On avian influenza epidemic models with time delay

Exponential stability and fixed-time control of a stochastic avian influenza model with spatial diffusion and nonlocal delay

Spatial heterogeneity, random disturbances in the external environment, and the incubation period of infected individuals collectively have a significant impact on the outbreak of avian influenza. In this paper, a stochastic susceptible-infective-susceptible-infected-recovered (SI-SIR) avian influenza model is established that incorporates spatial diffusion and nonlocal delay. The existence and uniqueness of mild solutions are established by applying the Banach fixed point theorem, the truncation method, and the semigroup approach. Based on the Borel-Cantelli lemma, the mean-square exponential stability and almost sure exponential stability of the mild solution are analyzed. Additionally, in combination with the Lyapunov theory, a fixed-time control strategy is proposed to achieve stability within the desired settling time. Numerical simulations are conducted to validate the impacts of key parameters and enhance the understanding of the results of the theory.

Stochastic dynamics in a time-delayed model for autoimmunity

In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance.

Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories

Since certain species of domestic poultry and poultry are the main food source in many countries, the outbreak of avian influenza, such as H7N9, is a serious threat to the health and economy of those countries. This can be considered as the main reason for considering the preventive ways of avian influenza. In recent years, the disease has received worldwide attention, and a large variety of different mathematical models have been designed to investigate the dynamics of the avian influenza epidemic problem. In this paper, two fractional models with logistic growth and with incubation periods were considered using the Liouville-Caputo and the new definition of a nonlocal fractional derivative with the Mittag-Leffler kernel. Local stability of the equilibria of both models has been presented. For the Liouville-Caputo case, we have some special solutions using an iterative scheme via Laplace transform. Moreover, based on the trapezoidal product-integration rule, a novel iterative method is utilized to obtain approximate solutions for these models. In the Atangana-Baleanu-Caputo sense, we studied the uniqueness and existence of the solutions, and their corresponding numerical solutions were obtained using a novel numerical method. The method is based on the trapezoidal product-integration rule. Also, we consider fractal-fractional operators to capture self-similarities for both models. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law and the Mittag-Leffler function. These models were solved numerically via the Adams-Bashforth-Moulton and Adams-Moulton scheme, respectively. We have performed many numerical simulations to illustrate the analytical achievements. Numerical simulations show very high agreement between the acquired and the expected results.

Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

In this paper, we analyze a delayed SEIR epidemic model in which the latent and infected states are infective. The model has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number R 0 , is less than or equal to unity. We investigate the effect of the time delay on the stability of endemic equilibrium when R 0 > 1 . We give criteria that ensure that endemic equilibrium is asymptotically stable for all time delays and a Hopf bifurcation occurs as time delay exceeds the critical value. We give formulae for the direction of Hopf bifurcations and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold reduction for functional differential equations. Numerical simulations are presented to illustrate the analytical results.

Transmission dynamics and control strategies assessment of avian influenza A (H5N6) in the Philippines

Due to the outbreaks of Highly Pathogenic Avian Influenza A (HPAI) H5N6 in the Philippines (particularly in Pampanga and Nueva Ecija) in August 2017, there has been an increase in the need to cull the domestic birds to control the spread of the infection. However, this control method poses a negative impact on the poultry industry. In addition, the pathogenicity and transmissibility of the H5N6 in both the birds and the humans remain largely unknown which call for the necessity to develop more strategic control methods for the virus. In this study, we constructed a mathematical model for the bilinear and half-saturated incidence to compare their corresponding effect on transmission dynamics of H5N6. The simulations of half-saturated incidence model were similar to what occurred during the H5N6 outbreak (2017) in the Philippines. Instead of culling the birds, we implemented other control strategies such as non-medicinal (personal protection and poultry isolation) and medicinal (poultry vaccination) ways to prevent, reduce, and control the rate of the H5N6 virus transmission. Among the proposed control strategies, we have shown that the poultry isolation strategy is still the most effective in reducing the infected birds.

Global Dynamics of an Avian Influenza A(H7N9) Epidemic Model with Latent Period and Nonlinear Recovery Rate

An SEIR type of compartmental model with nonlinear incidence and recovery rates was formulated to study the combined impacts of psychological effect and available resources of public health system especially the number of hospital beds on the transmission and control of A(H7N9) virus. Global stability of the disease-free and endemic equilibria is determined by the basic reproduction number as a threshold parameter and is obtained by constructing Lyapunov function and second additive compound matrix. The results obtained reveal that psychological effect and available resources do not change the stability of the steady states but can indeed diminish the peak and the final sizes of the infected. Our studies have practical implications for the transmission and control of A(H7N9) virus.

AVIAN INFLUENZA A H7N9 VIRUS HAS BEEN ESTABLISHED IN CHINA

In March 2013, a novel avian-origin influenza A H7N9 virus was identified among human patients in China and a total of 124 human cases with 24 related deaths were confirmed by May 2013. From November 2013 to July 2017, H7N9 broke out four more times in China. A deterministic model is proposed to study the transmission dynamics of the avian influenza A H7N9 virus between wild and domestic birds and from birds to humans, and is applied to simulate the open data on numbers of the infected human cases and related deaths reported from March to May 2013 and from November 2013 to June 2014 by the Chinese Center for Disease Control and Prevention. The basic reproduction number [Formula: see text] is estimated and sensitivity analysis of [Formula: see text] in terms of model parameters is performed. Taking into account the fact that it broke out again from November 2014 to June 2015, from November 2015 to July 2016, and from October 2016 to July 2017, we believe that H7N9 virus has been well established in birds and will likely cause regular outbreaks in humans again in the future. Control measures for the future spread of H7N9 include (i) reducing the transmission opportunities between wild birds and domestic birds, (ii) closing or monitoring the retail live-poultry markets in the infected areas, and (iii) culling the infected domestic birds in the epidemic regions.

On the global stability of an epidemic model of computer viruses

In this paper, we study the global properties of a computer virus propagation model. It is, interesting to note that the classical method of Lyapunov functions combined with the Volterra-Lyapunov matrix properties, can lead to the proof of the endemic global stability of the dynamical model characterizing the spread of computer viruses over the Internet. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in computer virus propagation model. A numerical study of the model is also carried out to investigate the analytical results.

ANALYSIS OF AVIAN INFLUENZA A (H7N9) MODEL BASED ON THE LOW PATHOGENICITY IN POULTRY

The avian influenza A (H7N9) virus is one subtype of influenza viruses, which has previously been isolated only in birds. Recently, an outbreak of a new avian influenza (H7N9) in China has resulted in numerous infections and high mortality in the humans. The H7N9 virus is low pathogenic in poultry and high pathogenic in human and that is critically different from other avian influenza viruses. An increasing number of the new H7N9 cases and the high mortality have caused a serious global concern. Here, based on the reported data, we propose and analyze an SE-SEIS avian–human influenza model. We prove the global stability results for both the disease-free equilibrium point and the endemic equilibrium point by using a general Bendixson–Dulac theorem. Our reported theoretical results of this paper are expected to help in exploring the development of efficient methods to controlling the spread of avian influenza A(H7N9).

Nonlinear dynamics of avian influenza epidemic models

Avian influenza is a zoonotic disease caused by the transmission of the avian influenza A virus, such as H5N1 and H7N9, from birds to humans. The avian influenza A H5N1 virus has caused more than 500 human infections worldwide with nearly a 60% death rate since it was first reported in Hong Kong in 1997. The four outbreaks of the avian influenza A H7N9 in China from March 2013 to June 2016 have resulted in 580 human cases including 202 deaths with a death rate of nearly 35%. In this paper, we construct two avian influenza bird-to-human transmission models with different growth laws of the avian population, one with logistic growth and the other with Allee effect, and analyze their dynamical behavior. We obtain a threshold value for the prevalence of avian influenza and investigate the local or global asymptotical stability of each equilibrium of these systems by using linear analysis technique or combining Liapunov function method and LaSalle's invariance principle, respectively. Moreover, we give necessary and sufficient conditions for the occurrence of periodic solutions in the avian influenza system with Allee effect of the avian population. Numerical simulations are also presented to illustrate the theoretical results.