33 research outputs found
Exact solutions for a Solow-Swan model with non-constant returns to scale
The Solow-Swan model is shortly reviewed from a mathematical point of view.
By considering non-constant returns to scale, we obtain a general solution
strategy. We then compute the exact solution for the Cobb-Douglas production
function, for both the classical model and the von Bertalanffy model. Numerical
simulations are provided.Comment: 10 pages, 4 figure
A minimal model for adaptive SIS epidemics
The interplay between disease spreading and personal risk perception is of
key importance for modelling the spread of infectious diseases. We propose a
planar system of ordinary differential equations (ODEs) to describe the
co-evolution of a spreading phenomenon and the average link density in the
personal contact network. Contrary to standard epidemic models,we assume that
the contact network changes based on the current prevalence of the disease in
the population, i.e.\ it adapts to the current state of the epidemic. We assume
that personal risk perception is described using two functional responses: one
for link-breaking and one for link-creation. The focus is on applying the model
to epidemics, but we highlight other possible fields of application. We derive
an explicit form for the basic reproduction number and guarantee the existence
of at least one endemic equilibrium, for all possible functional responses.
Moreover, we show that for all functional responses, limit cycles do not exist.Comment: 19 pages, 6 figure
A geometric analysis of the SIRS compartmental model with fast information and misinformation spreading
We propose an SIRS compartmental model with demography and fast information
and misinformation spreading in the population. The analysis of the complete
6-dimensional system shows the existence of seven equilibrium points. Since
under our assumptions the system evolves on two time scales, we completely
characterize the possible asymptotic behaviours with techniques of Geometric
Singular Perturbation Theory (GSPT). During our analysis of the fast dynamics,
we identify three branches of the critical manifold, which exist under
determined conditions. We perform a theoretical bifurcation analysis of the
fast system to understand the relation between these three equilibria when
varying specific parameters of the fast system. We then observed a delayed loss
of stability on the various branches of the critical manifold, as the slow
dynamics may cause the branches to lose their hyperbolicity. We emphasise how
the inclusion of (mis)information spreading, even in low dimensional
compartmental models, can radically alter the asymptotic behaviour of the
epidemic. We conclude with numerical simulations of various remarkable
scenarios.Comment: 27 pages, 8 figures, 1 tabl
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
A geometric analysis of the SIRS model with secondary infections
We propose a compartmental model for a disease with temporary immunity and
secondary infections. From our assumptions on the parameters involved in the
model, the system naturally evolves in three time scales. We characterize the
equilibria of the system and analyze their stability. We find conditions for
the existence of two endemic equilibria, for some cases in which . Then, we unravel the interplay of the three time scales, providing
conditions to foresee whether the system evolves in all three scales, or only
in the fast and the intermediate ones. We conclude with numerical simulations
and bifurcation analysis, to complement our analytical results.Comment: 31 pages, 9 figure
A geometric analysis of the impact of large but finite switching rates on vaccination evolutionary games
In contemporary society, social networks accelerate decision dynamics causing
a rapid switch of opinions in a number of fields, including the prevention of
infectious diseases by means of vaccines. This means that opinion dynamics can
nowadays be much faster than the spread of epidemics. Hence, we propose a
Susceptible-Infectious-Removed epidemic model coupled with an evolutionary
vaccination game embedding the public health system efforts to increase vaccine
uptake. This results in a global system ``epidemic model + evolutionary game''.
The epidemiological novelty of this work is that we assume that the switching
to the strategy ``pro vaccine'' depends on the incidence of the disease. As a
consequence of the above-mentioned accelerated decisions, the dynamics of the
system acts on two different scales: a fast scale for the vaccine decisions and
a slower scale for the spread of the disease. Another, and more methodological,
element of novelty is that we apply Geometrical Singular Perturbation Theory
(GSPT) to such a two-scale model and we then compare the geometric analysis
with the Quasi-Steady-State Approximation (QSSA) approach, showing a
criticality in the latter. Later, we apply the GSPT approach to the disease
prevalence-based model already studied in (Della Marca and d'Onofrio, Comm Nonl
Sci Num Sim, 2021) via the QSSA approach by considering medium-large values of
the strategy switching parameter.Comment: 26 pages, 6 figure
Transition from time-variant to static networks: timescale separation in NIMFA SIS epidemics
We extend the N-intertwined mean-field approximation (NIMFA) for the
Susceptible-Infectious-Susceptible (SIS) epidemiological process to
time-varying networks. Processes on time-varying networks are often analysed
under the assumption that the process and network evolution happen on different
timescales. This approximation is called timescale separation. We investigate
timescale separation between disease spreading and topology updates of the
network. We introduce the transition times and
as the boundaries between the intermediate regime
and the annealed (fast changing network) and quenched (static network) regimes,
respectively. By analysing the convergence of static NIMFA processes, we
analytically derive upper and lower bounds for . Our
results provide insights/bounds on the time of convergence to the steady state
of the static NIMFA SIS process. We show that, under our assumptions, the upper
transition time is almost entirely determined by the
basic reproduction number of the network. The value of the upper
transition time around the epidemic threshold is
large, which agrees with the current understanding that some real-world
epidemics cannot be approximated with the aforementioned timescale separation.Comment: 30 pages, 13 figure