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

fully in analyzing a whole class of compartmental models dealing with the transition from

the disease-free to the endemic state or the reverse situation leading to the disease elimina-

tion. The number R0 is used frequently as a rule-of-thumb for determining the presence of

an epidemic. The following threshold is established: whenever R0 > 1 the disease persists

and there is an epidemic, and whenever R0 < 1 there is no epidemic and the disease fades

away.

The transition between the elimination and endemic disease can also be studied by

bifurcation theory: in which case the threshold is fixed by a so called transcritical bifur-

cation. That is R0 = 1 at the TC-point and therefore the associated threshold values are

equal. The stable disease-free becomes unstable when a parameter is varied and there origi-

nates a stable (for a non-catastrophic or supercritical TC) or an unstable (for a catastrophic

or subcritical TC) endemic equilibrium. In the second case, the second rule that R0 < 1

implying no epidemic is violated. Therein the transcritical bifurcation produces an unsta-

ble endemic equilibrium, and the term used in the epidemiology literature is a backward

bifurcation [29, 53] instead of subcritical often used in the nonlinear dynamical system

theory literature. Consequently, in the parameter region where R0 < 1 a bistable regime of

another stable endemic equilibrium (or limit cycle) may coexist with the stable disease-free

equilibrium and an epidemic can occur despite R0 being under the threshold value [27].

Coming back to a multi-strain model, the characteristic long-term dynamics, in addi-

tion to an equilibrium could exhibit several features such as limit cycles or chaotic be-

havior, because of the large number of equations and highly non-linear couplings among

them. Besides the trivial disease-free equilibrium, endemic equilibria with a single strain

(also known as boundary or exclusion equilibria) and endemic equilibria where multiple

strains are present, are possible.

As an example, the presence of multiple strains in the model [25, 41] gives rise to multi-

ple equilibria. Depending on the parameter values, their local asymptotic stability changes.

In [25] the boundary equilibria are always locally asymptotically stable. While the general

structure of these models is broadly similar, the presence of intermediate recovered host

classes R1,R2 in the model [41] allows both boundary equilibria to be locally asymptoti-

cally unstable. This shows that the nonlinear interactions between the model compartments

are an important factor in driving the asymptotic behavior of the model.

Furthermore, in realistic dengue models with infection/recovery rates above the R0

threshold the endemic equilibrium becomes unstable leading to more complex long-term

dynamical behavior such as limit cycles and chaos. These types of dynamics can be studied

using bifurcation analysis. Limit cycles describe periodic behaviour and typically arise

from a Hopf bifurcation. For chaotic dynamics another tool for the study of non-linear

dynamical systems, namely the calculation of Lyapunov exponents can be used [52]. Many

of the multi-strain models show mathematical symmetry and this property governs specific

types of bifurcations [2, 1, 34, 41].

6.2.2

Time scale separation

Mathematical models for vector-borne diseases are natural candidates for time

scale separation analysis based on singular perturbation theory. Typically the vector