About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls
This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial st...
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| Language: | English |
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Wiley
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/5556897 |
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| author | Manuel De la Sen Asier Ibeas Raul Nistal |
| author_facet | Manuel De la Sen Asier Ibeas Raul Nistal |
| author_sort | Manuel De la Sen |
| collection | DOAJ |
| description | This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model. |
| format | Article |
| id | doaj-art-c471c383638841ceb933f93b645b8a25 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-c471c383638841ceb933f93b645b8a252025-02-03T06:43:34ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2021-01-01202110.1155/2021/55568975556897About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment ControlsManuel De la Sen0Asier Ibeas1Raul Nistal2Institute of Research and Development of Processes (IIDP), University of the Basque Country, Campus of Leioa, P.O. Box 48940, Leioa, Bizkaia, SpainDepartment of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona UAB, 08193 Barcelona, SpainInstitute of Research and Development of Processes (IIDP), University of the Basque Country, Campus of Leioa, P.O. Box 48940, Leioa, Bizkaia, SpainThis paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model.http://dx.doi.org/10.1155/2021/5556897 |
| spellingShingle | Manuel De la Sen Asier Ibeas Raul Nistal About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls Discrete Dynamics in Nature and Society |
| title | About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls |
| title_full | About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls |
| title_fullStr | About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls |
| title_full_unstemmed | About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls |
| title_short | About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls |
| title_sort | about partial reachability issues in an seir epidemic model and related infectious disease tracking in finite time under vaccination and treatment controls |
| url | http://dx.doi.org/10.1155/2021/5556897 |
| work_keys_str_mv | AT manueldelasen aboutpartialreachabilityissuesinanseirepidemicmodelandrelatedinfectiousdiseasetrackinginfinitetimeundervaccinationandtreatmentcontrols AT asieribeas aboutpartialreachabilityissuesinanseirepidemicmodelandrelatedinfectiousdiseasetrackinginfinitetimeundervaccinationandtreatmentcontrols AT raulnistal aboutpartialreachabilityissuesinanseirepidemicmodelandrelatedinfectiousdiseasetrackinginfinitetimeundervaccinationandtreatmentcontrols |