Local and Global Stability Analysis of Dengue Disease with Vaccination and Optimal Control

Abstract

Dengue fever is a disease that has spread all over the world, including Thailand. Dengue is caused by a virus and there are four distinct serotypes of the virus that cause dengue DENV-1, DENV-2, DENV-3, and DENV-4. The dengue viruses are transmitted by two species of the Aedes mosquitoes, the Aedes aegypti, and the Aedes albopictus. Currently, the dengue vaccine used in Thailand is chimeric yellow tetravalent dengue (CYD-TDV). This research presents optimal control which studies the vaccination only in individuals with a documented past dengue infection (seropositive), regardless of the serotypes of infection causing the initial infection by the disease. The analysis of dengue transmission model is used to establish the local asymptotically stabilities. The property of symmetry in the Lyapunov function an import role in achieving this global asymptotically stabilities. The optimal control systems are shown in numerical solutions and conclusions. The result shows that the control resulted in a significant reduction in the number of infected humans and infected vectors.

Effect of a Vaccination against the Dengue Fever Epidemic in an Age Structure Population: From the Perspective of the Local and Global Stability Analysis

“…It is obvious that all the terms appearing in Equation ( 41) are always nonpositive. Now, using LaSalle's extension to Lyapunov's theorem [21,[25][26][27][28][29], we have dθ dt ≤ 0 and so the function dθ dt is to be negative definite. The limit set of each solution is contained in the largest invariant set for which S H ia = S * H ia , S H ia V = S * H ia V and R H ia = 0 which is the singleton {E 1 * }.…”

“…, I H ia = 0, and I H ia V = 0. Therefore, by LaSalle's extension to Lyapunov's function theorem [21,[25][26][27][28][29], the endemic steady state E 2…”

“…It was generally found that if R 0 < 1, the disease-free equilibrium point is locally stable, and the endemic equilibrium point is stable at R 0 > 1. In 2021, the authors [21] studied the optimal control strategies through the SEIRS model by considering only the pre-exposure infectious transmission status and the impact of post-reinfection vaccination strategies without regarding the serotype sequence in infection. Sanusi et al [22] studied analyzing a SIRS model and also simulating the model to predict the number of dengue fever cases.…”

Sistematic Review: Mathematics Model Epidemiology of Dengue Fever

“…The steps taken in solving the dengue hemorrhagic fever disease model mathematically in the article discussed are carried out by determining the equilibrium point of the dengue hemorrhagic fever mathematical model, determining the basic reproduction number [31], determining local stability [42], [43], stability [9], [43] of the dengue hemorrhagic fever disease model at the disease-free equilibrium point and at the endemic equilibrium point using the eigenvalues of the Jacobian matrix. Furthermore, using optimal control [44], [45] from the dengue fever model, by first determining the state variables and state equations from the Differential Equation system, determining the performance index after being given the controller.…”

“…Optimal control can be used to determine alternative solutions after obtaining an epidemiological model in the form of a differential equation. Optimal control of vaccination in the model [56] and [43], but paper [43] with individuals who have previously been infected with the dengue virus, this model population (N) is divided into five classes, namely: Susceptible class (Sh) which indicates the number of humans potentially infected with the virus in healthy condition but can be infected with DHF, class Infected (Ih) states the number of humans infected with the virus including those who have become active DHF and can transmit DHF disease, Recovered (R h ) class states the number of humans who have recovered or recovered through vaccination which makes immunity permanent, Susceptible Vector (S v ) class represents the number of mosquitoes which have the potential to carry the virus, and the Infected (I v ) class indicates the number of infected mosquitoes. The size of the human population as a host is expressed as N h = S h +I h +R h and the size of the mosquito population as a vector is expressed as Nv = Sv+Iv.…”