Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections.

*(English)*Zbl 1349.34349This paper studies the global dynamics of a two-patch SIS model in which disease transmission occurs within patches, as well as during the migration of individuals between them. The patches are assumed to be characterized by different relevant parameters (transmission coefficient, dispersal, recovery and mortality rates). It is also assumed that the time required to complete an one-way trip between patches is fixed and the migrating individuals neither die nor recover during the trip.

First, the authors explicitly solve the equations which model the spread of infection during migration. The substitution of the solutions back into the systems describing the within-patch dynamics leads to the formulation of a delayed differential model, which the authors then restate in terms of \(N_j\) (total population within patch \(j\)) and \(I_j\), \(j=1,2\).

After determining the asymptotic behavior of \(N_j\), \(j=1,2\), the authors use the theory of asymptotically autonomous systems to obtain global stability results in terms of the basic reproduction number, defined as the spectral radius of a next generation matrix. This matrix is constructed using regional reproduction numbers (reproductive numbers for a patch in isolation) and the expected number of infective individuals appearing in one patch due to the migration of a single infective individual from the other patch.

If the migration between patches is unidirectional rather than bidirectional, the basic reproduction number is observed to partially lose its threshold characteristics, due to the onset of a semitrivial equilibrium, whose stability is then characterized in terms of the regional reproduction numbers.

The influence of the travel delay is also discussed, being shown that the basic reproduction number and all components of the endemic equilibrium are increasing functions of the travel delay. It is also shown via numerical simulations that ignoring the travel delay and migration-related infections may lead to an underestimation of the severity of an epidemics, while a travel delay larger than a given critical value leads to the persistency of a disease which would otherwise die out.

First, the authors explicitly solve the equations which model the spread of infection during migration. The substitution of the solutions back into the systems describing the within-patch dynamics leads to the formulation of a delayed differential model, which the authors then restate in terms of \(N_j\) (total population within patch \(j\)) and \(I_j\), \(j=1,2\).

After determining the asymptotic behavior of \(N_j\), \(j=1,2\), the authors use the theory of asymptotically autonomous systems to obtain global stability results in terms of the basic reproduction number, defined as the spectral radius of a next generation matrix. This matrix is constructed using regional reproduction numbers (reproductive numbers for a patch in isolation) and the expected number of infective individuals appearing in one patch due to the migration of a single infective individual from the other patch.

If the migration between patches is unidirectional rather than bidirectional, the basic reproduction number is observed to partially lose its threshold characteristics, due to the onset of a semitrivial equilibrium, whose stability is then characterized in terms of the regional reproduction numbers.

The influence of the travel delay is also discussed, being shown that the basic reproduction number and all components of the endemic equilibrium are increasing functions of the travel delay. It is also shown via numerical simulations that ignoring the travel delay and migration-related infections may lead to an underestimation of the severity of an epidemics, while a travel delay larger than a given critical value leads to the persistency of a disease which would otherwise die out.

Reviewer: Paul Georgescu (Iaşi)

##### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34K25 | Asymptotic theory of functional-differential equations |

92D30 | Epidemiology |