We give bounds for the global attractor of the delay differential equation $ \dot
x(t)=-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative
Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then,
regardless of the delay, all solutions enter the domain where $f$
is monotone decreasing and the powerful results for delayed
monotone feedback can be applied to describe the asymptotic
behaviour of solutions. In this situation we determine the sharpest interval that contains
the global attractor for any delay. In the absence of that condition, improving
earlier results, we show that if the delay is sufficiently small,
then all solutions enter the domain where $f'$ is negative. Our
theorems then are illustrated by numerical examples using
Nicholson's blowflies equation and the Mackey-Glass equation.
A new SEIR model with distributed infinite delay is derived when
the infectivity depends on the age of infection. The basic reproduction number
R0, which is a threshold quantity for the stability of equilibria, is calculated.
If $R_0$ < 1, then the disease-free equilibrium is globally asymptotically stable
and this is the only equilibrium. On the contrary, if $R_0$ > 1, then an endemic
equilibrium appears which is locally asymptotically stable. Applying a perma-
nence theorem for infinite dimensional systems, we obtain that the disease is
always present when $R_0$ > 1.
Finding optimal policies to reduce the morbidity and mortality of the ongoing pandemic is a top public health priority. Using a compartmental model with age structure and vaccination status, we examined the effect of age specific scheduling of vaccination during a pandemic influenza outbreak, when there is a race between the vaccination campaign and the dynamics of the pandemic. Our results agree with some recent studies on that age specificity is paramount to vaccination planning. However, little is known about the effectiveness of such control measures when they are applied during the outbreak. Comparing five possible strategies, we found that age specific scheduling can have a huge impact on the outcome of the epidemic. For the best scheme, the attack rates
were up to 10% lower than for other strategies. We demonstrate the importance of early start of the vaccination campaign, since ten days delay may increase the attack rate by up to 6%. Taking into account the delay between developing immunity and vaccination is a key factor in evaluating the impact of vaccination campaigns. We provide a general framework which will be
useful for the next pandemic waves as well.
We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.
In this paper, we study the global stability of a multistrain SIS model with superinfection. We present an iterative procedure to calculate a sequence of reproduction numbers, and we prove that it completely determines the global dynamics of the system. We show that for any number of strains with different infectivities, the stable coexistence of any subset of the strains is possible, and we completely characterize all scenarios. As an example, we apply our method to a three-strain model.
We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.