112

■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies

6.A

Time scale separation, example: SIR-UV model ............................

141

6.B

Parameter values ..........................................................

142

6.1

INTRODUCTION

Epidemiological models often must balance between accurate description of the dis-

ease dynamics, the different scales of modelling (within-host/micro scale, or popula-

tion/macro scale), and the associated levels of complexity which allow for establishing

tractable causal relationships in the health problems at hand. In the classical compartmen-

tal epidemiological models the human population is subdivided into two (susceptible and

infected, SI, SIS-models), or three (susceptible, infected and removed/ recovered, SIR-

models) compartments of individuals. This dichotomy is based on the importance of im-

munity against the disease in the situation of interest, and whether immunity is transient or

permanent.

Ordinary differential equations describe the changes in the sizes of the different com-

partments. This means that with the analysis of these models use is made of analytical

and numerical methods from nonlinear dynamical system theory. Epidemiological models

have properties in common and as a result specific nomenclature is introduced, we men-

tion the number R0, see for instance [18]. Continuous-time, deterministic compartmental

mathematical models are most common tools to simulate and analyze epidemic outbreak,

spread and course [5]. Stochastic models for epidemics also exist and they find application

to emerging fields in mathematical epidemiology such as the study of epidemics on net-

works [6, 33].

However, in cases of vector-borne diseases such as dengue, malaria, yellow fever, zika,

and so on, one must incorporate the presence of the vector population which plays a role

as carrier or vector of the pathogen between the human hosts. Dengue fever (DF) causes a

spectrum of diseases in humans ranging form clinically inapparent to severe and sometimes

fatal hemorrhagic form and the associated dengue shock syndrome. Generally, dengue

models are extensions of the classical models to account for multiple (two) serotypes of

the virus and their interaction. An important characteristic observed from serological tests

for dengue is that there are four different serotypes of the virus which can exist simulta-

neously and co-circulate in the human population (labelled in the literature as DENV-1 to

DENV-4). Hence, in this case a model must reflect this variety in the types of dengue virus

present both in the human and in the vector populations.

The main vectors of dengue transmission are the mosquitos of the Aedes genus (Ae.

aegypti and Ae. albopictus). These vectors are in the process of establishing themselves in

the Europe due to multiple factors (climate patterns, increased travel and trade) and this

has led to several dengue outbreaks in Southern Europe. Hence, study of the dengue epi-

demiology is important for health policymakers in Europe.