Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 482091, 12 pages

http://dx.doi.org/10.1155/2015/482091

## Equilibrium Analysis of a Yellow Fever Dynamical Model with Vaccination

^{1}School of Medicine, University of São Paulo and LIM01 HC-FMUSP, Avenida Doutor Arnaldo 455, Cerqueira César, 01246-903 São Paulo, SP, Brazil^{2}School of Veterinary Medicine and Zootechnology, University of São Paulo, Avenida Professor Doutor Orlando Marques de Paiva 87, Cidade Universitária, 05508 270 São Paulo, SP, Brazil^{3}London School of Hygiene and Tropical Medicine, London University, Keppel Street, London WC1E 7HT, UK

Received 6 October 2014; Revised 30 January 2015; Accepted 30 January 2015

Academic Editor: Chung-Min Liao

Copyright © 2015 Silvia Martorano Raimundo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose an equilibrium analysis of a dynamical model of yellow fever transmission in the presence of a vaccine. The model considers both human and vector populations. We found thresholds parameters that affect the development of the disease and the infectious status of the human population in the presence of a vaccine whose protection may wane over time. In particular, we derived a threshold vaccination rate, above which the disease would be eradicated from the human population. We show that if the mortality rate of the mosquitoes is greater than a given threshold, then the disease is naturally (without intervention) eradicated from the population. In contrast, if the mortality rate of the mosquitoes is less than that threshold, then the disease is eradicated from the populations only when the growing rate of humans is less than another threshold; otherwise, the disease is eradicated only if the reproduction number of the infection after vaccination is less than 1. When this reproduction number is greater than 1, the disease will be eradicated from the human population if the vaccination rate is greater than a given threshold; otherwise, the disease will establish itself among humans, reaching a stable endemic equilibrium. The analysis presented in this paper can be useful, both to the better understanding of the disease dynamics and also for the planning of vaccination strategies.

#### 1. Introduction

Yellow fever (YF), a hemorrhagic fever caused by a* Flavivirus*, family* Flaviviridae *[1, 2], is characterized by fever, chills, loss of appetite, nausea, muscle pains particularly in the back, and headaches [3]. There are more than 200,000 infections and 30,000 deaths every year [3]. About 90% of YF cases occur in Africa [4], and a billion people live in an area of the world where the disease is common [3]. It also affects tropical areas of South America, but not Asia [3, 5, 6]. The number of cases of yellow fever has been increasing in the last 30 years [3, 7], probably due to fewer people being immune, more people living in cities, people moving frequently, and changing climate [3]. The origin of the disease is Africa, from where it spread in South America through the slave trade in the 17th century [8, 9].

The yellow fever virus was the first human virus discovered [10], and its family comprises approximately 70 viruses [2], most of which are transmitted by arthropod insects (hence the name arthropod borne viruses or arboviruses).

A safe and effective vaccine against yellow fever exists and some countries require vaccinations for travelers [3]. In rare cases (less than one in 200,000 to 300,000 doses), the vaccination can cause yellow fever vaccine-associated viscerotropic disease (YEL-AVD), which is fatal in 60% of cases, probably due to the genetic morphology of the immune system. Another possible side effect is an infection of the nervous system, which occurs in one in 200,000 to 300,000 cases, causing yellow fever vaccine-associated neurotropic disease (YEL-AND), which can lead to meningoencephalitis, fatal in less than 5% of cases [6]. In some rare circumstances, however, the fatality rate of vaccine induced diseases can reach alarming proportions, as observed recently by Mascheretti et al. [11], who found 1 death per million doses applied in a Southeastern Brazilian region.

In this paper, we propose an equilibrium analysis of a dynamical model of yellow fever transmission in the presence of a vaccine. Such a kind of analysis can be useful, both to the better understanding of the disease dynamics and also for the planning of vaccination strategies.

#### 2. Model Formulation

The mathematical model described below addresses the transmission dynamics of an infectious agent in a homogeneous population in the presence of an imperfect vaccine. We consider a nonlinear system of ordinary differential equations involving the human and the vector—mosquitoes and their eggs—populations. The term “eggs” also includes the intermediate stages, such as larvae and pupae. It is also worth highlighting that the model proposed here is based on previous papers [12, 13], and we updated the model originally developed by Amaku et al., 2013 [14, 15], for the purpose of investigating the impact of vaccination on population. By including vaccination, in particular vaccines which may have serious adverse effects, the model may help the designing of realistic (from the cost and logistic point of view) vaccination strategies.

All variables and parameters in the human system will carry the subscript, while those in the vector system will carry one of the subscripts(mosquitoes) or(eggs). In our model the total human population, denoted by, is split into four subclasses which are susceptible humans (), vaccinated humans (), infected humans (), and recovered (and immune) humans (), so that. The total vector population, which is formed by both total mosquitoes population, denoted by, and the total eggs population, denoted by, are split into susceptible mosquitoes (), infected and latent mosquitoes (), infected and infectious mosquitoes (), and noninfected eggs (), so thatand.

A flow diagram of the model is depicted in Figure 1, and the associated variables and parameters are described in Tables 1 and 2, respectively (values from references [16–22]).