Bruno Buonomo

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 dOnofrio et al. [A. dOnofrio, 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 stability for an HIV-1 infection model including an eclipse stage of infected cells

n Abstractn n We consider the mathematical model for the viral dynamics of HIV-1 introduced in Rong et al. (2007) [37]. One main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class. The viral dynamics is described by four nonlinear ordinary differential equations. In Rong et al. (2007) [37], the stability of the infected equilibrium has been analyzed locally. Here, we perform the global stability analysis using two techniques, the Lyapunov direct method and the geometric approach to stability, based on the higher-order generalization of Bendixsonʼs criterion. We obtain sufficient conditions written in terms of the system parameters. Numerical simulations are also provided to give a more complete representation of the system dynamics.n n

Stability and bifurcation analysis of a vector-bias model of malaria transmission

The vector-bias model of malaria transmission, recently proposed by Chamchod and Britton, is considered. Nonlinear stability analysis is performed by means of the Lyapunov theory and the LaSalle Invariance Principle. The classical threshold for the basic reproductive number, R(0), is obtained: if R(0)>1, then the disease will spread and persist within its host population. If R(0)<1, then the disease will die out. Then, the model has been extended to incorporate both immigration and disease-induced death of humans. This modification has been shown to strongly affect the system dynamics. In particular, by using the theory of center manifold, the occurrence of a backward bifurcation at R(0)=1 is shown possible. This implies that a stable endemic equilibrium may also exists for R(0)<1. When R(0)>1, the endemic persistence of the disease has been proved to hold also for the extended model. This last result is obtained by means of the geometric approach to global stability.

A diffusive-convective model for the dynamics of population-toxicant interactions: some analytical and numerical results.

In this paper we consider a diffusive-convective model for the dynamics of a population living in a polluted environment. Threshold results are given concerning the effect of the toxicant on the living population. Some analytic results are proved and numerical experiments give suggestions in more general cases.

On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate

Abstract We study an epidemic model for infections with non permanent acquired immunity (SIRS). The incidence rate is assumed to be a general nonlinear function of the susceptibles and the infectious classes. By using a peculiar Lyapunov function, we obtain necessary and sufficient conditions for the local nonlinear stability of equilibria. Conditions ensuring the global stability are also obtained. Unlike the recent literature on this subject, here no restrictions are required about the monotonicity and concavity of the incidence rate with respect to the infectious class. Among the applications, the noteworthy case of a convex incidence rate is provided.

Analysis of a tuberculosis model with a case study in Uganda

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincaré–Bendixson criterion. An application is given for the case of Internally Displaced Peoples Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease.

Globally stable endemicity for infectious diseases with information-related changes in contact patterns

Abstract SIR and SIS epidemic models with information—related changes in contact patterns are introduced. The global stability analysis of the endemic equilibrium is performed by means of the Li–Muldowney geometric approach. Biological implications of the stability conditions are given.

Qualitative analysis and optimal control of an epidemic model with vaccination and treatment

We focus on an epidemic model which incorporates a non-linear force of infection and two controls: an imperfect preventive vaccine given to susceptible individuals and therapeutic treatment given to infectious. We study both the cases of constant and non constant controls. In the case of constant controls we perform a qualitative analysis based on Lyapunov stability which allows to integrate the bifurcation analysis performed in a previous paper. The occurrence of a backward bifurcation is discussed in the perspective of disease control. The case of time-dependent controls is studied by means of the optimal control theory. The strategy is to minimize both the disease burden and the intervention costs. We derive the optimality system and solve it numerically. The characterization of the optimal time profile of the controls, together with the qualitative analysis provides a rather complete picture of the possible outcomes of the model.

A simple analysis of vaccination strategies for rubella.

We consider an SEIR epidemic model with vertical transmission introduced by Li, Smith and Wang, [23], and apply optimal control theory to assess the effects of vaccination strategies on the model dynamics. The strategy is chosen to minimize the total number of infectious individuals and the cost associated with vaccination. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella in China.

Rational exemption to vaccination for non-fatal SIS diseases: globally stable and oscillatory endemicity.

Rational exemption to vaccination is due to a pseudo-rational comparison between the low risk of infection, and the perceived risk of side effects from the vaccine. Here we consider rational exemption in an SI model with information dependent vaccination where individuals use information on the diseases spread as their information set. Using suitable assumptions, we show the dynamic implications of the interaction between rational exemption, current and delayed information. In particular, if vaccination decisions are based on delayed informations, we illustrate both global attractivity to an endemic state, and the onset, through Hopf bifurcations, of general Yakubovich oscillations. Moreover, in some relevant cases, we plot the Hopf bifurcation curves and we give a behavioural interpretation of their meaning.