Bio-mathematics, Statistics, and Nano-Technologies Mosquito Control Strategies (Peyman Ghaffari)

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

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havior near its vicinity is demonstrated by numerical bifurcation analysis.

The author of [49] studies the effect of ADE (φ > 1) on the synchronization of in-

fected by the different pathogen strains. They ﬁnd that if seasonal forcing is included in

the model, it tends to stabilise the dynamic patters by synchronising them. Thus, periodic

or quasi-periodic cycles of each strain’s prevalence are observed unless the parameter φ is

large. In [43] the ADE is considered to act upon the force of infection via two pathways:

enhancement of transmission during a secondary dengue infection and increased suscep-

tibility due to presence of non-neutralising cross-reactive antibodies. The model exhibits

chaotic dynamics, and the simulations can reproduce the time series of observed dengue

fever outbreaks based on data from Vietnam.

The two-strain host-only model from [2] builds upon these previous works by adding

temporary cross-immunity to the model. It is of the SIRSIR-type: the recovered from a pri-

mary infection hosts after a period of cross immunity become susceptible to infection with

a different strain. The ﬂow of individuals from compartment Ri of those already recovered

from strain DENV-i enters an intermediate compartment Si of susceptible individuals to

the other strain. The individuals from S1 transition to class I12 and from S2 to class I21

during a secondary infection. The equations of [2] include a parameter α describing the

average duration of temporary cross-immunity (the time before transitioning from Ri to Si

is 1/α). In more detail the model is

˙S0 = −^β¹

N ^S⁰⁽^I¹⁺^I²⁺^φ⁽^I¹²⁺^I²¹⁾⁾⁺^µ⁽^N⁻^S⁰⁾^,

(6.6a)

˙I1 = ^β¹

N ^S⁰⁽^I¹⁺^φI²¹⁾⁻⁽^γ⁺^µ⁾^I¹^,

˙I2 = ^β¹

N ^S⁰⁽^I²⁺^φI¹²⁾⁻⁽^γ⁺^µ⁾^I²^,

(6.6b)

˙I12 = ^β²

N ^S¹⁽^I²⁺^φI¹²⁾⁻⁽^γ⁺^µ⁾^I¹²^,

˙I21 = ^β²

N ^S²⁽^I¹⁺^φI²¹⁾⁻⁽^γ⁺^µ⁾^I²¹^,

(6.6c)

˙S1 = αR1 −µS1 −^β²

N ^S¹⁽^I²⁺^φI¹²⁾^,

˙S2 = αR2 −µS2 −^β²

N ^S²⁽^I¹⁺^φI²¹⁾^,

(6.6d)

˙R1 = γI1 −(α+µ)R1,

˙R2 = γI2 −(α+µ)R2 .

(6.6e)

In particular, numerical bifurcation analysis shows that the ADE parameter φ could be

less than 1, but the system still exhibits a rich dynamic structure (Hopf bifurcations and

chaotic attractors) due to the presence of α > 0. The assumption φ < 1 is backed up by the

observation that virus carriers with a secondary infection are more likely to be hospitalized

and thus, less likely to contribute to the infectivity than carriers, with a primary infection

who are often asymptomatic.

In [1] the model from [2] is equipped with additional assumptions: a seasonal variation

in the force of infection and import of infected individuals from an external population.

The seasonal variation has the form of a cosine function and mirrors the seasonal ﬂuctua-

tions in vector prevalence due to climate factors. Chaotic dynamics is also observed in this

model. A common feature of the parameter sets employed in [12, 2, 1] is the epidemiologi-

cal symmetry between the virus strains; that is, the type of strain does not alter the force of

infection. A different approach is taken in [35] where asymmetry in the infectious between

the two strains is introduced. The numerical results demonstrate that the chaotic dynamics