49 research outputs found
On the dynamics of a class of multi-group models for vector-borne diseases
The resurgence of vector-borne diseases is an increasing public health
concern, and there is a need for a better understanding of their dynamics. For
a number of diseases, e.g. dengue and chikungunya, this resurgence occurs
mostly in urban environments, which are naturally very heterogeneous,
particularly due to population circulation. In this scenario, there is an
increasing interest in both multi-patch and multi-group models for such
diseases. In this work, we study the dynamics of a vector borne disease within
a class of multi-group models that extends the classical Bailey-Dietz model.
This class includes many of the proposed models in the literature, and it can
accommodate various functional forms of the infection force. For such models,
the vector-host/host-vector contact network topology gives rise to a bipartite
graph which has different properties from the ones usually found in directly
transmitted diseases. Under the assumption that the contact network is strongly
connected, we can define the basic reproductive number and show
that this system has only two equilibria: the so called disease free
equilibrium (DFE); and a unique interior equilibrium---usually termed the
endemic equilibrium (EE)---that exists if, and only if, . We
also show that, if , then the DFE equilibrium is globally
asymptotically stable, while when , we have that the EE is
globally asymptotically stable
State estimators for some epidemiological systems
International audienceWe consider a class of epidemiological models that includes most well-known dynamics for directly transmitted diseases, and some reduced models for indirectly transmitted diseases. We then propose a simple observer that can be applied to models in this class. The error analysis of this observer leads to a non-autonomous error equation, and a new bound for fundamental matrices is also presented. We analyse and implement this observer in two examples: the classical SIR model, and a reduced Bailey-Dietz model for vector-borne diseases. In both cases we obtain arbitrary exponential convergence of the observer. For the latter model, we also applied the observer to recover the number of susceptible using dengue infection data from a district in the city of Rio de Janeiro
Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model
Dengue fever is endemic in tropical and subtropical countries, and certain important features of the spread of dengue fever continue to pose challenges for mathematical modelling. Here we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involve multi-serotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody-dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that are completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics by constructing a suitable Lyapunov functional. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for a specific range of values of a key parameter representing ADE.Coordenação de Aperfeiçoamento de Pessoal de NÃvel Superior -
Brazil (CAPES) - Finance Code 001
LIAM - Laboratory for Industrial and Applied Mathematics,
Department of Mathematics and Statistics, York University-CA
A survey on Lyapunov functions for epidemic compartmental models
In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading
Epidemiological models and Lyapunov functions
International audienceWe give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio R0 \leq 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant
Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure
A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results
Algorithmic approach for an unique definition of the next generation matrix
The basic reproduction number R0 is a concept which originated in population
dynamics, mathematical epidemiology, and ecology and is closely related to the
mean number of children in branching processes.We offer below three new
contributions to the literature: 1) We order a universal algorithmic definition
of a (F, V) gradient decomposition (and hence of the resulting R0), which
requires a minimal input from the user, namely the specification of an
admissible set of disease/infection variables. We also present examples where
other choices may be more reasonable, with more terms in F, or more terms in V .
2) We glean out from the works of Bacaer a fixed point equation (8) for the
extinction probabilities of a stochastic model associated to a deterministic
ODE model, which may be expressed in terms of the (F, V ) decomposition. The
fact that both R0 and the extinction probabilities are functions of (F, V )
underlines the centrality of this pair, which may be viewed as more fundamental
than the famous next generation matrix FV^{-1}.
3) We suggest introducing a new concept of sufficient/minimal
disease/infection set (sufficient for determining R0). More precisely, our
universal recipe of choosing "new infections" once the "infections" are
specified suggests focusing on the choice of the latter, which is also not
unique. The maximal choice of choosing all compartments which become 0 at the
given boundary point seems to always work, but is the least useful for analytic
computations, therefore we propose to investigate the minimal one. As a bonus,
this idea seems useful for understanding the Jacobian factorization approach
for computing R0 . Last but not least, we offer Mathematica scripts and
implement them for a large variety of examples, which illustrate that our
recipe others always reasonable results, but that sometimes other reasonable
(F, V ) decompositions are available as well
Onset of a vector borne disease due to human circulation - uniform, local and network reproduction ratios
We study the effect of human circulation on the onset of an epidemics for a arboviral (mosquito-borne) illness such as dengue. The underlying dynamics on a metapopulation is given by a classical SIR (human)/SI (vector) model. We consider three concepts of reproduction numbers: local (for each isolated subsys- tem), uniform or mixing (disregarding movement and non-uniformity in the whole region), and network (coupling the patches via human circulation). Interrelations between the three concepts are obtained. Depending on the biological contact as- sumptions, two types of network models result. In destination type models, the force of infection depends on mosquito density, relative to human population or to area. In origin based models, it is assumed that the transmission is determined by the behaviour of susceptible humans. Archetypal examples can be found where each node has local reproduction ratio less than one, the uniform reproduction number is also less than one, but the network reproduction number is greater than one. This shows that the disease can propagate among the patches solely as a consequence of human circulation. An estimate about the effect of vector control on a given patch is given. The conceptual framework presented here may help decision makers to plan vector control policies and medical care in case of an outbreak