Page 421 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 421
MATHEMATICS IN BIOLOGY AND MEDICINE (MS-35)
by joining analytical and numerical results, the existence of periodic solutions, already shown
numerically in [1]. Finally, we studied an SIRS system resulting from pair approximation of a
network model; although the system is 5-dimensional, the method allows to reduce the study of
the system to that of a sequence of 2-dimensional maps; we are then able to prove existence of
periodic solutions of the system when the degree of network nodes is sufficiently small.
References
[1] M. P. Dafilis, F. Frascoli, J. G. Wood, and J. M. McCaw. (2012) The influence of increasing
life expectancy on the dynamics of SIRS systems with immune boosting. The ANZIAM
Journal, 54(1-2):50–63. https://doi.org/10.1017/S1446181113000023
[2] P. De Maesschalck and S. Schecter. (2016) The entry–exit function and geomet-
ric singular perturbation theory. Journal of Differential Equations, 260(8):6697–6715.
https://doi.org/10.1016/j.jde.2016.01.008
[3] H. Jardón-Kojakhmetov, C. Kuehn, A. Pugliese, and M. Sensi. (2021) A geometric anal-
ysis of the SIR, SIRS and SIRWS epidemiological models., Nonlinear Analysis RWA, 58,
103220. https://doi.org/10.1016/j.nonrwa.2020.103220
Basic Reproduction Number for Conservation laws
Jordi Ripoll, jripoll@imae.udg.edu
University of Girona, Spain
Coauthors: Carles Barril, Àngel Calsina, Sílvia Cuadrado
The famous basic reproduction number R0 plays a key-role in weighing birth/infection and
death/recovery processes in population models. In this work we focus on continuously struc-
tured models (e.g. where individuals are characterized by age or size, spatial position, etc. . . )
as given by conservation laws.
For constant and time periodic environments, R0 is defined as the spectral radius of the
so-called next generation operator, which let’s say, maps a distribution of population to the
distribution of population of their offspring along the whole life span of the former. Analogously
in epidemic models, it maps a distribution of infected population to the distribution of their
secondary cases.
On the one hand, for populations with concentrated state at birth, we have developed a sys-
tematic limit procedure to get to the basic reproduction number. Specifically, R0 is computed as
the limit of basic reproduction numbers of approximate models of populations with distributed
state at birth but tending to concentrate (e.g. from an interval to a singleton). So our approach
avoids the often computation of R0 in a heuristically way. We give several examples obtaining
explicit expressions.
On the other hand, for populations with distributed state at birth, the computation of the
spectral radius of next generation operators poses, in general, serious obstacles to the effective
and efficient determination of R0. Either we have an explicit formula (e.g. for rank one oper-
ators) or we address this problem numerically via suitable reductions of the relevant operators
to matrices, thus computing the sought quantity by solving generalized eigenvalue problems,
possibly of large dimension. We also give several examples obtaining good results where in
addition we proved the compactness of the corresponding next generation operator.
419
by joining analytical and numerical results, the existence of periodic solutions, already shown
numerically in [1]. Finally, we studied an SIRS system resulting from pair approximation of a
network model; although the system is 5-dimensional, the method allows to reduce the study of
the system to that of a sequence of 2-dimensional maps; we are then able to prove existence of
periodic solutions of the system when the degree of network nodes is sufficiently small.
References
[1] M. P. Dafilis, F. Frascoli, J. G. Wood, and J. M. McCaw. (2012) The influence of increasing
life expectancy on the dynamics of SIRS systems with immune boosting. The ANZIAM
Journal, 54(1-2):50–63. https://doi.org/10.1017/S1446181113000023
[2] P. De Maesschalck and S. Schecter. (2016) The entry–exit function and geomet-
ric singular perturbation theory. Journal of Differential Equations, 260(8):6697–6715.
https://doi.org/10.1016/j.jde.2016.01.008
[3] H. Jardón-Kojakhmetov, C. Kuehn, A. Pugliese, and M. Sensi. (2021) A geometric anal-
ysis of the SIR, SIRS and SIRWS epidemiological models., Nonlinear Analysis RWA, 58,
103220. https://doi.org/10.1016/j.nonrwa.2020.103220
Basic Reproduction Number for Conservation laws
Jordi Ripoll, jripoll@imae.udg.edu
University of Girona, Spain
Coauthors: Carles Barril, Àngel Calsina, Sílvia Cuadrado
The famous basic reproduction number R0 plays a key-role in weighing birth/infection and
death/recovery processes in population models. In this work we focus on continuously struc-
tured models (e.g. where individuals are characterized by age or size, spatial position, etc. . . )
as given by conservation laws.
For constant and time periodic environments, R0 is defined as the spectral radius of the
so-called next generation operator, which let’s say, maps a distribution of population to the
distribution of population of their offspring along the whole life span of the former. Analogously
in epidemic models, it maps a distribution of infected population to the distribution of their
secondary cases.
On the one hand, for populations with concentrated state at birth, we have developed a sys-
tematic limit procedure to get to the basic reproduction number. Specifically, R0 is computed as
the limit of basic reproduction numbers of approximate models of populations with distributed
state at birth but tending to concentrate (e.g. from an interval to a singleton). So our approach
avoids the often computation of R0 in a heuristically way. We give several examples obtaining
explicit expressions.
On the other hand, for populations with distributed state at birth, the computation of the
spectral radius of next generation operators poses, in general, serious obstacles to the effective
and efficient determination of R0. Either we have an explicit formula (e.g. for rank one oper-
ators) or we address this problem numerically via suitable reductions of the relevant operators
to matrices, thus computing the sought quantity by solving generalized eigenvalue problems,
possibly of large dimension. We also give several examples obtaining good results where in
addition we proved the compactness of the corresponding next generation operator.
419
