Toshikazu Kuniya

The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes.

In this paper, an Susceptible-Vaccines-Exposed-Infectious-Recovered model with continuous age-structure in the exposed and infectious classes is investigated. These two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the exposed and the infectious classes. We investigate the global dynamics of this model in the sense of basic reproduction number via constructing Lyapunov functions. The asymptotic smoothness of solutions and uniform persistence of the system is shown from reformulating the system as a system of Volterra integral equations.

A note on dynamics of an age-of-infection cholera model.

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if i(a,t)/i*(a)=0 on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.

Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients

Abstract In this paper, we are concerned with an age-structured SIR epidemic model with time periodic coefficients. We obtain the basic reproduction number ℛ 0 as the spectral radius of the next generation operator and show that it plays the role of a threshold value for the existence of a nontrivial periodic solution, that is, the model has a nontrivial periodic solution if ℛ 0 > 1 , while no nontrivial periodic solution if ℛ 0 1 . For the proof, we use a fixed point theorem of Inaba (1990) [7] based on the Krasnoselskii fixed point theorem.

Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

This paper deals with the problem of global asymptotic stability for equilibria of a spatially diffusive SIR epidemic model with homogeneous Neumann boundary condition. By discretizing the model with respect to the space variable, we first construct Lyapunov functions for the corresponding ODEs model, and then broaden the construction method into the PDEs model in which either susceptible or infective populations are diffusive. In both cases, we obtain the standard threshold dynamical behaviors, that is, if , then the disease-free equilibrium is globally asymptotically stable and if , then the (strictly positive) endemic equilibrium is so. Numerical examples are given to illustrate the effectiveness of the theoretical results.

Further stability analysis for a multi-group SIRS epidemic model with varying total population size

Abstract In this paper, we focus on a multi-group SIRS epidemic model with varying total population size and cross patch infection between different groups. By applying a monotone iterative approach to the model, we establish a new sufficient condition for large recovery rates δ k , k = 1 , 2 , … , n on the global asymptotic stability of endemic equilibrium of the model. By combining the sufficient condition for small δ k , k = 1 , 2 , … , n obtained by Lyapunov functional approach, we obtain new sufficient conditions which extend the known results in recent literature.

Global stability of a multi-group SIS epidemic model with varying total population size

In this paper, to analyze the effect of the cross patch infection between different groups to the spread of gonorrhea in a community, we establish the complete global dynamics of a multi-group SIS epidemic model with varying total population size by a threshold parameter. In the proof, we use special Lyapunov functional techniques, not only one proposed by the paper Pruss et?al., 2006, but also the other one for a varying total population size with some ideas specified to our model and no longer need a grouping technique derived from the graph theory which is commonly used for the global stability analysis of multi-group epidemic models.

Numerical approximation of the basic reproduction number for a class of age-structured epidemic models

Abstract We are concerned with the numerical approximation of the basic reproduction number R 0 , which is the well-known epidemiological threshold value defined by the spectral radius of the next generation operator. For a class of age-structured epidemic models in infinite-dimensional spaces, R 0 has the abstract form and cannot be explicitly calculated in general. We discretize the linearized equation for the infective population into a system of ordinary differential equations in a finite n -dimensional space and obtain a corresponding threshold value R 0 , n , which can be explicitly calculated as the positive dominant eigenvalue of the next generation matrix. Under the compactness of the next generation operator, we show that R 0 , n → R 0 as n → + ∞ in terms of the spectral approximation theory.

Reconsideration of r/K Selection Theory Using Stochastic Control Theory and Nonlinear Structured Population Models.

Despite the fact that density effects and individual differences in life history are considered to be important for evolution, these factors lead to several difficulties in understanding the evolution of life history, especially when population sizes reach the carrying capacity. r/K selection theory explains what types of life strategies evolve in the presence of density effects and individual differences. However, the relationship between the life schedules of individuals and population size is still unclear, even if the theory can classify life strategies appropriately. To address this issue, we propose a few equations on adaptive life strategies in r/K selection where density effects are absent or present. The equations detail not only the adaptive life history but also the population dynamics. Furthermore, the equations can incorporate temporal individual differences, which are referred to as internal stochasticity. Our framework reveals that maximizing density effects is an evolutionarily stable strategy related to the carrying capacity. A significant consequence of our analysis is that adaptive strategies in both selections maximize an identical function, providing both population growth rate and carrying capacity. We apply our method to an optimal foraging problem in a semelparous species model and demonstrate that the adaptive strategy yields a lower intrinsic growth rate as well as a lower basic reproductive number than those obtained with other strategies. This study proposes that the diversity of life strategies arises due to the effects of density and internal stochasticity.

Mathematical analysis for a multi-group SEIR epidemic model with age-dependent relapse

Abstract We consider a multi-group SEIR epidemic model in which recovered population relapse back to infectives depending on the time elapsed since the recovery. This leads to a hybrid system for which we can determine the basic reproduction number by the spectral radius of the next generation matrix and prove the threshold behaviors. The key idea to prove the global asymptotic stability of each equilibrium is the usage of the graph-theoretic approach to construct suitable Lyapunov functionals. The necessary arguments, including the existence of an endemic equilibrium, the asymptotic smoothness of the semiflow, the uniform persistence of the system, and the existence of a global attractor are also addressed.

Modelling infectious diseases with relapse: a case study of HSV-2

BackgroundHerpes Simplex Virus Type 2 (HSV-2) is one of the most common sexually transmitted diseases. Although there is still no licensed vaccine for HSV-2, a theoretical investigation of the potential effects of a vaccine is considered important and has recently been conducted by several researchers. Although compartmental mathematical models were considered for each special case in the previous studies, as yet there are few global stability results.ResultsIn this paper, we formulate a multi-group SVIRI epidemic model for HSV-2, which enables us to consider the effects of vaccination, of waning vaccine immunity, and of infection relapse. Since the number of groups is arbitrary, our model can be applied to various structures such as risk, sex, and age group structures. For our model, we define the basic reproduction number ℜ0 and prove that if ℜ0≤1, then the disease-free equilibrium is globally asymptotically stable, whereas if ℜ0>1, then the endemic equilibrium is so. Based on this global stability result, we estimate ℜ0 for HSV-2 by applying our model to the risk group structure and using US data from 2001 to 2014. Through sensitivity analysis, we find that ℜ0 is approximately in the range of 2-3. Moreover, using the estimated parameters, we discuss the optimal vaccination strategy for the eradication of HSV-2.ConclusionsThrough discussion of the optimal vaccination strategy, we come to the following conclusions. (1) Improving vaccine efficacy is more effective than increasing the number of vaccines. (2) Although the transmission risk in female individuals is higher than that in male individuals, distributing the available vaccines almost equally between female and male individuals is more effective than concentrating them within the female population.