Multi-Strain Host-Vector Dengue Modeling: Dynamics and Control

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susceptible human encountering an infected individual or a vector, respectively while β

and B capture the respective encounter and pathogen transmission rates.

In the literature the formulation of the force of infection is discussed in many papers,

we mention [20, 8]. Often the focus is on the comparison of density- (βI) versus frequency-

dependency (βI/N) in the case of the single-strain SIR-model. Note that density is here a

ratio of number of individuals and in this context different from the area-density mentioned

above.

In the case of a vector-borne disease, such as dengue, the situation is more complex

because contact between individuals belonging to different populations (host and vector)

is involved which are not required to live in the same area, these areas only have to overlap.

Here we used the same type of force of infection for both, namely frequency dependent.

Recently in [11, 10] the authors use multi-patch formulations to model spatial-distributions

of the dispersal of the individuals in the populations [11, 10] in order to take for instance

spatial-heterogeneity within populations into account.

In [4, 45, 48, 46] the same host-vector vector-borne dengue system is modeled but the

denominator of both force of infection terms is proportional to the human population num-

bers, N. In other words the force of infection of mosquitos upon hosts is taken as density

dependent, but the force of infection of host upon mosquitos is frequency-dependent. Only

when the sizes of the areas where host and vector live change proportionally, this model

formulation is the same as model (6.4).

This work considers recent papers focusing on modeling of the epidemiological mech-

anisms specific for dengue fever and control measures such as vaccination campaigns and

vector control.

6.2.1

Equilibria and basic reproduction number R0

In the study of disease dynamics described by a dynamical system one seeks to perform

a qualitative analysis of the states where the system is at rest, namely the equilibria. At such

equilibria the sizes of the compartments do not change over time. The equilibria analysis

is useful for studying the asymptotic behavior of the model, which includes conditions

under which the disease will be eliminated (the infected compartment will approach zero)

or whether it will persist and become endemic (the infected compartment will be strictly

positive). This fundamental question of determining existence and stability of disease-free

versus endemic states pervades the mathematical epidemiology literature.

In the epidemiological literature one searches for a measure that tells whether an epi-

demic will occur or not. This purpose is served by the basic reproduction number R0 repre-

senting the number of secondary cases that one case generates on average over the course

of the whole of its infectious period in an otherwise uninfected population [19, 18, 53].

It is a dimensionless quantity dependent on the parameters of the model equations. The

next-generation calculation approach of R0 [19, 22, 18, 53, 27] has been used success-