A nonautonomous epidemic model on time scales

Ferreira, Rui A. C.; Silva, César M.

doi:10.1080/10236198.2018.1479400

Cited by 4 publications

(10 citation statements)

References 13 publications

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“…Here C is an open subset of K n , f : T × C → K n is rd-continuous, f (t, •) : C → K n is differentiable for all t ∈ T. As in [10], we denote f x (t, x) := ∂f ∂x (t, x), (t, x) ∈ T × C. We assume that f x : T × C → K n×n , is rd-continuous (see Section 2.2 for the definitions). We denote by x(•, t 0 , x 0 ) the unique solution of ( 22) with initial condition x(t 0 ) = x 0 , see [37] for explicit conditions on the existence and uniqueness of solutions of (22). In certain applications (as in Section 8) the subset C is nonopen.…”

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confidence: 99%

“…For a non-open set C, as remarked in [57], differentiability of f with respect to x means that the vector field can be extended as a differentiable function to some open set that includes C. The continuity hypotheses hold on this open set. In what follows, we say that the set C is forward invariant for system (22) if, for all t 0 ∈ T and x 0 ∈ C, x(t, t 0 , x 0 ) ∈ C, for all t ∈ T t0 . In particular, this tacitly implies that the system is forward complete on C, i.e.…”

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confidence: 99%

“…Let |•| be a vector norm on K n with associated matrix measure m(•, •). Let C ⊆ K n be a convex forward invariant set for system (22). Assume that there exists a constantc > 0 such that for all ξ ∈ C and all t ∈ T we have m (f x (t, ξ), µ(t)) ≤ −c 2 .…”

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confidence: 99%

“…An epidemic model on time scales. We now consider an epidemic model on time scales originally introduced in [22] to generalize the classic deterministic SIQR model with standard incidence in continuous time (see e.g. [8] and references therein).…”

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confidence: 99%

“…In the model, the infected compartment includes not only individuals that have been tested and found positive but also individuals that have no symptoms, as well as individuals that have symptoms but have not been tested. The time scale SIQR dynamics devised in [22] and considered in this section is…”

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confidence: 99%

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Matrix measures, stability and contraction theory for dynamical systems on time scales

Russo¹,

Wirth²

2022

DCDS-B

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<p style='text-indent:20px;'>This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.</p>

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Matrix measures, stability and contraction theory for dynamical systems on time scales

Russo¹,

Wirth²

2022

DCDS-B

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Research on actual or industrial applications of difference equations: a chimerical task?

Lozi

2023

Journal of Difference Equations and Applications

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New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

2022

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Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the “R0 alternative” of Van den Driessche and Watmough, which states that the local stability condition of the disease-free equilibrium may be expressed as R0<1, where R0 is the famous basic reproduction number, which also plays a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite common that 1. When R0>1, there exists a unique fixed endemic point, and 2. the endemic point is locally stable when R0>1. One would like to establish these properties for a large class of realistic epidemic models (and we do not include here epidemics without casualties). We have introduced recently a “simple” but broad class of “SIR-PH models” with varying populations, with the express purpose of establishing for these processes the two properties above. Since that seemed still hard, we have introduced a further class of “SIR-PH-FA” models, which may be interpreted as approximations for the SIR-PH models, and which include simpler models typically studied in the literature (with constant population, without loss of immunity, etc.). For this class, the first “endemic law” above is “almost established”, as explicit formulas for a unique endemic point are available, independently of the number of infectious compartments, and it only remains to check its belonging to the invariant domain. This may yet turn out to be always verified, but we have not been able to establish that. However, the second property, the sufficiency of R0>1 for the local stability of an endemic point, remains open even for SIR-PH-FA models, despite the numerous particular cases in which it was checked to hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH and SIR-PH-FA, and also for a second, more refined “intermediate approximation” SIR-PH-IA. We illustrate the current status-quo by presenting new results on a generalization of the SAIRS epidemic model.

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A nonautonomous epidemic model on time scales

Cited by 4 publications

References 13 publications

Matrix measures, stability and contraction theory for dynamical systems on time scales

Matrix measures, stability and contraction theory for dynamical systems on time scales

Research on actual or industrial applications of difference equations: a chimerical task?

New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

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