Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations
In the case of Dengue transmission and control, the interaction of nature and society is captured by a system of difference equations. For the purpose of studying the dynamics of these interactions, four variables involved in a Dengue epidemic, proportion of infected people (P), number of mosquitoes...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
|
| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2014/101965 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849306155889721344 |
|---|---|
| author | T. Awerbuch-Friedlander Richard Levins M. Predescu |
| author_facet | T. Awerbuch-Friedlander Richard Levins M. Predescu |
| author_sort | T. Awerbuch-Friedlander |
| collection | DOAJ |
| description | In the case of Dengue transmission and control, the interaction of nature and society is captured by a system of difference equations. For the purpose of studying the dynamics of these interactions, four variables involved in a Dengue epidemic, proportion of infected people (P), number of mosquitoes involved in transmission (M), mosquito habitats (H), and population awareness (A), are linked in a system of difference equations: Pn+1=aPn+1-e-iMn1-Pn, Mn+1=lMne-An+bHn1-e-Mn, Hn+1=cHn/(1+pAn)+1/(1+qAn), and An+1=rAn+fPn, n=0,1,…. The constraints have socioecological meaning. The initial conditions are such that 0≤P0≤1, (M0,H0,A0)≥(0,0,0), the parameters l,a,c,r∈(0,1), and the parameters f, i, b, and p are positive. The paper is concerned with the analysis of solutions of the above system for p=q. We studied the global asymptotic stability of the degenerate equilibrium. We also propose extensions of the above model and some open problems. We explored the role of memory in community awareness by numerical simulations. When the memory parameter is large, the proportion of infected people decreases and stabilizes at zero. Below a critical point we observe periodic oscillations. |
| format | Article |
| id | doaj-art-385b69eb570344ddb88d60945ed71ec3 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-385b69eb570344ddb88d60945ed71ec32025-08-20T03:55:11ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2014-01-01201410.1155/2014/101965101965Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference EquationsT. Awerbuch-Friedlander0Richard Levins1M. Predescu2Department of Population and International Health, Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USADepartment of Population and International Health, Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USADepartment of Mathematical Sciences, Bentley University, 175 Forest Street, Waltham, MA 02452, USAIn the case of Dengue transmission and control, the interaction of nature and society is captured by a system of difference equations. For the purpose of studying the dynamics of these interactions, four variables involved in a Dengue epidemic, proportion of infected people (P), number of mosquitoes involved in transmission (M), mosquito habitats (H), and population awareness (A), are linked in a system of difference equations: Pn+1=aPn+1-e-iMn1-Pn, Mn+1=lMne-An+bHn1-e-Mn, Hn+1=cHn/(1+pAn)+1/(1+qAn), and An+1=rAn+fPn, n=0,1,…. The constraints have socioecological meaning. The initial conditions are such that 0≤P0≤1, (M0,H0,A0)≥(0,0,0), the parameters l,a,c,r∈(0,1), and the parameters f, i, b, and p are positive. The paper is concerned with the analysis of solutions of the above system for p=q. We studied the global asymptotic stability of the degenerate equilibrium. We also propose extensions of the above model and some open problems. We explored the role of memory in community awareness by numerical simulations. When the memory parameter is large, the proportion of infected people decreases and stabilizes at zero. Below a critical point we observe periodic oscillations.http://dx.doi.org/10.1155/2014/101965 |
| spellingShingle | T. Awerbuch-Friedlander Richard Levins M. Predescu Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations Discrete Dynamics in Nature and Society |
| title | Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations |
| title_full | Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations |
| title_fullStr | Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations |
| title_full_unstemmed | Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations |
| title_short | Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations |
| title_sort | codynamics of four variables involved in dengue transmission and its control by community intervention a system of four difference equations |
| url | http://dx.doi.org/10.1155/2014/101965 |
| work_keys_str_mv | AT tawerbuchfriedlander codynamicsoffourvariablesinvolvedindenguetransmissionanditscontrolbycommunityinterventionasystemoffourdifferenceequations AT richardlevins codynamicsoffourvariablesinvolvedindenguetransmissionanditscontrolbycommunityinterventionasystemoffourdifferenceequations AT mpredescu codynamicsoffourvariablesinvolvedindenguetransmissionanditscontrolbycommunityinterventionasystemoffourdifferenceequations |