Global stability for SIR and SIRS models with differential mortality

We consider SIR and SIRS models with differential mortality. Global stability of equilibria is established by using Lyapunov's methodComment: page 2, one figur

Multi-patch and multi-group epidemic models: A new framework

International audienceWe develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number R0 is derived. The effects of heterogeneity in groups, patches and mobility patterns on R0 and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number

On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients

International audienceThe aim of this paper is to give a method for the estimation of total parasite burden of the patient and the rate of infection in a malaria's intra-host model by using control theory tools. More precisely, we use an auxiliary system, called {\it observer} or {\it estimator}, whose solutions tend exponentially to those of the original model. This observer uses only the available measurable data, namely, the values of peripheral infected erythrocytes. It provides estimates of the {\it sequestered infected erythrocytes,} that cannot be measured by clinical methods. Therefore this method allows to estimate the total parasite burden within a malaria patient. Moreover, our constructed observer does not use the uncertain infection rate parameter $\beta$. In fact, we derive a simple method to estimate this parameter $\beta$. We apply this estimation method using real data that have been collected when malaria was used as therapy for neurosyphilis by the US Public Health Service

On the Dynamics of Dengue Virus type 2 with Residence Times and Vertical Transmission

A two-patch mathematical model of Dengue virus type 2 (DENV-2) that accounts for vectors' vertical transmission and between patches human dispersal is introduced. Dispersal is modeled via a Lagrangian approach. A host-patch residence-times basic reproduction number is derived and conditions under which the disease dies out or persists are established. Analytical and numerical results highlight the role of hosts' dispersal in mitigating or exacerbating disease dynamics. The framework is used to explore dengue dynamics using, as a starting point, the 2002 outbreak in the state of Colima, Mexico

On the estimation of sequestered parasite population in falciparum malaria patients

The aim of this paper is to give an estimation of total parasite burden of the patient and the rate of infection in a malaria's intra-host model by using control theory tools. More precisely, we use an auxiliary system called observer whose solutions tend exponentially to those of the original model. This observer uses only the available measurable data that are the values of peripheral infected erythrocytes and provides estimates of the sequestered ones that cannot be measured by clinical methods. Therefore this allows to estimate the total parasite burden within a malaria patient. Moreover the constructed observer does not use the generally unknown infection rate parameter $\beta$. We also derive a simple method to estimate this parameter $\beta$. We apply this estimation method using real data that have been collected when malaria was used as therapy for neurosyphilis by the US Public Health Service.L'objectif de ce papier est de donner une estimation de la charge parasitaire totale d'un patient á partir de sa charge parasitaire périphérique. On donne aussi une méthode d'estimation du taux d'infection $\beta$ dans un modéle intra-hôte du paludisme en utilisant les outils de la théorie du contrôle