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

Viability analysis of controlled trajectories, on the other hand, is a novel problem in

epidemiological models, especially in the case of vector-borne diseases. One example is

the Ross-MacDonald model with control on the mosquito mortality by fumigation [17].

A possible motivation behind viability analysis is that policy-makers could be interested

in maintaining prevalence of infection below a certain maximum level, which is linked to

capacity constraints in the national or regional healthcare system, availability of medical

workers, hospital beds, or medication.

The analysis in [40] establishes conditions for the existence of the initial points of so-

lution trajectories meeting partial state constrains for all future times, in particular, initial

conditions such that along the optimal controlled trajectory the proportion of infected indi-

viduals never exceeds a certain maximum level of infected individuals. The viability kernel,

or the largest closed controlled forward-invariant subset which satisfies this constraint, is

computed numerically using a variational approach leading to solution of an appropriate

Hamilton-Jacobi-Bellman equation [9].

6.6

CONCLUSIONS

Due to the high model dimension, multi-strain models for vector-borne diseases of-

ten do not yield to theoretical analysis, and are dependent on numerical simulations. For

models incorporating different types of controls aimed at reducing the disease burden, this

complexity bears particular relevance for sensitivity and controllability analyses. In partic-

ular, one must use appropriate numerical methods for the time integration of stiff systems

of ODEs, and robust methods for numerical bifurcation analysis.

In this work we combine epidemic modelling with model reduction techniques using

time-scale separation for host-vector models, making them into host-only models based on

QSSA. This approach provides a rigorous mechanism for checking that lower-dimensional

models preserve features of the original model. Sensitivity analysis gives more insight into

the effects of the different mechanisms included in the mathematical system or in the spe-

cific way they are modelled. This can be used for studying various scenarios for disease

control in order to find regions in the parameter space corresponding to disease elimination,

for example, and to study the effectiveness of varying those controls accordingly. How-

ever, as we have seen, introduction of control measures such as vaccination may rapidly

introduce substantial complexity leading to large-dimensional systems that do not yield to

theoretical analysis [3]. Hence, in the context of dengue simpler epidemiological models

with just a single strain can be used as a starting point of the investigation. These models

are useful in identifying transcritical bifurcation points which separate regions of disease

elimination from endemic states via analytic relationships with the control variables as ex-

emplified in Section 6.5. Furthermore, optimal control problems such as viability analysis

of simple epidemiological models [40] require nontrivial transformations in order to apply

theoretical results from the field of quasi-monotone dynamical systems.