Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Mean-field Approach for Controlling Singularly Perturbed Multi-population SIS Epidemics
375
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a multi-population epidemic model with one or more (almost) isolated communities and one mobile community. Each of the isolated communities has contact within itself and, in addition, contact with the outside world but only through the mobile community. The contact rate between the mobile community and the other communities is assumed to be controlled. We first derive a multidimensional ordinary differential equation (ODE) as a mean-field fluid approximation to the process of the number of infected nodes, after appropriate scaling. We show that the approximation becomes tight as the sizes of the communities grow. We then use a singular perturbation approach to reduce the dimension of the ODE and identify an optimal control policy on this system over a fixed time horizon via Pontryagins minimum principle. We then show that this policy is close to optimal, within a certain class, on the original problem for large enough communities. From a phenomenological perspective, we show that the epidemic may sustain in time in all communities (and thus the system has a nontrivial metastable regime) even though in the absence of the mobile nodes the epidemic would die out quickly within each of the isolated communities.
rate research
Read More
We study a multiscale approach for the control of agent-based, two-population models. The control variable acts over one population of leaders, which influence the population of followers via the coupling generated by their interaction. We cast a quadratic optimal control problem for the large-scale microscale model, which is approximated via a Boltzmann approach. By sampling solutions of the optimal control problem associated to binary two-population dynamics, we generate sub-optimal control laws for the kinetic limit of the multi-population model. We present numerical experiments related to opinion dynamics assessing the performance of the proposed control design.
This paper considers the susceptible-infected-susceptible (SIS) epidemic model with an underlying network structure among subpopulations and focuses on the effect of social distancing to regulate the epidemic level. We demonstrate that if each subpopulation is informed of its infection rate and reduces interactions accordingly, the fraction of the subpopulation infected can remain below half for all time instants. To this end, we first modify the basic SIS model by introducing a state dependent parameter representing the frequency of interactions between subpopulations. Thereafter, we show that for this modified SIS model, the spectral radius of a suitably-defined matrix being not greater than one causes all the agents, regardless of their initial sickness levels, to converge to the healthy state; assuming non-trivial disease spread, the spectral radius being greater than one leads to the existence of a unique endemic equilibrium, which is also asymptotically stable. Finally, by leveraging the aforementioned results, we show that the fraction of (sub)populations infected never exceeds half.
We study the synthesis of optimal control policies for large-scale multi-agent systems. The optimal control design induces a parsimonious control intervention by means of l-1, sparsity-promoting control penalizations. We study instantaneous and infinite horizon sparse optimal feedback controllers. In order to circumvent the dimensionality issues associated to the control of large-scale agent-based models, we follow a Boltzmann approach. We generate (sub)optimal controls signals for the kinetic limit of the multi-agent dynamics, by sampling of the optimal solution of the associated two-agent dynamics. Numerical experiments assess the performance of the proposed sparse design.
The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian $H_0(t)$, satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form $epsilon H_1(t)$. Here $epsilon to 0$ is the adiabaticity parameter and $H_1(t)$ is a self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$. Thus the total hamiltonian $H_0(t)+epsilon H_1(t)$ does not necessarily satisfy the gap assumption, $forall epsilon >0$. It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.
We consider smooth systems limiting as $epsilon to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < epsilon ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $epsilon-$dependent domain which shrinks to zero as $epsilon to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
