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

is less profound in the asymmetric case.

An extension of the model with age-structure of the host population and introduction

four dengue strains is analyzed in [3], where vaccination of different age groups is included

and simulations reveal that the vaccine is most effective in reducing the prevalence if only

individuals with a primary infection are vaccinated. A common feature of the host-only

models is to provide a mathematical model with reduced complexity which is able to re-

produce the observed irregular patterns of dengue seroprevalence and strain co-circulation

in tropical countries. Thus, a model with relatively few variables exhibits complex dynam-

ical behaviour ranging from limit cycles to chaotic trajectories.

6.3.2

Host-vector models

Host-vector models describe also the temporal dynamics of the vector population (in

the case of dengue or yellow fever, mosquitos Ae. egypti or Ae. albopictus) which trans-

mits the pathogen to humans. In this regard their structure is closer to the epidemiology of

the disease. However, as we have noted, because of the inherent difference in time scales

which characterise the dynamics of the host and the vector populations, these models are

described by stiff systems of ordinary differential equations. Thus, robust numerical meth-

ods must be used for their integration in time.

Model [22] considers a single strain dengue model of SIR type for the human host and

of SI type for the vector. The authors demonstrate that whenever R0 > 1 any oscillatory

dynamics is transient and the system converges to the endemic equilibrium. Optimal con-

trol is introduced into single-strain dengue models and publications analyze a multitude

of strategies targeting both humans or vectors, based on educational campaigns and ento-

mological surveillance [47], vector control via insecticides [46], use of bed nets [14] and

vaccination [48].

Explicit inclusion of the mosquito dynamics in multi-strain models for dengue in-

creases the model dimension by including susceptible and infected by DENV-i type vec-

tors. That explains why they are less frequent in the modelling literature. Furthermore, care

must be taken when interpreting the model parameters. In host-only models [26, 12, 1, 34],

infection rate per infected host is used, while host-vector models [44, 42, 41] employ in-

fection rate per infected vector. Thus, it is important to be able to compare the parameters

used in each type of model and assess the differences.

In [25] the vector was modeled explicitly and furthermore the effects of primary and

secondary infections was modeled with temporary cross-immunity. Similar to [26], the au-

thors in [25] do not introduce a separate class of hosts recovered from a primary infection.

Analysis of equilibria in [25] shows the existence of an unstable endemic steady state.

Hence, the asymptotic behavior of the model shows a long transient of coexisting strains

before one of the strains is displaced by the other.

In [24] the authors consider a model of the SISIR-type for the host and SI- type of the