Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Existence and uniqueness of endemic states for the age-structured S–I–R epidemic model

The existence and uniqueness of positive steady states for the age structured S–I–R epidemic model with intercohort transmission is considered. Threshold results for the existence of endemic states are established for most cases. Uniqueness is shown in each case. Threshold used are explicitly computable in terms of demographic and epidemiological parameters of the model.     

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

Journal of Mathematical Biology - Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many...     

An age-structured epidemic model of rotavirus with vaccination

The recent approval of a rotavirus vaccine in Mexico motivates this study on the potential impact of the use of such a vaccine on rotavirus prevention and control. An age-structured model that describes the rotavirus transmission dynamics of infections is introduced. Conditions that guarantee the local and global stability analysis of the disease-free steady state distribution as well as the existence of an endemic steady state distribution are established. The impact of maternal antibodies on the implementation of vaccine is evaluated. Model results are used to identify optimal age-dependent vaccination strategies. A convergent numerical scheme for the model is introduced but not implemented. This paper is dedicated to Prof. K. P. Hadeler, who continues to push the frontier of knowledge in mathematical biology.     

Analytical threshold and stability results on age-structured epidemic models with vaccination

This paper examines mathematical models for common childhood diseases such as measles and rubella and in particular the use of such models to predict whether or not an epidemic pattern of regular recurrent disease incidence will occur. We use age-structured compartmental models which divide the population amongst whom the disease is spreading into classes and use partial differential equations to model the spread of the disease. This paper is particularly concerned with an analytical investigation of the effects of different types of vaccination schemes. We examine possible equilibria and determine the stability of small oscillations about these equilibria. The results are important in predicting the long-term overall level of incidence of disease, in designing immunisation programs and in describing the variations of the incidence of disease about this equilibrium level.     

Equilibrium and local stability in a logistic matrix model for age-structured populations

A logistic matrix model for age-structured population dynamics is constructed. This model discretizes a continuous, density-dependent model with age structure, i.e. it is an extension of the logistic model to the case of age-dependence. We prove the existence and uniqueness of its equilibrium and give a necessary and sufficient condition for the local stability of the equilibrium.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

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.     

Senescence and antibiotic resistance in an age-structured population model

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.     

Global stability of an SIR epidemic model with information dependent vaccination

We study the global behavior of a non-linear susceptible-infectious-removed (SIR)-like epidemic model with a non-bilinear feedback mechanism, which describes the influence of information, and of information-related delays, on a vaccination campaign. We upgrade the stability analysis performed in d’Onofrio et al. [A. d’Onofrio, P. Manfredi, E. Salinelli, Vaccinating behavior, information, and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol. 71 (2007) 301] and, at same time, give a special example of application of the geometric method for global stability, due to Li and Muldowney. Numerical investigations are provided to show how the stability properties depend on the interplay between some relevant parameters of the model.     

Global asymptotic stability of a periodic solution to an epidemic model

In this paper a periodic delay differential equation with spatial spread is investigated. This equation can be used to model the growth of malaria which is transmitted by a mosquito. Using monotone techniques, it is shown that the following bifurcation holds: either the disease dies out or the density of infectious people tends to a spatially homogeneous, time periodic and positive solution.Research partially supported by NSF Grant MCS 810-4837     

Absolute stability and transitions in ecosystems with a multiplicity of stable states and dispersal

The principle that for a large class of spatially extended dynamic systems with several locally stable states there exists at most one absolutely stable state is applied to population dynamical models with diffusive spatial dispersal. The basic concepts and methods to characterize absolute stability and its limits are presented. Some interesting results concerning the significance of dispersal ability in competing and mutualistic systems are found. The concepts appear of practical importance since only absolutely stable states are resilient against all localized fluctuations and disturbances.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator-prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator-prey system, Science 290 (2000), pp. 1358-1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

The Chapman-Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections

We consider a mathematical model of an age-structured population of some fisheries (for example, anchovies, sardines or soles). Two time scales are involved in the problem: the fast time scale for the migration dynamics and the slow time scale for the demographic process. At a first step, we study the so called 'aggregated' system by means of the semigroups theory. Then, we study the asymptotic behaviour of the model by using the Chapman-Enskog procedure. In particular, we study initial, boundary and corner layer effects in order to obtain the exact initial and boundary conditions the approximated solution has to satisfy.     

Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix

Journal of Mathematical Biology - The dynamics of an SIS epidemic patch model with asymmetric connectivity matrix is analyzed. It is shown that the basic reproduction number $$R_0$$ is strictly...