spatially inhomogeneous pattern formations. It turns out that the new model issues some new challenges due to the enhancement infection term in the global stability problem and Turing instability analysis for the high dimensional system. Our results reveal that the inflammatory cytokines can make the CTLs level increase, which is a new phenomenon not presented in the existing literature.

Effect of pulse vaccination on dynamics of dengue with periodic transmission functions

Pulse vaccination is a repeated vaccination policy, which plays a tremendous role in the global fight against communicable diseases in terms of saving medical resources and decreasing the economic burden. In this article, we propose a dynamic model of dengue infection with periodic transmission functions and seasonality in vector population. Furthermore, we introduce a pulse vaccination strategy in the susceptible host population to examine how frequency and intensity of implementation of this strategy affect the dynamics of dengue infection. We successfully obtained the threshold dynamics by defining the basic reproduction number R0, which is the spectral radius of the next generation operator and governs whether the disease dies out or not. It has been established that the infection-free periodic solution of the proposed impulsive system is globally asymptotically stable if mbox{R}_{0}<1 and is unstable otherwise. Moreover, we found that the dengue infection is uniformly persistent for the proposed system if mbox{R}_{0}>1. Finally, we execute the system numerically to illustrate the piecewise solutions of the proposed system with impulsive vaccination measure and to investigate the influence of different control parameters on the basic reproduction. The finding indicates that a frequent implementation of the vaccination strategy with great intensity and the use of mosquito nets can essentially lead to a decline of new infections.

一类具有脉冲免疫治疗的HIV-1感染模型的动力学分析

Based on a class of HIV-1 infection immunotherapy models, an HIV-1 infection model with pulsed immunotherapy was addressed. The non-negativity and uniform boundedness of the solutions to the pulsed immunotherapy model were studied with the pulsed differential equation theory. According to the Floquet multiplier theory and the comparison theorem for differential equations, the threshold conditions for the local and global asymptotic stability of the infection-free periodic solution as well as uniform persistence of HIV-1 were obtained. Through numerical simulation, the therapeutic effects of 3 different treatment regimens were compared, and the effectiveness of the pulsed immunotherapy was verified. The numerical results show that, the virus can be effectively controlled or eliminated theoretically when the drug input is large enough or the dosing interval is short properly.

Dynamics of cholera epidemics with impulsive vaccination and disinfection

Waterborne diseases have a tremendous influence on human life. The contaminated drinking water causes water-borne disease like cholera. Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. A waterborne disease like cholera can also be controlled by using impulse technique. In this paper, we have proposed a delayed SEIRB epidemic model with impulsive vaccination and disinfection. We have studied the pulse vaccination strategy and sanitation to control the cholera disease. The existence and stability of the disease-free and endemic periodic solution are investigated both analytically and numerically. It is shown that there exists an infection-free periodic solution, using the impulsive dynamical system defined by the stroboscopic map. It is observed that the infection-free periodic solution is globally attractive when the impulse period is less than some critical value. From the analysis of the model, we have obtained a sufficient condition for the permanence of the epidemic with pulse vaccination. The main highlight of this paper is to introduce impulse technique along with latent period into the SEIRB epidemic model to investigate the role of pulse vaccination and disinfection on the dynamics of the cholera epidemics.

Dynamics of a delayed SEIR epidemic model with pulse vaccination and restricting the infected dispersal

In this work, we propose a delayed SEIR epidemic model with pulse vaccination and restricting the infected dispersal. By the stroboscopic map of the discrete dynamical system, we obtain infection-free boundary periodic solution. Further, we prove that the infection-free boundary periodic solution is globally attractive. By the theory on the delay and impulsive differential equation, we prove that the investigated system is permanent. Our results indicate that the time delay, pulse vaccination and impulsive dispersal have influence to the dynamical behaviors of the investigated system. https://doi.org/10.28919/cmbn/3383

Dynamics of an SIR epidemic model with stage structure and pulse vaccination

Pulse vaccination is an important strategy to eradicate an infectious disease. In this paper, we investigate an SIR epidemic model with stage structure and pulse vaccination. By using the discrete dynamical system determined by stroboscopic map, we obtain the conditions for the global asymptotical stability of the infection-free periodic solution of the studied system. The permanent conditions of the investigated system are also given. The results indicate that a large pulse vaccination rate is a sufficient condition to eradicate the disease. It provides a reliable tactic basis for preventing the epidemic outbreak.

A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination

ABSTRACTIn this paper, we have considered a dynamical model of diseases that spread by droplet infection and also through direct contact with varying total population size and discrete time delay to become infectious. It is assumed that there is a latent period of the disease and an immune period of the recovered individuals. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analysed this model with pulse vaccination and saturation incidence rate. It is also assumed that the time lag due to lose of immunity of recovered individuals is equal to the interval between two pulses. We have defined two positive numbers and . It is proved that there exists an infection-free periodic solution which is globally attractive if and the disease is permanent if The important mathematical findings for the dynamical behaviour of the model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

Nonsynchronous bifurcation of SIRS epidemic model with birth pulse and pulse vaccination

In this paper, we construct an SIRS epidemic model with birth pulse and pulse vaccination at the different fixed moment. The stability of the infection-free periodic solution is obtained by using the Poincare map. The existence of nontrivial periodic solution bifurcated from the infection-free periodic solution is discussed by means of the bifurcation theory. It is shown that once a threshold is reached, a nontrivial periodic solution emerges via a supercritical bifurcation. Furthermore, some numerical simulations are given, which are in good accordance with the theoretical results.

Analysis of a delayed epidemic model of diseases through droplet infection and direct contact with pulse vaccination

In this paper, we have considered a dynamical model of diseases that spread by droplet infection and also through direct contact. It is assumed that there is a time lag due to incubation period of pathogens, i.e. the development of an infection from the time the pathogen enters the body until signs or symptoms first appear. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination and saturation incidence rate. We have defined two positive numbers \(R_{1}\) and \(R_{2}\). It is proved that there exists an infection-free periodic solution which is globally attractive if \(R_{1} 1.\) The important mathematical findings for the dynamical behaviour of the model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

A delayed hand–foot–mouth disease model with pulse vaccination strategy

In this paper, we have considered a dynamical model of hand–foot–mouth disease (HFMD) with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a latent period of the disease and an immune period of the recovered individuals. It is reported that the first vaccine to protect children against enterovirus 71 or EV71 has been invented (Zhang and Teng, J Biol Dyn 2(1):64-84, 2008). Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. It is also assumed that the time lag due to lose of immunity of recovered individuals is equal to the interval between two pulses. We have defined two positive numbers $$R_{1}$$ and $$R_{2}$$ . It is proved that there exists an infection-free periodic solution which is globally attractive if $$R_{1}<1$$ and the disease is permanent if $$R_{2}>1$$ . It is derived that the infectious population extinct if the pulse vaccination rate is larger than a critical vaccination proportion. Our analysis indicates that a small pulse vaccination rate will rise the permanence of the disease. The important mathematical findings for the dynamical behaviour of the HFMD model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

Mathematical analysis of a Chlamydia epidemic model with pulse vaccination strategy.

In this paper, we have considered a dynamical model of Chlamydia disease with varying total population size, bilinear incidence rate and pulse vaccination strategy. We have defined two positive numbers R₀ and (R₁≤ R₀). It is proved that there exists an infection-free periodic solution which is globally attractive if R₀ < 1 and the disease is permanent if R₁> 1 The important mathematical findings for the dynamical behaviour of the Chlamydia disease model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

Analysis of a delayed epidemic model with pulse vaccination

In this paper, we have considered a dynamical model of infectious disease that spread by asymptomatic carriers and symptomatically infectious individuals with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a time lag (τ) to account for the fact that an individual infected with bacteria or virus is not infectious until after some time after exposure. The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for 0⩽t⩽τ and value zero for t>τ. The probability that an individual in the latency period has survived is given by e-μτ, where μ denotes the natural mortality rate in all epidemiological classes. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. We have defined two positive numbers R1 and R2. It is proved that there exists an infection-free periodic solution which is globally attractive if R1<1 and the disease is permanent if R2>1. The important mathematical findings for the dynamical behaviour of the infectious disease model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

A delayed SEIRS epidemic model with impulsive vaccination and nonlinear incidence rate

In this paper, a delayed SEIRS epidemic model with nonlinear incidence rate and impulsive vaccination is investigated. In vaccination strategy, we perform impulsive vaccination of newborn infants. Using the discrete dynamic system determined by stroboscopic map, we obtain an infection-free periodic solution and establish conditions, on which the solution is globally attractive. We also conclude that the disease is permanent if the parameters of the model satisfy appropriate conditions. Finally, we illustrate the effectiveness of our theorems with numerical simulation. The results obtained in this paper are a good extension of the results obtained in [J. Hou and Z. Teng, Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rate, Math. Comput. Simulat.79 (2009) 3038–3054] to the corresponding delayed SEIRS epidemic model with nonlinear incidence rate and impulsive vaccination.

Analysis of a delayed hand–foot–mouth disease epidemic model with pulse vaccination

In this paper, we have considered a dynamical model of hand–foot–mouth disease (HFMD) with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a time lag (τ) to account for the fact that an individual infected with virus is not infectious until after some time after exposure. The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for 0≤t<τ and value zero for t>τ. The probability that an individual in the latency period has survived is given by e−μ τ, where μ denotes the natural mortality rate in all epidemiological classes. It is reported that the first vaccine to protect children against enterovirus 71, or EV71 has been discovered [Zhu, F. C., Meng, F. Y., Li, J. X., Li, X. L., Mao, A. Y., Tao, H., …, Shen, X. L. (2013, May 29). Efficacy, safety, and immunology of an inactivated alum-adjuvant enterovirus 71 vaccine in children in China: A multicentre, randomised, double-blind, placebo-controlled, phase 3 trial. The Lancet, 381, 2024–2032. doi:10.1016/S0140-6736(13)61049-1]. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. We have defined two positive numbers R1 and R2. It is proved that there exists an infection-free periodic solution which is globally attractive if R1<1 and the disease is permanent if R2>1. The important mathematical findings for the dynamical behavior of the HFMD model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.