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

over time can be considered to account for seasonally changing climatic factors such as

temperature or rainfall that affect the vector’s breeding pattern, [51]. A common simpli-

fication in the modelling is to incorporate the seasonally changing vector population as a

periodic forcing (often sinusoidal) onto the population size.

The earlier models were extensions of the epidemiological models used to model and

simulate epidemics. Specific mechanisms for the dengue fever are modeled by adding other

compartments. To reflect the epidemiological data we consider susceptible individuals

without a previous dengue infection S0, primary infected with i-th strain Ii, and recov-

ered Ri from a primary infection respectively with strain i = 1,2,3,4 depending on the

total number of strains considered. After a period of a temporary cross immunity, recov-

ered individuals in class Ri move to the class Si of susceptible individuals with a history

of a primary dengue infection with i-th strain i = 1,2,3,4.

The underlying assumption of multi-strain dengue models is that infection with a given

strain DENV-i,i = 1,2,3,4 confers life-long strain-specific immunity. However, there are

multiple approaches to modelling the transitions between the compartments depending on

whether the model includes a period of cross-immunity. These will be described in detail

in the following Section. In general multi-strain models assume that reinfection with a

different strain is possible: the individuals in class Si are then prone to reinfection and

transition to the class Iij with a history of a primary dengue infection with i-th strain and

secondary infection with j-th strain, i , j. Because tertiary infections are rare, individuals

from classes Iij develop life-long immunity (global removed class R) and do not contribute

to the force of infected thereafter. For vectors, we denote by U the susceptible and by V

the infected individuals.

In a review paper on models and analysis for dengue [7] a systematic overview of all

types of models is given with a “phylogenetic tree” of selected articles till 2012. We fo-

cus here on two-strain models only, but we mention that also multiple strain models with

1,2,3,4 or n serotypes are formulated and analyzed (see [32, Table 1]). Furthermore, we

do not claim completeness and we focus on the most important mechanisms involved in

modeling dengue fever: demographics (rate of birth and death), two-way vector-host trans-

mission (and its respective approximation as host-to-host transmission), rate of infection

or force of infection, anti-body-enhancement, (temporary) cross immunity, seasonal fluc-

tuations and the import of infected individuals.

Transmission of the disease occurs when a diseased individual meets another host indi-

vidual in the host-only model, or by an encounter with a individual vector in the host-vector

model. The number of host individuals is denoted by N and the one of the vector population

by M. We assume that the mosquitos live in an environment of fixed size which is propor-

tional to M. In this way the area-density of the vector-population remains constant when

the size of the environment changes over time. Then, for host-only models [26, 12, 1, 34]

the force of infection is βI/N and for the host-vector models [42, 41] it is BV/M. This

is based on the law of mass action, where I/N and V/M represent the probability of an