Disease-free Periodic Solution Research Articles

DYNAMICS OF AN IMPULSIVE STOCHASTIC SIR EPIDEMIC MODEL WITH SATURATED INCIDENCE RATE

In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.

Analysis of a vector-borne disease model with impulsive perturbation and reinfection

We formulate a mathematical model for vector-borne disease with impulsive perturbation based on indoor residual spraying. The dynamical properties are studied theoretically and numerically. It is shown that with consideration of impulsive spraying at fixed times, there exists a disease-free periodic solution that is locally stable when the threshold $${\mathcal {R}}_0$$ is less than unity, otherwise the disease is uniformly persistent if $${\mathcal {R}}_0>1$$. Furthermore, the bifurcation analysis is performed, revealing the possible existence of nontrivial periodic solution bifurcated from the disease-free periodic solution at $${\mathcal {R}}_0=1$$. In addition to simulations of parameter sensitivity, when implementing impulsive spraying once the number of infected humans exceeds a threshold level, the effectiveness of such state-dependent control is also conducted numerically.

A two-group age of infection epidemic model with periodic behavioral changes

In this paper we propose a two-group SIR age of infection epidemic model by incorporating periodical behavioral changes for both susceptible and infected individuals. Our model allows different incubation periods for the two groups. It is proved in this paper that the persistence and extinction of the disease are determined by a threshold condition given in term of the basic reproductive number $ R_0 $. That is, the disease is uniformly persistent if $ R_0 >1 $ with the existence of a positive periodic solution, while the disease goes to extinction if $ R_0< 1 $ with the global asymptotic stability of the disease free periodic solution. The model we have proposed is general and can be applied to a wide class of diseases.

A West Nile Virus Model with Vertical Transmission and Periodic Time Delays

Seasonal change has played a critical role in the evolution dynamics of West Nile virus transmission. In this paper, we formulate and analyze a novel delay differential equation model, which incorporates seasonality, the vertical transmission of the virus, the temperature-dependent maturation delay and the temperature-dependent extrinsic incubation period in mosquitoes. We first introduce the basic reproduction ratio $$R_0$$ for this model and then show that the disease is uniformly persistent if $$R_0>1$$. It is also shown that the disease-free periodic solution is attractive if $$R_0<1$$, provided that there is only a small invasion. In the case where all coefficients are constants and the disease-induced death rate of birds is zero, we establish a threshold result on the global attractivity in terms of $$R_0$$. Numerically, we study the West Nile virus transmission in Orange County, California, USA.

Dynamical Behavior and Bifurcation Analysis of the SIR Model with Continuous Treatment and State-Dependent Impulsive Control

In this study, we propose a state-dependent impulsive model describing the susceptible individuals-triggered interventions. We find that the model with susceptible individuals-guided impulsive interventions can exhibit very complex dynamical behaviors with rich biological meanings. We note that this formulated impulsive model has disease-free periodic solution, and we can investigate the threshold dynamics by defining the control reproduction number. We study the existence and stability of the disease-free periodic solution (DFPS) for [Formula: see text]. Our results show that, even if the basic reproduction number [Formula: see text], the DFPS can still be stable when the threshold level of susceptible population [Formula: see text], indicating that with a proper chosen [Formula: see text], the state-dependent impulsive strategy can effectively control the development of the infectious disease and eradicate the disease eventually. By employing the bifurcation theory, we investigate the bifurcation phenomenon near the DFPS with respect to some key parameters, and observe that a positive order-1 periodic solution can bifurcate from the DFPS via a transcritical bifurcation. By utilizing numerical simulation, we further explore the existence and stability of the positive order-[Formula: see text] periodic solutions, and found the feasibility of stable positive order-1, order-2 and order-3 periodic solutions, that imply the existence of chaos. In particular, we find that there can be three positive order-1 periodic solutions simultaneously, in which one is stable and the other two are unstable. Our finding indicates that the comprehensive strategy combining continuous treatment with state-dependent impulsive vaccination and isolation plays a crucial role in controlling the prevalence and further spread of the infectious diseases.

Global dynamical analysis of plant-disease models with nonlinear impulsive cultural control strategy.

To eradicate plant diseases and maintain the number of infected plants below an economic threshold, two impulsive plant-disease models with periodic and state-dependent nonlinear cultural control are established. We focus on saturated nonlinear roguing (identifying and removing infected plants), with three situations for healthy plants: constant replanting, proportional replanting and proportional incidental removal. The global dynamics of the model with periodic impulsive effects are investigated. We establish conditions for the existence and stability of the disease-free periodic solution, the existence of a positive periodic solution and permanence. Latin Hypercube Sampling is used to perform a sensitivity analysis on the threshold of disease extinction to determine the significance of each parameter. Disease extinction will follow from increasing the harvesting rate, increasing the intervention period or decreasing the replanting number. The global behavior of the model with an economic threshold is established, including the existence and global stability of periodic solutions. The results imply that the control methods have an important effect on disease development, and the density-dependent parameter can decelerate extinction and accelerate the growth of healthy plants. These findings suggest that we can successfully eradicate the disease or maintain the infections below a certain level under suitable control measures. The analytic methods developed here provide a general framework for exploring plant-disease models with nonlinear impulsive control strategies.

Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period

Coinfection of hosts with multiple strains or serotypes of the same agent, such as different influenza virus strains, different human papilloma virus strains, and different dengue virus serotypes, is not only a very serious public health issue but also a very challenging mathematical modeling problem. In this paper, we study a time-periodic two-strain SIS epidemic model with diffusion and latent period. We first define the basic reproduction number R0i and introduce the invasion number Rˆ0i for each strain i(i=1,2), which can determine the ability of each strain to invade the other single-strain. The main question that we investigate is the threshold dynamics of the model. It is shown that if R0i⩽1(i=1,2), then the disease-free periodic solution is globally attractive; if R0i>1⩾R0j(i≠j,i,j=1,2), then competitive exclusion, where the jth strain dies out and the ith strain persists, is a possible outcome; and if Rˆ0i>1(i=1,2), then the disease persists uniformly. Finally we present the basic framework of threshold dynamics of the system by using numerical simulations, some of which are different from that of the corresponding multi-strain SIS ODE models.

Modeling the Potential Role of Engineered Symbiotic Bacteria in Malaria Control.

Recent experimental study suggests that the engineered symbiotic bacteria Serratia AS1 may provide a novel, effective and sustainable biocontrol of malaria. These recombinant bacteria have been shown to be able to rapidly disseminate throughout mosquito populations and to efficiently inhibit development of malaria parasites in mosquitoes in controlled laboratory experiments. In this paper, we develop a climate-based malaria model which involves both vertical and horizontal transmissions of the engineered Serratia AS1 bacteria in mosquito population. We show that the dynamics of the model system is totally determined by the vector reproduction ratio [Formula: see text], and the basic reproduction ratio [Formula: see text]. If [Formula: see text], then the mosquito-free state is globally attractive. If [Formula: see text] and [Formula: see text], then the disease-free periodic solution is globally attractive. If [Formula: see text] and [Formula: see text], then the positive periodic solution is globally attractive. Numerically, we verify the obtained analytic result and evaluate the effects of releasing the engineered Serratia AS1 bacteria in field by conducting a case study for Douala, Cameroon. We find that ideally, by using Serratia AS1 alone, it takes at least 25 years to eliminate malaria from Douala. This implies that continued long-term investment is needed in the fight against malaria and confirms the necessity of integrating multiple control measures.

Modeling hantavirus infections in mainland China

Since the repeated outbreaks of hantavirus infections have been reported in mainland China, especially in Shannxi province, they have led to public concern and controlling the spread of hantavirus becomes challenging. To understand the transmission dynamics of hantaviruses and identify the key factors that significantly affect hemorrhagic fever with renal syndrome (HFRS) infections, we propose a periodic dynamic model, which representing seasonal variation of mice population and related transmission. The model does include the dynamics of field mice and house mice with their contaminated environments and the dynamics of human, in which indirect transmission between human and mice via contaminated environment is involved. We analyzed the periodic system and examined the threshold dynamics. We obtained the basic reproduction number R0 for the periodic system, and proved that the disease-free periodic solution is globally asymptotically stable if R0 < 1 and the hantavirus infections uniformly persist in human population for R0 > 1. By fitting data on HFRS cases in the province of Shaanxi from 2006 to 2012 to our proposed model, we were able to obtain estimates of the unknown parameter values, we further use the case data from 2013 to 2016 to verify the formulated model. We obtain the basic reproduction numbers for HFRS infection as R0=1.01. The finding indicated that immune strategy, cleaning environment and limiting contact with mice are important measures to reduce HFRS infections.

Analysis of a SIR model with pulse vaccination and temporary immunity: Stability, bifurcation and a cylindrical attractor

A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical level. Using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution as the vaccination coverage is decreased and a Hopf point is detected. This leads to a bifurcation to an attracting, invariant cylinder. As the vaccination coverage is further decreased, the geometry of the cylinder contracts along its length until it finally collapses to a periodic orbit when the vaccination coverage goes to zero. In the intermediate regime, phase locking on the cylinder is observed.

Threshold Dynamics and Bifurcation of a State-Dependent Feedback Nonlinear Control Susceptible–Infected–Recovered Model1

A classic susceptible–infected–recovered (SIR) model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease-free periodic solution (DFPS) are addressed, and the threshold condition is provided, which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincaré map of the proposed model, and the main results indicate that under certain conditions, a stable or unstable interior periodic solution could be generated through transcritical, pitchfork, and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve, which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency, and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.

A periodic SEIRS epidemic model with a time-dependent latent period.

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. In order to investigate the impact of time-dependent delays on disease control, we propose an SEIRS epidemic model with a periodic latent period. We introduce the basic reproduction ratio [Formula: see text] for this model and establish a threshold type result on its global dynamics in terms of [Formula: see text]. More precisely, we show that the disease-free periodic solution is globally attractive if [Formula: see text]; while the system admits a positive periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations are also carried out to illustrate the analytic results. In addition, we find that the use of the temporal average of the periodic delay may underestimate or overestimate the real value of [Formula: see text].

Analysis of a mathematical model with nonlinear susceptibles-guided interventions.

In this paper, we considered a mathematical model describing the nonlinear susceptibles-guided vaccination and isolation strategies, incorporating the continuously saturated treatment. In this strategy, we find that the disease-free periodic solution can always exist, and consequently the control reproduction number can be defined through analyzing the stability of the disease-free periodic solution. Also, we discussed the existence and stability of the positive order-1 periodic solution from two points of view. Initially, we investigated the transcritical and pitchfork bifurcation of the Poincaré map with respect to key parameters, and proved the existence of a stable or an unstable positive order-1 periodic solution near the disease-free periodic solution. For another aspect, by studying the properties of the Poincaré map, we verified the existence of the positive order-1 periodic solution in a large range of the control parameters, especially, we verified the co-existence of finite or infinite countable different positive order-1 periodic solutions. Furthermore, numerical simulations show that the unstable order-1 periodic solution can co-exist with the stable order-1, or order-2, or order-3 periodic solution. The finding implies that the nonlinear susceptibles-triggered feedback control strategy can induce much rich dynamics, which suggests us to carefully choose key parameters to ensure the stability of the disease-free periodic solution, indicating that infectious diseases die out.

Modelling the Effects of Snail Control and Health Education in Clonorchiasis Infection in Foshan, China

Clonorchiasis is the most important food‐borne parasitic disease in China. In this paper, a mathematical model of clonorchiasis sinensis is proposed to mimic its transmission dynamics to assess the effects of intervention strategies such as snail control, health education, and chemotherapy. A threshold dynamics in terms of the basic reproductive number R0 has been established; that is, if R0 &lt; 1, the disease dies out and the disease‐free periodic solution is globally asymptotically stable, and if R0 &gt; 1, then the disease breaks out. The effects of different control measures are compared by numerical simulations. The numerical results suggest that it is necessary to strengthen health education and improve faeces management and illustrate that snail control is the most effective way to be implemented in the clonorchiasis sinensis control in Foshan.

Dynamics of a Nonautonomous Plant Disease Model with General Nonlinear Incidence Rate and Time-Varying Impulse

In this paper, for controlling the spread of plant diseases, a nonautonomous SEIS (Susceptible → Exposed → Infectious → Susceptible) epidemic model with a general nonlinear incidence rate and time-varying impulsive control strategy is proposed and investigated. This novel model could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Using the method of small amplitude perturbation, the sufficient conditions under which guarantee the globally attractive of the disease-free periodic solution and the permanence of the disease are obtained, that is, the disease dies out if R12>1.

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.

A reaction-diffusion malaria model with seasonality and incubation period.

In this paper, we propose a time-periodic reaction-diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number [Formula: see text] is derived, and it is shown that the disease-free periodic solution is globally attractive if [Formula: see text], while there is an endemic periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

A climate-based malaria model with the use of bed nets.

Insecticide-treated bed nets (ITNs) are among the most important and effective intervention measures against malaria. In order to investigate the impact of bed net use on disease control, we formulate a periodic vector-bias malaria model incorporating the juvenile stage of mosquitoes and the use of ITNs. We derive the vector reproduction ratio [Formula: see text] and the basic reproduction ratio [Formula: see text]. We show that the global dynamics of the model is completely determined by these two reproduction ratios. More precisely, the mosquito-free periodic solution is globally attractive if [Formula: see text]; the unique disease-free periodic solution is globally attractive if [Formula: see text] and [Formula: see text]; and the model admits a unique positive periodic solution and it is globally attractive if [Formula: see text] and [Formula: see text]. Numerically, we study the malaria transmission case in Port Harcourt, Nigeria. Our findings show that the use of ITNs has a positive effect on reducing [Formula: see text], and that malaria may be eliminated from this area if over 75% of the human population were to use ITNs. The simulation about the long term behavior of solutions has good agreement with the obtained analytic result. Moreover, we find that the ignorance of the vector-bias effect may result in underestimation of the basic reproduction ratio [Formula: see text]. Another notable result is that the infection risk would be underestimated if the basic reproduction ratio [Formula: see text] of the time-averaged autonomous system were used.

SWITCHING VACCINATION SCHEMES FOR VECTOR-BORNE DISEASES WITH SEASONAL FLUCTUATIONS

Theory is developed for an epidemic model of a seasonally-spreading vector-borne disease using a hybrid system framework. Applicable to diseases spread by mosquitoes (e.g., chikungunya and Zika virus via Aedes albopictus), seasonal variations in transmission are modeled using switching parameters to represent term-time forcing. The vector agent is assumed to exhibit a period of incubation upon infection, modeled using a distribution. Three hybrid control strategies are analyzed in detail: switching cohort immunization, pulse vaccination at pre-specified times, and state-dependent pulse vaccination. Methods from switched systems theory are used to derive threshold disease eradication conditions involving the model parameters; convergence of solutions to a disease-free set or periodic solution is shown. A comprehensive analysis is performed to compare and contrast the different control schemes.

A Malaria Transmission Model with Temperature-Dependent Incubation Period.

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio [Formula: see text] and establish a threshold type result on the global dynamics in terms of [Formula: see text], that is, the unique disease-free periodic solution is globally asymptotically stable if [Formula: see text]; and the model system admits a unique positive periodic solution which is globally asymptotically stable if [Formula: see text]. Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.