Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation

Copyright © 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells

The transmission dynamics of a within- and between-hosts malaria model

In this paper, we developed a novel deterministic coupled model tying together the effects of within-host and population level dynamics on malaria transmission dynamics. We develop within-host and within-vector dynamic models, population level between-hosts models, and a nested coupled model combining these levels. The unique feature of this work is the way the coupling and feedback for the model use the various life stages of the malaria parasite both in the human host and the mosquito vector. Analysis of the coupled and the within-human host models indicate the existence of locally asymptotically stable infection- and parasite-free equilibria when the associated reproduction numbers are less than one. The population-level model, on the other hand, exhibits backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. A global sensitivity analysis was carried out to measure the effects of the sensitivity and uncertainty in the various model parameters estimates. The results indicate that the most important parameters driving the pathogen level within an infected human are the production rate of the red blood cells from the bone marrow, the infection rate, the immunogenicity of the infected red blood cells, merozoites and gametocytes, and the immunosensitivity of the merozoites and gametocytes. The key parameters identified at the population level are the human recovery rate, the death rate of the mosquitoes, the recruitment rate of susceptible humans into the population, the mosquito biting rate, the transmission probabilities per contact in mosquitoes and in humans, and the parasite production and clearance rates in the mosquitoes. Defining the feedback functions as a linear function of the mosquito biting rate, numerical exploration of the coupled model reveals oscillations in the parasite populations within a human host in the presence of the host immune response. These oscillations dampen as the mosquito biting rate increases. We also observed that the oscillation and damping effect seen in the within-human host dynamics fed back into the population level dynamics; this in turn amplifies the oscillations in the parasite population within the mosquito-host