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

sembles models of airborne diseases such as flu, measles, SARS. These models consider

multiple virus strains and two characteristics of dengue fever which are important for the

host epidemiology: antibody-dependent enhancement and temporary cross-immunity be-

tween the two strains.

In [43] models based on immunological serotype (strain) interactions are studied. They

define two mechanisms: the increase in transmissibility during secondary infection (in-

cluded as a factor in the force of infection terms for secondary infections) and enhance-

ment of the susceptibility to secondary infection (included as an extra loss in the recovery

rate terms by a factor for the force of infection). Figure 6.4 in [43] presents the analysis re-

sults by changing the two parameters presenting the two mechanisms show that both have

qualitatively similar effects.

In this review, we only consider further the enhancement by increased transmissibil-

ity during secondary infection by a different strain. We also consider neutralization where

the effect is a decrease of the transmissibility. In [26] a two-strain model with an anti-

body effect is proposed, where pre-existing immunity to dengue virus could be a fac-

tor for the development of severe dengue in a subsequent infection. The model includes

a parameter φ representing the degree of antibody-dependent enhancement (φ > 1) or

neutralization (φ < 1) caused by the presence of cross-reactive antibodies. The enhance-

ment/neutralization is interpreted as an increase/decrease in the transmission probability of

the dengue virus. The model structure is of the SEII-type (susceptible-exposed-infected-

infected), because an immediate transition occurs between the compartments of exposed

to one strain hosts to infected by a different strain hosts, in other words, there is no period

of recovery after the primary infection. Thus, [26] implicitly assumes co-infection with

both strains as there is no distinction between the compartments. If the antibody effect is

strongly neutralizing (φ = 0), the authors in [26] demonstrate that the strain with the higher

basic reproduction number R0 will displace the other. However, the model exhibits peri-

odic and chaotic behavior in large parameter ranges, under the assumption of strong ADE

for one or both strains.

The models proposed in [49, 12, 43] including a compartment for the recovered from

the primary or from both infections and can be thought of an extensions of the model [26].

The compartments of primary recovered hosts is susceptible to an infection with the differ-

ent strain. These models are of the SIRIR-type (susceptible-infected-recovered-infected-

recovered) type. The model equations [12, Eq.1] are easily generalized to a disease with n

strains (serotypes).

In contrast to [26], the authors of [12] assume there is only antibody-dependent en-

hancement, so the compartments of hosts with secondary infection are more infectious.

Analysis of the basic reproduction numbers Rⁱ

0^{,i = 1,2 for each strain shows that if both}

Rⁱ

0 ^{> 1, the strain with the higher Ri}

0 ^{will displace the other, and the boundary endemic}

equilibrium is asymptotically stable. In case R¹

0 ^{= R2}

0^{, both boundary endemic equilibria}

are unstable, there is a coexistence equilibrium, and periodic or deterministic chaotic be-