Raul Nistal

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

This paper studies the nonnegativity and local and global stability properties of the solutions of a newly proposed SEIADR model which incorporates asymptomatic and dead-infective subpopulations into the standard SEIR model and, in parallel, it incorporates feedback vaccination plus a constant term on the susceptible and feedback antiviral treatment controls on the symptomatic infectious subpopulation. A third control action of impulsive type (or “culling”) consists of the periodic retirement of all or a fraction of the lying corpses which can become infective in certain diseases, for instance, the Ebola infection. The three controls are allowed to be eventually time varying and contain a total of four design control gains. The local stability analysis around both the disease-free and endemic equilibrium points is performed by the investigation of the eigenvalues of the corresponding Jacobian matrices. The global stability is formally discussed by using tools of qualitative theory of differential equations by using Gauss-Stokes and Bendixson theorems so that neither Lyapunov equation candidates nor the explicit solutions are used. It is proved that stability holds as a parallel property to positivity and that disease-free and the endemic equilibrium states cannot be simultaneously either stable or unstable. The periodic limit solution trajectories and equilibrium points are analyzed in a combined fashion in the sense that the endemic periodic solutions become, in particular, equilibrium points if the control gains converge to constant values and the control gain for culling the infective corpses is asymptotically zeroed.

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points for of the solution. Two examples are provided.

A nonlinear SEIR epidemic model with feedback vaccination control

A mathematical model describing a generic disease is introduced and the dynamics of the subpopulations are studied. Through linearization of the continuous-time model, the stability of the equilibrium points and the characteristics of the disease are defined properly. Due to the nature of the disease, the model is discretized in order to apply some adaptive vaccination strategies involving feedback loops. Furthermore, such techniques are compared to the traditional regular-vaccination strategies. The obtained results indicate a possible improvement in the use of the adaptive strategies for vaccination.

An Observer-Based Vaccination Law for a SEIR Epidemic Model

—This paper presents a simple continuous-time linear vaccination-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously the remaining population tends to zero.

A Strategy for Minimum Time Equilibrium Targetting in Epidemic Diseases

This paper relies on a minimum-time vaccination control strategy for a class of epidemic models. A targeted state final value is defined as a certain accuracy closed ball around some point being a reasonable approximate measure of both disease- free equilibrium points associated with the two vaccination levels used for the optimal- time control.

On Vaccination Strategies for a SISV Epidemic Model Guaranteeing the Nonexistence of Endemic Solutions

A vaccination strategy based on the state feedback control theory is proposed. The objective is to fight against the propagation of an infectious disease within a host population. This propagation is modelled by means of a SISV (susceptible-infectious-susceptible-vaccinated) epidemic model with a time-varying whole population and with a mortality directly associated with the disease. The vaccination strategy adds four free-design parameters, with three of them being the feedback gains of the vaccination control law. The other one is used to switch off the vaccination if the proportion of susceptible individuals is smaller than a prescribed threshold. The paper analyses the positivity of such a model under the proposed vaccination strategy as well as the conditions for the existence of the different equilibrium points of its normalized model. The fact that an appropriate adjustment of the control gains avoids the existence of endemic equilibrium points in the normalized SISV model while guaranteeing the existence of a unique disease-free equilibrium point being globally exponentially stable is proved. This is a relevant novelty dealt with in this paper. The persistence of the infectious disease within a host population irrespective of the growing properties of the whole population can be avoided in this way. Such theoretical results are mathematically proved and, also, they are illustrated by means of simulation examples. Moreover, the performance of the proposed vaccination strategy in several real situations is studied in some simulation examples. One of them deals with the presence of uncertainties, which affects the synthesis of the vaccination control law, in the measures of the subpopulations of the model.

Discretization and control of an SEIR epidemic model under equilibrium Wiener noise disturbances

A discretized SEIR epidemic model, subject to Wiener noise disturbances of the equilibrium points, is studied. The discrete-time model is got from a general discretization technique applied to its continuous-time counterpart so that its behaviour be close to its continuous-time counterpart irrespective of the size of the discretization period. The positivity and stability of a normalized version of such a discrete-time model are emphasized. The paper also proposes the design of a periodic impulsive vaccination which is periodically injected to the susceptible subpopulation in order to eradicate the propagation of the disease or, at least, to reduce its unsuitable infective effects within the potentially susceptible subpopulation. The existence and asymptotic stability of a disease-free periodic solution are proved. In particular, both the exposed and infectious subpopulations converge asymptotically to zero as time tends to infinity while the normalized subpopulations of susceptible and recovered by immun...

On a new model for Ebola disease

This paper states and studies an epidemic model where the dead corpses are also infective while the asymptomatic population is also present. There are several vaccination ways including susceptible vaccination, infectious antiviral control and culling of infectious corpses.

Equilibrium, stability and discretization of some basic epidemic models

Epidemic basic models will be analysed. The goal is to study the stability of equilibrium points using the Lyapunovs indirect method, through the linearisation of the systems to study local stability, and the Lyapunovs direct method to study the global stability of each equilibrium point. A simulation example will show that, despite the simplicity of the models, they can give a good representation of how will be the dynamics of the disease.

On some properties of a nonlinear delay integral equation used in epidemiology

This paper discusses some relevant properties of a nonlinear integral equation such as the positivity and boundedness of its solution as well as its controllability under impulsive controls. The model can describe an infection evolution and the acting impulsive controls can be interpreted in particular as either vaccination actions or as variations of the infected population due to infected immigration or lost of infective numbers due to lost of infected population.