Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations

Abstract

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), andAn+1=rAn+fPn,n=0,1,…. The constraints have socioecological meaning. The initial conditions are such that0≤P0≤1, (M0,H0,A0)≥(0,0,0), the parametersl,a,c,r∈(0,1), and the parametersf, i, b, andpare positive. The paper is concerned with the analysis of solutions of the above system forp=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.