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

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vector with 2 dengue strains with cross-immunity. Their results state that if both strains

confer total cross-immunity, competitive exclusion is always the result and coexistence is

possible under the assumption of increased susceptibility to the strain in a secondary infec-

tion. This model with optimal control has been studied in [54].

In [41] the authors extend the model of [2] by including a vector population. The model

is of the SIRSIR-type for the host and SI for the vector. The authors propose a model for

dengue fever with two strains and temporary cross-immunity (α) and different likelihood

of transmission from hosts with primary or secondary infections to vectors (φ). The vector

is included explicitly and furthermore the different effects of primary and secondary infec-

tions due to ADE are modeled.

The model equations of [41] for the host dynamics are recalled here:

˙S0 = −^B1

M ^{S0(V1 +V2)+µ(N −S) ,}

(6.7a)

˙I1 = ^B1

M ^{S0V1 −(γ +µ)I1 ,}

˙I2 = ^B1

M ^{S0V2 −(γ +µ)I2 ,}

(6.7b)

˙R1 = γI1 −(α+µ)R1 ,

˙R2 = γI2 −(α+µ)R2 ,

(6.7c)

˙S1 = −^B2

M ^{S1V2 +αR1 −µS1 ,}

˙S2 = −^B2

M ^{S2V1 +αR2 −µS2 ,}

(6.7d)

˙I12 = ^B2

M ^{S1V2 −(γ +µ)I12 ,}

˙I21 = ^B2

M ^{S2V1 −(γ +µ)I21}

(6.7e)

and for the vector dynamics:

˙V1 = ^ϑ

N ^{(M −V1 −V2)(I1 +φI21)−νV1 ,}

(6.7f)

˙V2 = ^ϑ

N ^{(M −V1 −V2)(I2 +φI12)−νV2 .}

(6.7g)

Comparison with the host-only model shows that due to a different virus transmission

mechanism the force of infection terms and the infection rates differ. The discussion in

Sec. 6.2.2 of system with two time scales for the host and vector dynamics gives a possi-

bility to connect the parameter β in Eq. (6.6b) with B in Eq. (6.7b). In [42] the following

equation was derived using the linearized expression in (6.5) for the force of infection

where V = V ^S

QSSA^:

Bi

M ^{V = Bi}

M

ϑM

νN ^{I = βi}

N ^I

with

βi = ^ϑ

ν ^{Bi .}

(6.8)

Using the parameter values from [2, 34] yields Bi = βi/2 and makes it possible to com-

pare the results of the host-vector model (6.7) from [41] with those from for the host-only

model (6.6) in [34].

Temporary cross-immunity is considered alongside a parameter that quantifies the dif-

ferences in likelihood for host-to-vector transmission between infected individuals in a

primary and a secondary dengue infection. This is motivated by the fact that individuals