# Modeling the Impact of Climate Change on the Dynamics of Rift Valley Fever

Authors: Saul C. Mpeshe; Livingstone S. Luboobi; Yaw Nkansah-Gyekye
Journal: Computational and Mathematical Methods in Medicine
 (2014)

---

## Abstract

A deterministic SEIR model of rift valley fever (RVF) with climate change parameters was considered to compute the basic reproduction numberℛ
0 and investigate the impact of temperature and precipitation on ℛ
0. To study the effect of model parameters to ℛ
0, sensitivity and elasticity analysis of ℛ
0 were performed. When temperature and precipitation effects are not considered, ℛ
0 is more sensitive to the expected number of infected Aedes spp. due to one infected livestock and more elastic to the expected number of infected livestock due to one infected Aedes spp. When climatic data are used, ℛ
0 is found to be more sensitive and elastic to the expected number of infected eggs laid by Aedes spp. via transovarial transmission, followed by the expected number of infected livestock due to one infected Aedes spp. and the expected number of infected Aedes spp. due to one infected livestock for both regions Arusha and Dodoma. These results call for attention to parameters regarding incubation period, the adequate contact rate of Aedes spp. and livestock, the infective periods of livestock and Aedes spp., and the vertical transmission in Aedes species.


---

## Body

## 1. Introduction

Rift valley fever (RVF) is a viral disease that primarily affects animals (such as sheep, horses, cattle, goats, camels, and buffalos) and has the capacity to affect human beings. Rift valley fever virus (RVFV) is a member of thePhlebovirus genus family Bunyaviridae which has been isolated from at least 40 mosquito species in the filed and other arthropods [1, 2]. RVFV infection can cause severe disease in both animals and humans, leading to high disease induced death rate in livestock, long-term health effects in humans, and economic destruction of people [3, 4]. Currently, two types of vaccine for animals exist: a live vaccine and inactivated vaccine. However, the current live vaccine cannot be used for prevention, and prevention using the inactivated vaccine is difficult to sustain in RVF affected countries for economic reasons [3, 5].RVF can be transmitted through an initial aerosol release and subsequent transmission through the mosquito vector. RVFV can remain dormant inAedes spp. mosquito eggs in dry soil for years. During periods of heavy rainfall, larval habitats frequently become flooded, enabling the eggs to hatch and the mosquito population to rapidly increase, spreading the virus to animals on which they feed [6, 7]. Among animals, RVFV is spread primarily by the bite of infected mosquitoes, mainlyAedes andCulex spp. which can acquire the virus from feeding on an infected animal [8–10]. The femaleAedes spp. mosquito is also capable of transmitting the virus directly to her offspring (vertical transmission) via eggs leading to new generations of infected mosquitoes hatching from eggs [11, 12]. This is not the case forCulex spp. mosquito.RVFV can be transmitted to humans through the handling of animal tissue during slaughtering or butchering, assisting with animal births, conducting veterinary procedures, or from the disposal of carcasses or fetuses. Human infections have also resulted from the bites of infected mosquito vector, and by ingesting unpasteurized or uncooked milk and meat of infected animals [8–10, 13]. Transmission of RVFV by blood feeding flies is also possible. To date no human-to-human transmission of RVF has been documented [12].RVF was first reported in Kenya (Africa) in 1931 [8], and it was primarily considered to be of sub-Saharan Africa until September, 2000, when RVF cases were confirmed in Saudi Arabia and Yemen (outside Africa) [14]. The recent outbreak in East Africa is that of 2006-2007 where 684 cases and 155 deaths were confirmed in Kenya, and 264 cases and 109 deaths in Tanzania. There were outbreaks also in Somalia and Sudan in the same period [12].RVF outbreaks in East Africa have been largely correlated with the unusual heavy rainfall associated with El Nin
¨o [15], which consequently flooded manyAedes spp. breeding habitats. The hatching dynamics ofAedes spp. mosquitoes, the main reservoir of RVF in Africa, strongly depends on the rainfall pattern [1]. Eggs need to be flooded to hatch; thus, heavy rainfall results in a massive hatching episode and, consequently, the development of a large vector population. Once infection has been amplified in livestock, secondary vectors such asCulex spp. and other biting flies, which breed in semipermanent pools of water, become involved in the transmission of the virus [16].Global temperature change, on the other hand, would affect the biology of the vectors, including feeding rate and egg production, and the length of the development cycle and the extrinsic incubation period. This may result in high vector density, an increased vector capacity to transmit the virus and a higher transmission rate [16]. When temperature rises above the biological maximum threshold for a species, it may decrease the vector population. Sustained climate shifts may lead to changes in the RVF burden in endemic areas and new outbreaks in areas of similar conditions. Thus, modeling the impact of climate change in the dynamics of RVF and its interventions is important for understanding of the disease.Mathematical epidemiological models have been developed to assess the dynamics of RVF. Gaff et al. [17] proposed a theoretical model in a closed system which included two mosquito populationsAedes andCulex spp. and a livestock population. Their proposed model was a system of ordinary differential equations developed to explain the behaviour of the RVF transmission. The result of the development process was the production of a first-time model of this disease. The model was later modified by Gaff et al. [18] to assess the relative effectiveness of RVF countermeasures such as vector adulticide, vector larvicide, livestock vaccination, and livestock culling.A theoretical model involving mosquito population, livestock and human population has been developed to study the dynamics of the disease using nonlinear differential equations [19]. The results show that the disease prevalence in both human and livestock is more sensitive to livestock and human recruitment rates suggesting isolation of livestock from human as a viable measure during the outbreak. The initial transmission and disease prevalence were found to be highly linked to mosquito population suggesting control measures such as vector adulticides and larvicides to be applied to reduce the mosquito population.Fischer et al. [20] investigated the transmission potential of RVFV among livestock in the Netherlands. The model included the effect of temperature on the biting rate, mosquito population size, and the mortality of the vectors. The results show that high degree of vaccination and vector control strategy are needed to prevent RVF outbreaks. Other studies include that of Xue et al. [21] who developed a network-based metapopulation model approach to RVF epidemics to assess the disease spread in both time and space using network theory, Xue et al. [22] who investigate the spread of RVFV when introduced in United States, Chitnis et al. [23] who developed a model to assess the effect of vertical transmission in vector-borne disease with applications to RVF, and Niu et al. [24] who developed an epidemiological model of RVF with spatial dynamics to study the spatial effects.In this paper, we propose a model that assesses the impact of climate change on the dynamics of RVF. The approach is based on the previous model of RVF transmission by Mpeshe et al. [19] and modifications have been made to incorporate vertical transmission and climate-driven parameters. To simplify the model, only temperature and precipitation are considered in this study. WhileAedes spp. mosquito eggs are naturally infected by RVF virus via vertical transmission, this is not a case forCulex spp. mosquito and, therefore, we assume vertical transmission in our model only forAedes species. To accommodate the impact of climate change we assert that temperature and precipitation can affect the laying and hatching of the eggs as well as the death rate, the effective contact rate, and the incubation period of the mosquitoes. When the epizootic is very high human can also be a source of infection for mosquitoes [25] and, therefore, also we assert in our model the human-to-mosquito transmission when the mosquitoes feed on an infected human.


## 2. Materials and Methods

### 2.1. Model Formulation

The model considers three populations: mosquitoes, livestock, and humans with disease-dependent death rate for livestock and humans. The mosquito population is subdivided into two:Aedes species andCulex species. Due to vertical transmission inAedes spp., we include both infected and uninfected eggs in the model for determining the effect of vertical transmission in the initial transmission of RVF. The mode of transmission of RVF virus from vector to host, host to host, and host to vector is shown in the model flowchart shown by Figure 1. The egg population ofAedes spp. consists of uninfected eggs (
X
a
) and infected eggs (
Y
a
). The population for adultAedes spp. consists of susceptible adults (
S
a
), latently infected adults (
E
a
), and infectious adults (
I
a
). The egg population ofCulex spp. consists of uninfected eggs (
X
c
) only and the population for adultCulex spp. consists of susceptible adults (
S
c
), latently infected adults (
E
c
), and infectious adults (
I
c
). The livestock population consists of susceptible livestock (
S
l
), latently infected livestock (
E
l
), infectious livestock (
I
l
), and recovered livestock (
R
l
). The human population consists of susceptible humans (
S
h
), latently infected humans (
E
h
), infectious humans (
I
h
) and recovered humans (
R
h
). Table 1 shows the model parameters and their description as they have been used in this work. T and P represent temperature and precipitation, respectively.Table 1
Parameters used in the model formulation and their description.
Parameter
Description
Dependent on climate change
1
/
h
a
(
T
,
P
)
Development time ofAedes mosquitoes
Temperature and precipitation
1
/
h
c
(
T
,
P
)
Development rate ofCulex mosquitoes
Temperature and precipitation
b
a
(
T
,
P
)
Number ofAedes eggs laid per day
Temperature and precipitation
b
c
(
T
,
P
)
Number ofCulex eggs laid per day
Temperature and precipitation
b
h
Daily birth rate in humans
Not considered
b
l
Daily birth rate in livestock
Not considered
1
/
μ
a
(
T
)
Lifespan ofAedes mosquitoes
Temperature
1
/
μ
c
(
T
)
Lifespan ofCulex mosquitoes
Temperature
1
/
μ
h
Lifespan of humans
Not considered
1
/
μ
l
Lifespan of livestock
Not considered
1
/
ε
a
(
T
)
Incubation period ofAedes mosquitoes
Temperature
1
/
ε
c
(
T
)
Incubation period ofCulex mosquitoes
Temperature
1
/
ε
h
Incubation period of humans
Not considered
1
/
ε
l
Incubation period of livestock
Not considered
ϕ
l
Death rate of livestock due to disease
Not considered
ϕ
h
Death rate of humans due to disease
Not considered
1
/
γ
l
Infectious period in livestock
Not considered
1
/
γ
h
Infectious period in humans
Not considered
λ
al
(
T
)
Adequate contact rate:Aedes to livestock
Temperature
λ
cl
⁡
(
T
)
Adequate contact rate:Culex to livestock
Temperature
λ
la
(
T
)
Adequate contact rate: livestock toAedes
Temperature
λ
lc
(
T
)
Adequate contact rate: livestock toCulex
Temperature
λ
ah
(
T
)
Adequate contact rate:Aedes to humans
Temperature
λ
ch
(
T
)
Adequate contact rate:Culex to humans
Temperature
λ
ha
(
T
)
Adequate contact rate: humans toAedes
Temperature
λ
hc
(
T
)
Adequate contact rate: humans toCulex
Temperature
λ
lh
Adequate contact rate: livestock to humans
Not considered
f
a
Vertical transmission rate inAedes
Not consideredFigure 1
Flow diagram for the RVF model.The epidemiology cycle of RVF presented by Balenghien et al. [26] and Chevalier et al. [27] is here applied to develop the flow diagram shown by Figure 1. The inclusion of the transmission dynamics of RVF fromAedes spp. to human and vice versa is due to the fact that someAedes spp. such asAedes vexans,Aedes aegpti,Aedes albopictus,Ae. ochraceus,Ae. mcintonshi, andAe. dalzieli and many others numerously feed on humans, and therefore has the capacity to cause infection to human [26–31].Using the parameters in Table1 and the model flow diagram in Figure 1, an SEIR model is derived on the basis of the explanations above using first-order nonlinear ordinary differential equations as follows:Aedes Mosquito (1a)
d
X
a
d
t
=
b
a
(
T
,
P
)
(
N
a
-
f
a
I
a
)
-
h
a
(
T
,
P
)
X
a
,
(1b)
d
Y
a
d
t
=
b
a
(
T
,
P
)
f
a
I
a
-
h
a
(
T
,
P
)
Y
a
,
(1c)
d
S
a
d
t
=
h
a
(
T
,
P
)
X
a
-
μ
a
(
T
)
S
a
-
λ
l
a
(
T
)
I
l
N
l
S
a
-
λ
h
a
(
T
)
I
h
N
h
S
a
,
(1d)
d
E
a
d
t
=
λ
l
a
(
T
)
I
l
N
l
S
a
+
λ
h
a
(
T
)
I
h
N
h
S
a
-
(
ε
a
(
T
)
+
μ
a
(
T
)
)
E
a
,
(1e)
d
I
a
d
t
=
h
a
(
T
,
P
)
Y
a
+
ε
a
(
T
)
E
a
-
μ
a
(
T
)
I
a
,
(1f)
d
N
a
d
t
=
h
a
(
T
,
P
)
(
X
a
+
Y
a
)
-
μ
a
(
T
)
N
a
.Culex Mosquito(2a)
d
X
c
d
t
=
b
c
(
T
,
P
)
N
c
-
h
c
(
T
,
P
)
X
c
,
(2b)
d
S
c
d
t
=
h
c
(
T
,
P
)
X
c
-
μ
c
(
T
)
S
c
-
λ
l
c
(
T
)
I
l
N
l
S
c
-
λ
h
c
(
T
)
I
h
N
h
S
c
,
(2c)
d
E
c
d
t
=
λ
l
c
(
T
)
I
l
N
l
S
c
+
λ
h
c
(
T
)
I
h
N
h
S
c
-
(
ε
c
(
T
)
+
μ
c
(
T
)
)
E
c
,
(2d)
d
I
c
d
t
=
ε
c
(
T
)
E
c
-
μ
c
(
T
)
I
c
,
(2e)
d
N
c
d
t
=
h
c
(
T
,
P
)
X
c
-
μ
c
(
T
)
N
c
.Livestock(3a)
d
S
l
d
t
=
b
l
N
l
-
μ
l
S
l
-
λ
a
l
(
T
)
I
a
N
a
S
l
-
λ
c
l
(
T
)
I
c
N
c
S
l
,
(3b)
d
E
l
d
t
=
λ
a
l
(
T
)
I
a
N
a
S
l
+
λ
c
l
(
T
)
I
c
N
c
S
l
-
(
ε
l
+
μ
l
)
E
l
,
(3c)
d
I
l
d
t
=
ε
l
E
l
-
(
μ
l
+
ϕ
l
+
γ
l
)
I
l
,
(3d)
d
R
l
d
t
=
γ
l
I
l
-
μ
l
R
l
,
(3e)
d
N
l
d
t
=
(
b
l
-
μ
l
)
N
l
-
ϕ
l
I
l
.Humans(4a)
d
S
h
d
t
=
b
h
N
h
-
μ
h
S
h
-
λ
l
h
I
l
N
l
S
h
-
λ
a
h
(
T
)
I
a
N
a
S
h
-
λ
c
h
(
T
)
I
c
N
c
S
h
,
(4b)
d
E
h
d
t
=
λ
l
h
I
l
N
l
S
h
+
λ
a
h
(
T
)
I
a
N
a
S
h
+
λ
c
h
(
T
)
I
c
N
c
S
h
-
(
ε
h
+
μ
h
)
E
h
,
(4c)
d
I
h
d
t
=
ε
h
E
h
-
(
μ
h
+
ϕ
h
+
γ
h
)
I
h
,
(4d)
d
R
h
d
t
=
γ
h
I
h
-
μ
h
R
h
,
(4e)
d
N
h
d
t
=
(
b
h
-
μ
h
)
N
h
-
ϕ
h
I
h
.To test whether the model is well posed epidemiologically and mathematically, we need to investigate the feasibility of the model solution. SinceR
l and R
h can be determined when S
l
,
S
h
,
E
l
,
E
h
,
I
l, and I
h are known, without loss of generality, we omit the expression for d
R
l
/
d
t and d
R
h
/
d
t and write the system in compact form as
(5)
d
X
d
t
=
M
(
x
)
X
+
F
,
where X
=
(
X
a
,
Y
a
,
S
a
,
E
a
,
I
a
,
X
c
,
S
c
,
E
c
,
I
c
,
S
l
,
E
l
,
I
l
,
S
h, E
h
,
I
h
)
T,   M
(
x
) is a 15 by 15 matrix, and F is a column matrix.SubstitutingI
a
=
N
a
-
(
S
a
+
E
a
) in d
X
a
/
d
t, we have
(6)
d
X
a
d
t
=
b
a
(
T
,
P
)
N
a
(
1
-
f
a
)
+
b
a
f
a
(
S
a
+
E
a
)
-
h
a
(
T
,
P
)
X
a
,
and, therefore,
(7)
M
(
x
)
=
[
M
1
(
x
)
0
0
0
M
2
(
x
)
0
0
0
M
3
(
x
)
]
,
where(8)
M
1
(
x
)
=
[
-
h
a
(
T
,
P
)
0
b
a
(
T
,
P
)
f
a
b
a
(
T
,
P
)
f
a
0
0
-
h
a
(
T
,
P
)
0
0
b
a
(
T
,
P
)
f
a
h
a
(
T
,
P
)
0
-
(
μ
a
+
A
)
0
0
0
0
A
-
(
ε
a
(
T
,
P
)
+
μ
a
(
T
,
P
)
)
0
0
h
a
(
T
,
P
)
0
ε
a
(
T
,
P
)
-
μ
a
(
T
,
P
)
]
,
M
2
(
x
)
=
[
-
h
c
(
T
,
P
)
0
0
0
h
c
(
T
,
P
)
-
(
μ
c
(
T
,
P
)
+
B
)
0
0
0
B
-
(
ε
c
(
T
,
P
)
+
μ
c
(
T
,
P
)
)
0
0
0
ε
c
(
T
,
P
)
-
μ
c
(
T
,
P
)
]
,
M
3
(
x
)
=
[
-
(
μ
l
+
C
)
0
0
0
0
0
C
-
(
ε
l
+
μ
l
)
0
0
0
0
0
ε
l
-
(
μ
l
+
ϕ
l
+
γ
l
)
0
0
0
0
0
0
-
(
μ
h
+
D
)
0
0
0
0
0
D
-
(
ε
h
+
μ
h
)
0
0
0
0
0
ε
h
-
(
μ
h
+
ϕ
h
+
γ
h
)
]with (9a)
A
=
λ
l
a
(
T
)
I
l
N
l
+
λ
h
a
(
T
)
I
h
N
h
,
(9b)
B
=
λ
l
c
(
T
)
I
l
N
l
+
λ
h
c
(
T
)
I
h
N
h
,
(9c)
C
=
λ
a
l
(
T
)
I
a
N
a
+
λ
c
l
(
T
)
I
c
N
c
,
(9d)
D
=
λ
l
h
(
T
)
I
l
N
l
+
λ
a
h
(
T
)
I
a
N
a
+
λ
c
h
(
T
)
I
c
N
c
,
(10)
F
=
(
b
a
(
T
,
P
)
N
a
(
1
-
f
a
)
,
0,0
,
0,0
,
b
c
(
T
,
P
)
N
c
,
0,0
,
0
,
b
l
N
l
,
0,0
,
b
h
N
h
,
0,0
)
T
≥
0
.Combining all together, the matrixM
(
x
) is a Metzler matrix for all ℝ
+
15. Therefore, the model system is positively invariant in ℝ
+
15, and F is Lipschitz continuous. Thus, the feasible region for the model system is the set
(11)
𝒟
=
{
(
X
a
,
Y
a
,
S
a
,
E
a
,
I
a
,
X
c
,
S
c
,
E
c
,
I
c
,
S
l
,
E
l
,
I
l
,
S
h
,
E
h
,
I
h
)
ℝ
+
15
≥
0
∈
ℝ
+
15
}
.
That is, the solution remains in the feasible region 𝒟 if it starts in this region. Hence, it is sufficient to study the dynamics of the model in 𝒟.


### 2.2.  Climate Driven Parameters

Several parameters related to mosquito vector, such as the hatching rate, vector mortality and longevity, biting rate, and extrinsic incubation period, depend on the temperature and precipitation. Using the existing studies and information fromAedes vexans,Aedes aegypti,Culex pipiens, andCulex quinquefasciatus [20, 32–34] which are potential vectors of RVF, we generalise the following relations forAedes andCulex spp. mosquitoes.


#### 2.2.1. Hatching Rate or Mosquito Birth Rate,h
(
T
,
P
)

This is the number of eggs hatching into adult mosquitoes at a certain period of time which we also refer to as the mosquito birth rate. It will depend on the the daily survival probabilityρ from eggs to adults and the duration d it takes to develop from eggs to adults. The daily survival probability is assumed to depend independently on temperature, precipitation/rainfall, and prolonged period of desiccation. Thus,
(12)
ρ
(
T
,
P
,
D
)
=
ρ
(
T
)
ρ
(
P
)
ρ
(
D
)
,
where ρ
(
T
) is the daily survival probability of immaturity due to temperature effect T
;
ρ
(
P
) is the daily survival probability of immaturity due to precipitation effect P; and ρ
(
D
) is the daily survival probability of immaturity due to desiccation effect D. The duration of maturation d is assumed to depend on temperature. Therefore, the hatching rate is given by
(13)
h
(
T
,
P
,
D
)
=
ρ
(
T
,
P
,
D
)
d
(
T
)
.


#### 2.2.2.  Survival Probability due to Temperature Effectρ
(
T
)

The daily survival probabilityρ
(
T
) is affected by the duration of maturation d
(
T
) in exponential form, that is,
(14)
ρ
(
T
)
=
exp
⁡
{
-
1
d
(
T
)
}
.
Fitting the data from [34] we obtain that 1
/
d
(
T
)
=
α
1
T
2
+
α
2
T
+
α
3 forCulex spp. and 1
/
d
(
T
)
=
α
1
T
3
+
α
2
T
2
+
α
3
T
+
α
4 forAedes spp., where α
1
=
0.0095
,
α
2
=
-
0.4684
,
α
3
=
5.8343 forCulex spp. and α
1
=
-
0.0025
,
α
2
=
0.2069
,
α
3
=
-
5.5285
,
α
4
=
48.2951 forAedes spp.


#### 2.2.3.  Survival Probability due to Precipitation Effectρ
(
P
)

Precipitation or rainfall is important in creating breeding sites for mosquitoes and causing massive hatching. But excessive rainfall increases mortality of immature due to flushing effect. Since rainfall has two effects, that is, positive and negative effect, we use the idea from [35] and assume the daily survival probability of immaturity due to precipitation effect to be
(15)
ρ
(
P
)
=
(
1
-
exp
⁡
{
-
β
1
(
P
-
P
1
)
}
)
(
1
-
exp
⁡
{
-
β
2
(
P
2
-
P
)
}
)
,
where β
i
(
i
=
1,2
) are the sensitivity parameters; P
1 is the minimum amount of rainfall to support maturity; and P
2 is the maximum amount of rainfall which reduces their survival. For computational purposes we set ρ
(
P
)
=
0 for P
<
P
1 and for P
>
P
2.


#### 2.2.4.  Survival Probability due to Desiccation Effectρ
(
D
)

Lack of precipitation affects the development of the immature. Following the approach by [32] we define the daily survival probability due desiccation as
(16)
ρ
(
D
t
)
=
exp
⁡
(
-
ω
D
t
)
c
+
exp
⁡
(
-
ω
D
t
)
,
where D depends on precipitation P and is defined as
(17)
D
t
=
{
D
t
-
1
+
1
P
t
≤
P
th
0
otherwise
,
where P
th is the threshold precipitation; D
t is the number of consecutive days up to time t when the precipitation P
t was below the threshold P
th;   ω is the sensitivity parameter; and c is the constant that ensures that ρ
(
D
t
) is close to 1 at small values of D
t.


#### 2.2.5.  Daily Egg Laying Rateb
(
T
)

The egg laying rate is assumed to depend on the moisture index. High moisture index correlates with high egg laying rate [33]. To model the daily egg laying rate we employ the equation derived by Gong et al. [33] that
(18)
b
(
T
,
P
)
=
Baseline
Egg
rate
+
E
max
⁡
1
+
exp
⁡
⁡
{
-
(
Moisture
Index
-
E
mean
)
/
E
var
}
,
where Baseline Egg rate is the baseline for fecundity, E
max
⁡ is the maximum daily egg laying rate, E
mean is the mean of daily egg laying rate, and E
var is the variance function.To compute the moisture index, we apply Thornthwaite's moisture index [36] that
(19)
Moisture
Index
(
I
m
)
=
100
(
r
E
0
-
1
)
,
where r is the precipitation rate, and E
0 is the potential evapotranspiration. In absence of the potential evapotranspiration, Linacre's method [37] can be applied. That is,
(20)
E
0
=
700
T
m
/
(
100
-
A
)
+
15
(
T
-
T
d
)
(
80
-
T
)
m
mday
-
1
,
where, T
m
=
T
+
0.00
h with h being the elevation (metres), T is the mean temperature, A is the latitude (degrees), and T
d is the mean dew-point.


#### 2.2.6.  Longevity of Mosquitoes1
/
μ
(
T
)

Different studies show that the longevity of mature mosquitoes also depends on the temperature. To model the longevity, equations deduced by Fischer et al. [20] are applied. That is,
(21)
1
μ
(
T
)
=
a
0
-
a
1
T
,
where a
0
=
25.8
,
a
1
=
0.45 forAedes spp., and a
0
=
69.1
,
a
1
=
2.14 forCulex spp.


#### 2.2.7.  Extrinsic Incubation Period of Mosquitoes1
/
ɛ
(
T
)

Extrinsic incubation period is the time between a blood meal on an infections host and the first successful transmission from vector to host during another blood meal. We also adapt the expressions by Fischer et al. [20]. That is,
(22)
1
ɛ
(
T
)
=
ε
max
⁡
-
ε
slope
T
,
where ε
max
⁡
=
18.9
,
ε
slope
=
0.30 forAedes spp., and ε
max
⁡
=
11.3
,
ε
slope
=
0.30 forCulex spp.


#### 2.2.8.  Adequate Contact Rateλ
(
T
)

Adequate contact rate is contact which is sufficient for transmission of the infection from an infective to a susceptible. Thus, in this study(23)
adequate
contact
rate
=
biting
rate
×
probability
of
transmission
.
The biting rate depends on temperature, and we assume a linear relationship as in Fischer et al. [20]. That is,
(24)
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
,
where a
slope
=
0.0173
,
T
min
⁡
=
9.60 for all mosquito species. Assume that the probability of transmission is independent to temperature, we have (25a)
λ
a
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
l
,
ρ
a
l
=
0.70
,
(25b)
λ
c
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
l
,
ρ
c
l
=
0.78
,
(25c)
λ
l
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
a
,
ρ
l
a
=
0.38
,
(25d)
λ
l
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
c
,
ρ
l
c
=
0.22
,
(25e)
λ
a
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
h
,
ρ
a
h
=
0.01
,
(25f)
λ
h
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
a
,
ρ
h
a
=
0.05
,
(25g)
λ
c
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
h
,
ρ
c
h
=
0.01
,
(25h)
λ
h
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
c
,
ρ
h
c
=
0.015
.


### 2.3.  The Basic Reproduction Number

The basic reproduction numberℛ
0 is computed using the method of next generation matrix as outlined by [38]. Let k
i
j be the expected number of the new cases of type i caused by one infected individual of type j, during the entire period of infectiousness. Define a matrix K whose entries are k
i
j, that is, K
=
[
k
i
j
]. Then, ℛ
0
=
ρ
(
K
), where ρ
(
K
) is spectral radius of K. For our model, we define four type-at-infection consisting of two vectors and two hosts, namely,Aedes spp. (type 1),Culex spp. (type 2), livestock (type 3), and humans (type 4). The resulting next generation matrix is
(26)
K
=
[
k
11
k
12
k
13
k
14
k
21
k
22
k
23
k
24
k
31
k
32
k
33
k
34
k
41
k
42
k
43
k
44
]
,
where k
11 is the expected number of infected eggs laid byAedes spp. via transovarial transmission, k
12 is the expected number of infectedAedes spp. due to one infectedCulex, k
21 is the expected number of infectedCulex spp. due to one infectedAedes spp., k
13 is the expected number of infectedAedes spp. due to one infected livestock, k
31 is the expected number of infected livestock due to one infectedAedes spp., k
14 is the expected number of infectedAedes spp. due to one infected human, k
41 is the expected number of infected humans due to one infectedAedes spp., k
22 is the expected number of infected eggs laid byCulex spp. via transovarial transmission, k
23 is the expected number of infectedCulex spp. due to one infected livestock, k
32 is the expected number of infected livestock due to one infectedCulex spp., k
24 is the expected number of infectedCulex spp. due to one infected human, k
42 is the expected number of infected humans due to one infectedCulex spp., k
33 is the expected number of infected livestock due to one infected livestock, k
34 is the expected number of infected livestock due to one infected human, k
43 is the expected number of infected humans due to one infected livestock, and and k
44 is the expected number of infected humans due to one infected human.Since there is no vertical transmission inCulex spp., then k
22
=
0. The same applies for k
33 and k
44. AlsoAedes spp. cannot infectCulex spp. and vice versa; therefore, k
12
=
k
21
=
0. Humans cannot infect livestock, so k
34
=
0. Hence, we have
(27)
K
=
[
k
11
0
k
13
k
14
0
0
k
23
k
24
k
31
k
32
0
0
k
41
k
42
k
43
0
]
.
The entry k
i
j depends on the probability that the individual of type j survives the incubation, the adequate contact rate: individual type j to individual type i, and the infective period of individual of type j. For example, k
13 will depend on the probability that livestock survives the incubation period, the adequate contact rate from livestock toAedes spp., and the infective period of livestock. We therefore derive the k
i
j values as follows: (28a)
k
11
=
b
a
(
T
,
P
)
f
a
μ
a
,
k
13
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
a
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
,
(28b)
k
14
=
(
ε
h
ε
h
+
μ
h
)
(
λ
h
a
(
T
)
μ
h
+
ϕ
h
+
γ
h
)
,
k
23
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
c
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
,
(28c)
k
24
=
(
ε
h
ε
h
+
μ
h
)
(
λ
h
c
(
T
)
μ
h
+
ϕ
h
+
γ
h
)
,
k
31
=
(
ε
a
(
T
)
ε
a
(
T
)
+
μ
a
(
T
)
)
(
λ
a
l
(
T
)
μ
a
(
T
)
)
,
(28d)
k
32
=
(
ε
c
(
T
)
ε
c
(
T
)
+
μ
c
(
T
)
)
(
λ
c
l
(
T
)
μ
c
(
T
)
)
,
k
41
=
(
ε
a
(
T
)
ε
a
(
T
)
+
μ
a
(
T
)
)
(
λ
a
h
(
T
)
μ
a
(
T
)
)
,
(28e)
k
42
=
(
ε
c
(
T
)
ε
c
(
T
)
+
μ
c
(
T
)
)
(
λ
c
h
(
T
)
μ
c
(
T
)
)
,
k
43
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
h
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
.


### 2.4. Sensitivity and Elasticity Analyses ofℛ
0

Sensitivities quantify howℛ
0 changes in response to the small shifts in the value of a parameter, while elasticities quantify the proportional change in ℛ
0 in response to the proportional change in a parameter. Both sensitivity and elasticity values can be used to judge which parameters are important to measure accurately and where variation in parameters will translate into variation in ℛ
0.Caswell [39] developed a way to quantify sensitivity and elasticity of the growth rate λ to changes in vital rates a
i
j where a
i
j are the entries of population matrix A. That is, the sensitivity of the growth rate λ to changes in vital rates a
i
j is given by
(29)
s
i
j
=
∂
λ
∂
a
i
j
=
v
i
w
j
〈
w
,
v
〉
,
wherew andv are the right and left eigenvectors, respectively, corresponding to the dominant eigenvalue λ of the matrix A, and 〈
w
,
v
〉 is the dot product ofw andv. In case a
i
j is a function of other lower-level parameters, then, the chain rule can be applied to estimate the sensitivity of λ to changes in any model parameter p as
(30)
s
(
p
)
=
∂
λ
∂
p
=
∑
i
j
‍
∂
λ
∂
a
i
j
∂
a
i
j
∂
p
.The elasticity of the growth rateλ to changes in vital rates a
i
j, the entries of population matrix A, is given by
(31)
e
i
j
=
∂
log
⁡
λ
∂
log
⁡
a
i
j
=
a
i
j
λ
∂
λ
∂
a
i
j
.
For a
i
j a function of other lower-level parameters p, the elasticity is given by
(32)
e
(
p
)
=
p
λ
∂
λ
∂
p
=
p
λ
∑
i
j
‍
∂
λ
∂
a
i
j
∂
a
i
j
∂
p
.The theory of sensitivity analysis developed for the matrix models by Caswell [39] can be extended to the disease models to study the sensitivity and elasticity of ℛ
0 to the changes in the reproduction numbers k
i
j or the parameters defining them. Thus, the sensitivity s
i
j of a matrix element k
i
j is defined as the change in the eigenvalue (ℛ
0) due to change in k
i
j given by
(33)
s
i
j
=
∂
ℛ
0
∂
k
i
j
.
For individual parameter, the sensitivity s
(
p
) is given by
(34)
s
(
p
)
=
∑
i
j
‍
∂
ℛ
0
∂
k
i
j
∂
k
i
j
∂
p
.
The elasticity e
i
j of a matrix element k
i
j is defined as
(35)
e
i
j
=
k
i
j
ℛ
0
∂
ℛ
0
∂
k
i
j
.
For individual parameters p, the elasticity is given by
(36)
e
(
p
)
=
p
ℛ
0
∑
i
j
‍
∂
ℛ
0
∂
k
i
j
∂
k
i
j
∂
p
.In order to study the impact of climate change to climate-driven parameter in the distribution ofℛ
0 we use climate data from two different regions in Tanzania, namely, Arusha and Dodoma for the 2006-2007 outbreak. According to WHO [12], RVF was reported in 10 out of the 21 regions of Tanzania where 12 cases were reported in Arusha region, 1 in Dar es Salaam, 156 in Dodoma, 4 in Iringa, 6 in Manyara, 50 in Morogoro, 5 in Mwanza, 5 in the Pwani, 24 in Singida, and 1 in Tanga regions. From the data we find that Dodoma has more than 50% of the total cases giving a justification for being a case of study, and Arusha is considered in this study because the first case was reported in January 2007 in this region.


## 2.1. Model Formulation

The model considers three populations: mosquitoes, livestock, and humans with disease-dependent death rate for livestock and humans. The mosquito population is subdivided into two:Aedes species andCulex species. Due to vertical transmission inAedes spp., we include both infected and uninfected eggs in the model for determining the effect of vertical transmission in the initial transmission of RVF. The mode of transmission of RVF virus from vector to host, host to host, and host to vector is shown in the model flowchart shown by Figure 1. The egg population ofAedes spp. consists of uninfected eggs (
X
a
) and infected eggs (
Y
a
). The population for adultAedes spp. consists of susceptible adults (
S
a
), latently infected adults (
E
a
), and infectious adults (
I
a
). The egg population ofCulex spp. consists of uninfected eggs (
X
c
) only and the population for adultCulex spp. consists of susceptible adults (
S
c
), latently infected adults (
E
c
), and infectious adults (
I
c
). The livestock population consists of susceptible livestock (
S
l
), latently infected livestock (
E
l
), infectious livestock (
I
l
), and recovered livestock (
R
l
). The human population consists of susceptible humans (
S
h
), latently infected humans (
E
h
), infectious humans (
I
h
) and recovered humans (
R
h
). Table 1 shows the model parameters and their description as they have been used in this work. T and P represent temperature and precipitation, respectively.Table 1
Parameters used in the model formulation and their description.
Parameter
Description
Dependent on climate change
1
/
h
a
(
T
,
P
)
Development time ofAedes mosquitoes
Temperature and precipitation
1
/
h
c
(
T
,
P
)
Development rate ofCulex mosquitoes
Temperature and precipitation
b
a
(
T
,
P
)
Number ofAedes eggs laid per day
Temperature and precipitation
b
c
(
T
,
P
)
Number ofCulex eggs laid per day
Temperature and precipitation
b
h
Daily birth rate in humans
Not considered
b
l
Daily birth rate in livestock
Not considered
1
/
μ
a
(
T
)
Lifespan ofAedes mosquitoes
Temperature
1
/
μ
c
(
T
)
Lifespan ofCulex mosquitoes
Temperature
1
/
μ
h
Lifespan of humans
Not considered
1
/
μ
l
Lifespan of livestock
Not considered
1
/
ε
a
(
T
)
Incubation period ofAedes mosquitoes
Temperature
1
/
ε
c
(
T
)
Incubation period ofCulex mosquitoes
Temperature
1
/
ε
h
Incubation period of humans
Not considered
1
/
ε
l
Incubation period of livestock
Not considered
ϕ
l
Death rate of livestock due to disease
Not considered
ϕ
h
Death rate of humans due to disease
Not considered
1
/
γ
l
Infectious period in livestock
Not considered
1
/
γ
h
Infectious period in humans
Not considered
λ
al
(
T
)
Adequate contact rate:Aedes to livestock
Temperature
λ
cl
⁡
(
T
)
Adequate contact rate:Culex to livestock
Temperature
λ
la
(
T
)
Adequate contact rate: livestock toAedes
Temperature
λ
lc
(
T
)
Adequate contact rate: livestock toCulex
Temperature
λ
ah
(
T
)
Adequate contact rate:Aedes to humans
Temperature
λ
ch
(
T
)
Adequate contact rate:Culex to humans
Temperature
λ
ha
(
T
)
Adequate contact rate: humans toAedes
Temperature
λ
hc
(
T
)
Adequate contact rate: humans toCulex
Temperature
λ
lh
Adequate contact rate: livestock to humans
Not considered
f
a
Vertical transmission rate inAedes
Not consideredFigure 1
Flow diagram for the RVF model.The epidemiology cycle of RVF presented by Balenghien et al. [26] and Chevalier et al. [27] is here applied to develop the flow diagram shown by Figure 1. The inclusion of the transmission dynamics of RVF fromAedes spp. to human and vice versa is due to the fact that someAedes spp. such asAedes vexans,Aedes aegpti,Aedes albopictus,Ae. ochraceus,Ae. mcintonshi, andAe. dalzieli and many others numerously feed on humans, and therefore has the capacity to cause infection to human [26–31].Using the parameters in Table1 and the model flow diagram in Figure 1, an SEIR model is derived on the basis of the explanations above using first-order nonlinear ordinary differential equations as follows:Aedes Mosquito (1a)
d
X
a
d
t
=
b
a
(
T
,
P
)
(
N
a
-
f
a
I
a
)
-
h
a
(
T
,
P
)
X
a
,
(1b)
d
Y
a
d
t
=
b
a
(
T
,
P
)
f
a
I
a
-
h
a
(
T
,
P
)
Y
a
,
(1c)
d
S
a
d
t
=
h
a
(
T
,
P
)
X
a
-
μ
a
(
T
)
S
a
-
λ
l
a
(
T
)
I
l
N
l
S
a
-
λ
h
a
(
T
)
I
h
N
h
S
a
,
(1d)
d
E
a
d
t
=
λ
l
a
(
T
)
I
l
N
l
S
a
+
λ
h
a
(
T
)
I
h
N
h
S
a
-
(
ε
a
(
T
)
+
μ
a
(
T
)
)
E
a
,
(1e)
d
I
a
d
t
=
h
a
(
T
,
P
)
Y
a
+
ε
a
(
T
)
E
a
-
μ
a
(
T
)
I
a
,
(1f)
d
N
a
d
t
=
h
a
(
T
,
P
)
(
X
a
+
Y
a
)
-
μ
a
(
T
)
N
a
.Culex Mosquito(2a)
d
X
c
d
t
=
b
c
(
T
,
P
)
N
c
-
h
c
(
T
,
P
)
X
c
,
(2b)
d
S
c
d
t
=
h
c
(
T
,
P
)
X
c
-
μ
c
(
T
)
S
c
-
λ
l
c
(
T
)
I
l
N
l
S
c
-
λ
h
c
(
T
)
I
h
N
h
S
c
,
(2c)
d
E
c
d
t
=
λ
l
c
(
T
)
I
l
N
l
S
c
+
λ
h
c
(
T
)
I
h
N
h
S
c
-
(
ε
c
(
T
)
+
μ
c
(
T
)
)
E
c
,
(2d)
d
I
c
d
t
=
ε
c
(
T
)
E
c
-
μ
c
(
T
)
I
c
,
(2e)
d
N
c
d
t
=
h
c
(
T
,
P
)
X
c
-
μ
c
(
T
)
N
c
.Livestock(3a)
d
S
l
d
t
=
b
l
N
l
-
μ
l
S
l
-
λ
a
l
(
T
)
I
a
N
a
S
l
-
λ
c
l
(
T
)
I
c
N
c
S
l
,
(3b)
d
E
l
d
t
=
λ
a
l
(
T
)
I
a
N
a
S
l
+
λ
c
l
(
T
)
I
c
N
c
S
l
-
(
ε
l
+
μ
l
)
E
l
,
(3c)
d
I
l
d
t
=
ε
l
E
l
-
(
μ
l
+
ϕ
l
+
γ
l
)
I
l
,
(3d)
d
R
l
d
t
=
γ
l
I
l
-
μ
l
R
l
,
(3e)
d
N
l
d
t
=
(
b
l
-
μ
l
)
N
l
-
ϕ
l
I
l
.Humans(4a)
d
S
h
d
t
=
b
h
N
h
-
μ
h
S
h
-
λ
l
h
I
l
N
l
S
h
-
λ
a
h
(
T
)
I
a
N
a
S
h
-
λ
c
h
(
T
)
I
c
N
c
S
h
,
(4b)
d
E
h
d
t
=
λ
l
h
I
l
N
l
S
h
+
λ
a
h
(
T
)
I
a
N
a
S
h
+
λ
c
h
(
T
)
I
c
N
c
S
h
-
(
ε
h
+
μ
h
)
E
h
,
(4c)
d
I
h
d
t
=
ε
h
E
h
-
(
μ
h
+
ϕ
h
+
γ
h
)
I
h
,
(4d)
d
R
h
d
t
=
γ
h
I
h
-
μ
h
R
h
,
(4e)
d
N
h
d
t
=
(
b
h
-
μ
h
)
N
h
-
ϕ
h
I
h
.To test whether the model is well posed epidemiologically and mathematically, we need to investigate the feasibility of the model solution. SinceR
l and R
h can be determined when S
l
,
S
h
,
E
l
,
E
h
,
I
l, and I
h are known, without loss of generality, we omit the expression for d
R
l
/
d
t and d
R
h
/
d
t and write the system in compact form as
(5)
d
X
d
t
=
M
(
x
)
X
+
F
,
where X
=
(
X
a
,
Y
a
,
S
a
,
E
a
,
I
a
,
X
c
,
S
c
,
E
c
,
I
c
,
S
l
,
E
l
,
I
l
,
S
h, E
h
,
I
h
)
T,   M
(
x
) is a 15 by 15 matrix, and F is a column matrix.SubstitutingI
a
=
N
a
-
(
S
a
+
E
a
) in d
X
a
/
d
t, we have
(6)
d
X
a
d
t
=
b
a
(
T
,
P
)
N
a
(
1
-
f
a
)
+
b
a
f
a
(
S
a
+
E
a
)
-
h
a
(
T
,
P
)
X
a
,
and, therefore,
(7)
M
(
x
)
=
[
M
1
(
x
)
0
0
0
M
2
(
x
)
0
0
0
M
3
(
x
)
]
,
where(8)
M
1
(
x
)
=
[
-
h
a
(
T
,
P
)
0
b
a
(
T
,
P
)
f
a
b
a
(
T
,
P
)
f
a
0
0
-
h
a
(
T
,
P
)
0
0
b
a
(
T
,
P
)
f
a
h
a
(
T
,
P
)
0
-
(
μ
a
+
A
)
0
0
0
0
A
-
(
ε
a
(
T
,
P
)
+
μ
a
(
T
,
P
)
)
0
0
h
a
(
T
,
P
)
0
ε
a
(
T
,
P
)
-
μ
a
(
T
,
P
)
]
,
M
2
(
x
)
=
[
-
h
c
(
T
,
P
)
0
0
0
h
c
(
T
,
P
)
-
(
μ
c
(
T
,
P
)
+
B
)
0
0
0
B
-
(
ε
c
(
T
,
P
)
+
μ
c
(
T
,
P
)
)
0
0
0
ε
c
(
T
,
P
)
-
μ
c
(
T
,
P
)
]
,
M
3
(
x
)
=
[
-
(
μ
l
+
C
)
0
0
0
0
0
C
-
(
ε
l
+
μ
l
)
0
0
0
0
0
ε
l
-
(
μ
l
+
ϕ
l
+
γ
l
)
0
0
0
0
0
0
-
(
μ
h
+
D
)
0
0
0
0
0
D
-
(
ε
h
+
μ
h
)
0
0
0
0
0
ε
h
-
(
μ
h
+
ϕ
h
+
γ
h
)
]with (9a)
A
=
λ
l
a
(
T
)
I
l
N
l
+
λ
h
a
(
T
)
I
h
N
h
,
(9b)
B
=
λ
l
c
(
T
)
I
l
N
l
+
λ
h
c
(
T
)
I
h
N
h
,
(9c)
C
=
λ
a
l
(
T
)
I
a
N
a
+
λ
c
l
(
T
)
I
c
N
c
,
(9d)
D
=
λ
l
h
(
T
)
I
l
N
l
+
λ
a
h
(
T
)
I
a
N
a
+
λ
c
h
(
T
)
I
c
N
c
,
(10)
F
=
(
b
a
(
T
,
P
)
N
a
(
1
-
f
a
)
,
0,0
,
0,0
,
b
c
(
T
,
P
)
N
c
,
0,0
,
0
,
b
l
N
l
,
0,0
,
b
h
N
h
,
0,0
)
T
≥
0
.Combining all together, the matrixM
(
x
) is a Metzler matrix for all ℝ
+
15. Therefore, the model system is positively invariant in ℝ
+
15, and F is Lipschitz continuous. Thus, the feasible region for the model system is the set
(11)
𝒟
=
{
(
X
a
,
Y
a
,
S
a
,
E
a
,
I
a
,
X
c
,
S
c
,
E
c
,
I
c
,
S
l
,
E
l
,
I
l
,
S
h
,
E
h
,
I
h
)
ℝ
+
15
≥
0
∈
ℝ
+
15
}
.
That is, the solution remains in the feasible region 𝒟 if it starts in this region. Hence, it is sufficient to study the dynamics of the model in 𝒟.


## 2.2.  Climate Driven Parameters

Several parameters related to mosquito vector, such as the hatching rate, vector mortality and longevity, biting rate, and extrinsic incubation period, depend on the temperature and precipitation. Using the existing studies and information fromAedes vexans,Aedes aegypti,Culex pipiens, andCulex quinquefasciatus [20, 32–34] which are potential vectors of RVF, we generalise the following relations forAedes andCulex spp. mosquitoes.


### 2.2.1. Hatching Rate or Mosquito Birth Rate,h
(
T
,
P
)

This is the number of eggs hatching into adult mosquitoes at a certain period of time which we also refer to as the mosquito birth rate. It will depend on the the daily survival probabilityρ from eggs to adults and the duration d it takes to develop from eggs to adults. The daily survival probability is assumed to depend independently on temperature, precipitation/rainfall, and prolonged period of desiccation. Thus,
(12)
ρ
(
T
,
P
,
D
)
=
ρ
(
T
)
ρ
(
P
)
ρ
(
D
)
,
where ρ
(
T
) is the daily survival probability of immaturity due to temperature effect T
;
ρ
(
P
) is the daily survival probability of immaturity due to precipitation effect P; and ρ
(
D
) is the daily survival probability of immaturity due to desiccation effect D. The duration of maturation d is assumed to depend on temperature. Therefore, the hatching rate is given by
(13)
h
(
T
,
P
,
D
)
=
ρ
(
T
,
P
,
D
)
d
(
T
)
.


### 2.2.2.  Survival Probability due to Temperature Effectρ
(
T
)

The daily survival probabilityρ
(
T
) is affected by the duration of maturation d
(
T
) in exponential form, that is,
(14)
ρ
(
T
)
=
exp
⁡
{
-
1
d
(
T
)
}
.
Fitting the data from [34] we obtain that 1
/
d
(
T
)
=
α
1
T
2
+
α
2
T
+
α
3 forCulex spp. and 1
/
d
(
T
)
=
α
1
T
3
+
α
2
T
2
+
α
3
T
+
α
4 forAedes spp., where α
1
=
0.0095
,
α
2
=
-
0.4684
,
α
3
=
5.8343 forCulex spp. and α
1
=
-
0.0025
,
α
2
=
0.2069
,
α
3
=
-
5.5285
,
α
4
=
48.2951 forAedes spp.


### 2.2.3.  Survival Probability due to Precipitation Effectρ
(
P
)

Precipitation or rainfall is important in creating breeding sites for mosquitoes and causing massive hatching. But excessive rainfall increases mortality of immature due to flushing effect. Since rainfall has two effects, that is, positive and negative effect, we use the idea from [35] and assume the daily survival probability of immaturity due to precipitation effect to be
(15)
ρ
(
P
)
=
(
1
-
exp
⁡
{
-
β
1
(
P
-
P
1
)
}
)
(
1
-
exp
⁡
{
-
β
2
(
P
2
-
P
)
}
)
,
where β
i
(
i
=
1,2
) are the sensitivity parameters; P
1 is the minimum amount of rainfall to support maturity; and P
2 is the maximum amount of rainfall which reduces their survival. For computational purposes we set ρ
(
P
)
=
0 for P
<
P
1 and for P
>
P
2.


### 2.2.4.  Survival Probability due to Desiccation Effectρ
(
D
)

Lack of precipitation affects the development of the immature. Following the approach by [32] we define the daily survival probability due desiccation as
(16)
ρ
(
D
t
)
=
exp
⁡
(
-
ω
D
t
)
c
+
exp
⁡
(
-
ω
D
t
)
,
where D depends on precipitation P and is defined as
(17)
D
t
=
{
D
t
-
1
+
1
P
t
≤
P
th
0
otherwise
,
where P
th is the threshold precipitation; D
t is the number of consecutive days up to time t when the precipitation P
t was below the threshold P
th;   ω is the sensitivity parameter; and c is the constant that ensures that ρ
(
D
t
) is close to 1 at small values of D
t.


### 2.2.5.  Daily Egg Laying Rateb
(
T
)

The egg laying rate is assumed to depend on the moisture index. High moisture index correlates with high egg laying rate [33]. To model the daily egg laying rate we employ the equation derived by Gong et al. [33] that
(18)
b
(
T
,
P
)
=
Baseline
Egg
rate
+
E
max
⁡
1
+
exp
⁡
⁡
{
-
(
Moisture
Index
-
E
mean
)
/
E
var
}
,
where Baseline Egg rate is the baseline for fecundity, E
max
⁡ is the maximum daily egg laying rate, E
mean is the mean of daily egg laying rate, and E
var is the variance function.To compute the moisture index, we apply Thornthwaite's moisture index [36] that
(19)
Moisture
Index
(
I
m
)
=
100
(
r
E
0
-
1
)
,
where r is the precipitation rate, and E
0 is the potential evapotranspiration. In absence of the potential evapotranspiration, Linacre's method [37] can be applied. That is,
(20)
E
0
=
700
T
m
/
(
100
-
A
)
+
15
(
T
-
T
d
)
(
80
-
T
)
m
mday
-
1
,
where, T
m
=
T
+
0.00
h with h being the elevation (metres), T is the mean temperature, A is the latitude (degrees), and T
d is the mean dew-point.


### 2.2.6.  Longevity of Mosquitoes1
/
μ
(
T
)

Different studies show that the longevity of mature mosquitoes also depends on the temperature. To model the longevity, equations deduced by Fischer et al. [20] are applied. That is,
(21)
1
μ
(
T
)
=
a
0
-
a
1
T
,
where a
0
=
25.8
,
a
1
=
0.45 forAedes spp., and a
0
=
69.1
,
a
1
=
2.14 forCulex spp.


### 2.2.7.  Extrinsic Incubation Period of Mosquitoes1
/
ɛ
(
T
)

Extrinsic incubation period is the time between a blood meal on an infections host and the first successful transmission from vector to host during another blood meal. We also adapt the expressions by Fischer et al. [20]. That is,
(22)
1
ɛ
(
T
)
=
ε
max
⁡
-
ε
slope
T
,
where ε
max
⁡
=
18.9
,
ε
slope
=
0.30 forAedes spp., and ε
max
⁡
=
11.3
,
ε
slope
=
0.30 forCulex spp.


### 2.2.8.  Adequate Contact Rateλ
(
T
)

Adequate contact rate is contact which is sufficient for transmission of the infection from an infective to a susceptible. Thus, in this study(23)
adequate
contact
rate
=
biting
rate
×
probability
of
transmission
.
The biting rate depends on temperature, and we assume a linear relationship as in Fischer et al. [20]. That is,
(24)
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
,
where a
slope
=
0.0173
,
T
min
⁡
=
9.60 for all mosquito species. Assume that the probability of transmission is independent to temperature, we have (25a)
λ
a
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
l
,
ρ
a
l
=
0.70
,
(25b)
λ
c
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
l
,
ρ
c
l
=
0.78
,
(25c)
λ
l
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
a
,
ρ
l
a
=
0.38
,
(25d)
λ
l
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
c
,
ρ
l
c
=
0.22
,
(25e)
λ
a
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
h
,
ρ
a
h
=
0.01
,
(25f)
λ
h
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
a
,
ρ
h
a
=
0.05
,
(25g)
λ
c
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
h
,
ρ
c
h
=
0.01
,
(25h)
λ
h
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
c
,
ρ
h
c
=
0.015
.


## 2.2.1. Hatching Rate or Mosquito Birth Rate,h
(
T
,
P
)

This is the number of eggs hatching into adult mosquitoes at a certain period of time which we also refer to as the mosquito birth rate. It will depend on the the daily survival probabilityρ from eggs to adults and the duration d it takes to develop from eggs to adults. The daily survival probability is assumed to depend independently on temperature, precipitation/rainfall, and prolonged period of desiccation. Thus,
(12)
ρ
(
T
,
P
,
D
)
=
ρ
(
T
)
ρ
(
P
)
ρ
(
D
)
,
where ρ
(
T
) is the daily survival probability of immaturity due to temperature effect T
;
ρ
(
P
) is the daily survival probability of immaturity due to precipitation effect P; and ρ
(
D
) is the daily survival probability of immaturity due to desiccation effect D. The duration of maturation d is assumed to depend on temperature. Therefore, the hatching rate is given by
(13)
h
(
T
,
P
,
D
)
=
ρ
(
T
,
P
,
D
)
d
(
T
)
.


## 2.2.2.  Survival Probability due to Temperature Effectρ
(
T
)

The daily survival probabilityρ
(
T
) is affected by the duration of maturation d
(
T
) in exponential form, that is,
(14)
ρ
(
T
)
=
exp
⁡
{
-
1
d
(
T
)
}
.
Fitting the data from [34] we obtain that 1
/
d
(
T
)
=
α
1
T
2
+
α
2
T
+
α
3 forCulex spp. and 1
/
d
(
T
)
=
α
1
T
3
+
α
2
T
2
+
α
3
T
+
α
4 forAedes spp., where α
1
=
0.0095
,
α
2
=
-
0.4684
,
α
3
=
5.8343 forCulex spp. and α
1
=
-
0.0025
,
α
2
=
0.2069
,
α
3
=
-
5.5285
,
α
4
=
48.2951 forAedes spp.


## 2.2.3.  Survival Probability due to Precipitation Effectρ
(
P
)

Precipitation or rainfall is important in creating breeding sites for mosquitoes and causing massive hatching. But excessive rainfall increases mortality of immature due to flushing effect. Since rainfall has two effects, that is, positive and negative effect, we use the idea from [35] and assume the daily survival probability of immaturity due to precipitation effect to be
(15)
ρ
(
P
)
=
(
1
-
exp
⁡
{
-
β
1
(
P
-
P
1
)
}
)
(
1
-
exp
⁡
{
-
β
2
(
P
2
-
P
)
}
)
,
where β
i
(
i
=
1,2
) are the sensitivity parameters; P
1 is the minimum amount of rainfall to support maturity; and P
2 is the maximum amount of rainfall which reduces their survival. For computational purposes we set ρ
(
P
)
=
0 for P
<
P
1 and for P
>
P
2.


## 2.2.4.  Survival Probability due to Desiccation Effectρ
(
D
)

Lack of precipitation affects the development of the immature. Following the approach by [32] we define the daily survival probability due desiccation as
(16)
ρ
(
D
t
)
=
exp
⁡
(
-
ω
D
t
)
c
+
exp
⁡
(
-
ω
D
t
)
,
where D depends on precipitation P and is defined as
(17)
D
t
=
{
D
t
-
1
+
1
P
t
≤
P
th
0
otherwise
,
where P
th is the threshold precipitation; D
t is the number of consecutive days up to time t when the precipitation P
t was below the threshold P
th;   ω is the sensitivity parameter; and c is the constant that ensures that ρ
(
D
t
) is close to 1 at small values of D
t.


## 2.2.5.  Daily Egg Laying Rateb
(
T
)

The egg laying rate is assumed to depend on the moisture index. High moisture index correlates with high egg laying rate [33]. To model the daily egg laying rate we employ the equation derived by Gong et al. [33] that
(18)
b
(
T
,
P
)
=
Baseline
Egg
rate
+
E
max
⁡
1
+
exp
⁡
⁡
{
-
(
Moisture
Index
-
E
mean
)
/
E
var
}
,
where Baseline Egg rate is the baseline for fecundity, E
max
⁡ is the maximum daily egg laying rate, E
mean is the mean of daily egg laying rate, and E
var is the variance function.To compute the moisture index, we apply Thornthwaite's moisture index [36] that
(19)
Moisture
Index
(
I
m
)
=
100
(
r
E
0
-
1
)
,
where r is the precipitation rate, and E
0 is the potential evapotranspiration. In absence of the potential evapotranspiration, Linacre's method [37] can be applied. That is,
(20)
E
0
=
700
T
m
/
(
100
-
A
)
+
15
(
T
-
T
d
)
(
80
-
T
)
m
mday
-
1
,
where, T
m
=
T
+
0.00
h with h being the elevation (metres), T is the mean temperature, A is the latitude (degrees), and T
d is the mean dew-point.


## 2.2.6.  Longevity of Mosquitoes1
/
μ
(
T
)

Different studies show that the longevity of mature mosquitoes also depends on the temperature. To model the longevity, equations deduced by Fischer et al. [20] are applied. That is,
(21)
1
μ
(
T
)
=
a
0
-
a
1
T
,
where a
0
=
25.8
,
a
1
=
0.45 forAedes spp., and a
0
=
69.1
,
a
1
=
2.14 forCulex spp.


## 2.2.7.  Extrinsic Incubation Period of Mosquitoes1
/
ɛ
(
T
)

Extrinsic incubation period is the time between a blood meal on an infections host and the first successful transmission from vector to host during another blood meal. We also adapt the expressions by Fischer et al. [20]. That is,
(22)
1
ɛ
(
T
)
=
ε
max
⁡
-
ε
slope
T
,
where ε
max
⁡
=
18.9
,
ε
slope
=
0.30 forAedes spp., and ε
max
⁡
=
11.3
,
ε
slope
=
0.30 forCulex spp.


## 2.2.8.  Adequate Contact Rateλ
(
T
)

Adequate contact rate is contact which is sufficient for transmission of the infection from an infective to a susceptible. Thus, in this study(23)
adequate
contact
rate
=
biting
rate
×
probability
of
transmission
.
The biting rate depends on temperature, and we assume a linear relationship as in Fischer et al. [20]. That is,
(24)
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
,
where a
slope
=
0.0173
,
T
min
⁡
=
9.60 for all mosquito species. Assume that the probability of transmission is independent to temperature, we have (25a)
λ
a
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
l
,
ρ
a
l
=
0.70
,
(25b)
λ
c
l
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
l
,
ρ
c
l
=
0.78
,
(25c)
λ
l
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
a
,
ρ
l
a
=
0.38
,
(25d)
λ
l
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
l
c
,
ρ
l
c
=
0.22
,
(25e)
λ
a
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
a
h
,
ρ
a
h
=
0.01
,
(25f)
λ
h
a
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
a
,
ρ
h
a
=
0.05
,
(25g)
λ
c
h
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
c
h
,
ρ
c
h
=
0.01
,
(25h)
λ
h
c
(
T
)
=
a
slope
(
T
-
T
min
⁡
)
ρ
h
c
,
ρ
h
c
=
0.015
.


## 2.3.  The Basic Reproduction Number

The basic reproduction numberℛ
0 is computed using the method of next generation matrix as outlined by [38]. Let k
i
j be the expected number of the new cases of type i caused by one infected individual of type j, during the entire period of infectiousness. Define a matrix K whose entries are k
i
j, that is, K
=
[
k
i
j
]. Then, ℛ
0
=
ρ
(
K
), where ρ
(
K
) is spectral radius of K. For our model, we define four type-at-infection consisting of two vectors and two hosts, namely,Aedes spp. (type 1),Culex spp. (type 2), livestock (type 3), and humans (type 4). The resulting next generation matrix is
(26)
K
=
[
k
11
k
12
k
13
k
14
k
21
k
22
k
23
k
24
k
31
k
32
k
33
k
34
k
41
k
42
k
43
k
44
]
,
where k
11 is the expected number of infected eggs laid byAedes spp. via transovarial transmission, k
12 is the expected number of infectedAedes spp. due to one infectedCulex, k
21 is the expected number of infectedCulex spp. due to one infectedAedes spp., k
13 is the expected number of infectedAedes spp. due to one infected livestock, k
31 is the expected number of infected livestock due to one infectedAedes spp., k
14 is the expected number of infectedAedes spp. due to one infected human, k
41 is the expected number of infected humans due to one infectedAedes spp., k
22 is the expected number of infected eggs laid byCulex spp. via transovarial transmission, k
23 is the expected number of infectedCulex spp. due to one infected livestock, k
32 is the expected number of infected livestock due to one infectedCulex spp., k
24 is the expected number of infectedCulex spp. due to one infected human, k
42 is the expected number of infected humans due to one infectedCulex spp., k
33 is the expected number of infected livestock due to one infected livestock, k
34 is the expected number of infected livestock due to one infected human, k
43 is the expected number of infected humans due to one infected livestock, and and k
44 is the expected number of infected humans due to one infected human.Since there is no vertical transmission inCulex spp., then k
22
=
0. The same applies for k
33 and k
44. AlsoAedes spp. cannot infectCulex spp. and vice versa; therefore, k
12
=
k
21
=
0. Humans cannot infect livestock, so k
34
=
0. Hence, we have
(27)
K
=
[
k
11
0
k
13
k
14
0
0
k
23
k
24
k
31
k
32
0
0
k
41
k
42
k
43
0
]
.
The entry k
i
j depends on the probability that the individual of type j survives the incubation, the adequate contact rate: individual type j to individual type i, and the infective period of individual of type j. For example, k
13 will depend on the probability that livestock survives the incubation period, the adequate contact rate from livestock toAedes spp., and the infective period of livestock. We therefore derive the k
i
j values as follows: (28a)
k
11
=
b
a
(
T
,
P
)
f
a
μ
a
,
k
13
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
a
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
,
(28b)
k
14
=
(
ε
h
ε
h
+
μ
h
)
(
λ
h
a
(
T
)
μ
h
+
ϕ
h
+
γ
h
)
,
k
23
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
c
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
,
(28c)
k
24
=
(
ε
h
ε
h
+
μ
h
)
(
λ
h
c
(
T
)
μ
h
+
ϕ
h
+
γ
h
)
,
k
31
=
(
ε
a
(
T
)
ε
a
(
T
)
+
μ
a
(
T
)
)
(
λ
a
l
(
T
)
μ
a
(
T
)
)
,
(28d)
k
32
=
(
ε
c
(
T
)
ε
c
(
T
)
+
μ
c
(
T
)
)
(
λ
c
l
(
T
)
μ
c
(
T
)
)
,
k
41
=
(
ε
a
(
T
)
ε
a
(
T
)
+
μ
a
(
T
)
)
(
λ
a
h
(
T
)
μ
a
(
T
)
)
,
(28e)
k
42
=
(
ε
c
(
T
)
ε
c
(
T
)
+
μ
c
(
T
)
)
(
λ
c
h
(
T
)
μ
c
(
T
)
)
,
k
43
=
(
ε
l
ε
l
+
μ
l
)
(
λ
l
h
(
T
)
μ
l
+
ϕ
l
+
γ
l
)
.


## 2.4. Sensitivity and Elasticity Analyses ofℛ
0

Sensitivities quantify howℛ
0 changes in response to the small shifts in the value of a parameter, while elasticities quantify the proportional change in ℛ
0 in response to the proportional change in a parameter. Both sensitivity and elasticity values can be used to judge which parameters are important to measure accurately and where variation in parameters will translate into variation in ℛ
0.Caswell [39] developed a way to quantify sensitivity and elasticity of the growth rate λ to changes in vital rates a
i
j where a
i
j are the entries of population matrix A. That is, the sensitivity of the growth rate λ to changes in vital rates a
i
j is given by
(29)
s
i
j
=
∂
λ
∂
a
i
j
=
v
i
w
j
〈
w
,
v
〉
,
wherew andv are the right and left eigenvectors, respectively, corresponding to the dominant eigenvalue λ of the matrix A, and 〈
w
,
v
〉 is the dot product ofw andv. In case a
i
j is a function of other lower-level parameters, then, the chain rule can be applied to estimate the sensitivity of λ to changes in any model parameter p as
(30)
s
(
p
)
=
∂
λ
∂
p
=
∑
i
j
‍
∂
λ
∂
a
i
j
∂
a
i
j
∂
p
.The elasticity of the growth rateλ to changes in vital rates a
i
j, the entries of population matrix A, is given by
(31)
e
i
j
=
∂
log
⁡
λ
∂
log
⁡
a
i
j
=
a
i
j
λ
∂
λ
∂
a
i
j
.
For a
i
j a function of other lower-level parameters p, the elasticity is given by
(32)
e
(
p
)
=
p
λ
∂
λ
∂
p
=
p
λ
∑
i
j
‍
∂
λ
∂
a
i
j
∂
a
i
j
∂
p
.The theory of sensitivity analysis developed for the matrix models by Caswell [39] can be extended to the disease models to study the sensitivity and elasticity of ℛ
0 to the changes in the reproduction numbers k
i
j or the parameters defining them. Thus, the sensitivity s
i
j of a matrix element k
i
j is defined as the change in the eigenvalue (ℛ
0) due to change in k
i
j given by
(33)
s
i
j
=
∂
ℛ
0
∂
k
i
j
.
For individual parameter, the sensitivity s
(
p
) is given by
(34)
s
(
p
)
=
∑
i
j
‍
∂
ℛ
0
∂
k
i
j
∂
k
i
j
∂
p
.
The elasticity e
i
j of a matrix element k
i
j is defined as
(35)
e
i
j
=
k
i
j
ℛ
0
∂
ℛ
0
∂
k
i
j
.
For individual parameters p, the elasticity is given by
(36)
e
(
p
)
=
p
ℛ
0
∑
i
j
‍
∂
ℛ
0
∂
k
i
j
∂
k
i
j
∂
p
.In order to study the impact of climate change to climate-driven parameter in the distribution ofℛ
0 we use climate data from two different regions in Tanzania, namely, Arusha and Dodoma for the 2006-2007 outbreak. According to WHO [12], RVF was reported in 10 out of the 21 regions of Tanzania where 12 cases were reported in Arusha region, 1 in Dar es Salaam, 156 in Dodoma, 4 in Iringa, 6 in Manyara, 50 in Morogoro, 5 in Mwanza, 5 in the Pwani, 24 in Singida, and 1 in Tanga regions. From the data we find that Dodoma has more than 50% of the total cases giving a justification for being a case of study, and Arusha is considered in this study because the first case was reported in January 2007 in this region.


## 3. Results and Discussion

In this section we first present the result forℛ
0 when the parameters are assumed to be independent of climate change. Then, we will compute the numerical value for ℛ
0 when climate change is considered to climate-driven parameters. Sensitivity and elasticity analysis results in both cases will be presented. Table 2 shows the parameter values for low range and high range which are used to compute the numerical value for ℛ
0 when temperature and precipitation effects are not considered.Table 2
Parameters with their estimated lower and higher values without considering impact of climate change.
Parameter
low value
high value
Reference
1
/
b
a
100
200
Assumed
1
/
μ
a
3 days
60 days
[17]
1
/
μ
c
3 days
60 days
[17]
1
/
μ
h
40 yrs
60 yrs
[19]
1
/
μ
l
1 yr
10 yrs
[40]
1
/
ε
a
4 days
8 days
[19]
1
/
ε
c
4 days
8 days
[19]
1
/
ε
h
2 day
6 days
[19]
1
/
ε
l
1 day
6 days
[19]
ϕ
l
0.025
0.10
[19]
ϕ
h
0.01
0.10
[19]
f
a
0.05
0.1
[18]
1
/
γ
l
1 day
5 days
[19]
1
/
γ
h
4 days
7 days
[19]
λ
al
0.15
0.48
[18]
λ
cl
⁡
0.05
0.13
[18]
λ
la
0.15
0.395
[18]
λ
lc
0.15
0.56
[18]
λ
ah
0.001
0.002
[22]
λ
ch
0.0005
0.001
[22]
λ
ha
0.001
0.0015
Assumed
λ
hc
0.0015
0.002
Assumed
λ
lh
0.001
0.002
[19]When we substitute the values in Table2 to the expressions of the elements of matrix K and compute ℛ
0, we obtain that for low parameter values ℛ
0
=
0.1941 and for high parameter values ℛ
0
=
6.8071.When climate change parameters were evaluated using the climate variable the value ofℛ
0 change from 0.4747 to 14.2007 in Arusha with the highest value marked in November 2006 (=
14.2007) followed by December 2006 (=
14.1530). The value of ℛ
0 dropped below 1 in January 2007 and February 2007, but it rose again in March, April, and May. Figure 2(a) shows the distribution of ℛ
0 from July 2006 to June 2007 in Arusha region.Distribution ofℛ
0 for climatic data in Arusha and Dodoma.
(a)
Distribution ofℛ
0 for Arusha
(b)
Distribution ofℛ
0 for DodomaIn Dodoma, the highestℛ
0 was marked February 2007 (=
12.7438) followed by January 2007 (=
12.7368) then March 2007 (=
7.9899) and December 2006 (=
1.5088) as Figure 2(b) indicates.While it is clear thatℛ
0 increases with increase in rainfall, it is not the case for temperature where we experience high ℛ
0 for low temperatures. Figure 3 shows the plots for ℛ
0 and precipitation over months, and Figure 4 shows the plots for ℛ
0 and temperature over months.ℛ
0 and precipitation for climatic data in Arusha and Dodoma.
(a)
ℛ
0 and precipitation for Arusha
(b)
ℛ
0 and precipitation for Dodomaℛ
0 temperature for climatic data in Arusha and Dodoma.
(a)
ℛ
0 and temperature for Arusha
(b)
ℛ
0 and temperature for DodomaTable3 shows the sensitivity and elasticity values of ℛ
0, to both low and high parameter values. For both low and high parameter values, ℛ
0 is most sensitive to k
13, the expected number of infectedAedes spp. Due to one infected livestock, and to and most elastic to k
31, the expected number of infected livestock due to one infectedAedes spp. Table 3 shows the sensitivity and elasticity values of ℛ
0 for low and high parameter values, and Figure 5 shows the plots of sensitivity and elasticity values plotted against the parameter k
i
j. The results suggest that attention should be given to parameters regarding incubation period, the adequate contact rate, and the infective period of livestock andAedes spp.Table 3
Sensitivity and elasticity ofℛ
0 for low and high parameter values.
Parameter
Sensitivity
Elasticity
Low parameter values
k
11
0.37750
0.00292
k
13
0.49913
0.37438
k
14
0.00629
0.00012
k
23
0.16513
0.12386
k
24
0.00208
0.00006
k
31
0.37679
0.37443
k
32
0.37399
0.12388
k
41
0.01118
0.00007
k
42
0.01109
0.00003
k
43
0.01497
0.00008
High parameter values
k
11
0.36274
0.00160
k
13
1.87148
0.36107
k
14
0.01149
0.00001
k
23
0.50469
0.13804
k
24
0.00310
0.00000
k
31
0.09672
0.36107
k
32
0.13653
0.13804
k
41
0.00045
0.00001
k
42
0.00063
0.00000
k
43
0.00231
0.00000Sensitivity and elasticity ofℛ
0 plotted against the low and high parameters values.
(a)
Sensitivity and elasticity ofℛ
0 for low parameter values
(b)
Sensitivity and elasticity ofℛ
0 for high parameter valuesWhen climatic data are used,ℛ
0 is found to be more sensitive and elastic to k
11, the expected number of infected eggs laid byAedes spp. via transovarial transmission, followed by k
13 and k
31 for both regions Arusha and Dodoma. Table 4 shows the sensitivity and elasticity values of ℛ
0 for Dodoma and Arusha climate data, and Figure 6 shows the plots of sensitivity and elasticity against the parameters k
i
j. The results call for attention to parameters regarding incubation period, the adequate contact rate ofAedes spp. and livestock, the infective periods of livestock andAedes spp., and the vertical transmission inAedes spp.Table 4
Sensitivity and elasticity ofℛ
0 for Dodoma and Arusha climate data.
Parameter
Sensitivity
Elasticity
Dodoma
k
11
0.99874
0.99748
k
13
0.12996
0.00126
k
14
0.00020
0.00000
k
23
0.00010
0.00000
k
24
0.00000
0.00000
k
31
0.00971
0.00126
k
32
0.00001
0.00000
k
41
0.00010
0.00000
k
42
0.00000
0.00000
k
43
0.00001
0.00000
Arusha
k
11
0.99921
0.99841
k
13
0.10903
0.00079
k
14
0.00016
0.00000
k
23
0.00012
0.00000
k
24
0.00000
0.00000
k
31
0.00727
0.00079
k
32
0.00000
0.00000
k
41
0.00008
0.00000
k
42
0.00000
0.00000
k
43
0.00001
0.00000Sensitivity and elasticity ofℛ
0 plotted against the parameters k
i
j for climatic data in Arusha and Dodoma.
(a)
Sensitivity and elasticity ofℛ
0 for Arusha
(b)
Sensitivity and elasticity ofℛ
0 for Dodoma


## 4. Conclusion

A deterministic SEIR model of RVF has been presented to study the impact of climate change variables mainly temperature and precipitation. The model presented here is just a simple representation of the complex ecological situation involved in the epidemiology of RVF. The formulation of the model, computation ofℛ
0, and sensitivity and elasticity analyses of ℛ
0 are based on the assumptions made to build the model as well as the chosen parameter values. Real climate data from Dodoma and Arusha where outbreak occured in 2006-2007 have been used to study the distribution of ℛ
0 in the whole period of the outbreak. Though the current analysis presented in this work may not be exhaustible, it remains, however, an important step toward the study of the impact of climate change on the dynamics of RVF.



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*Source: 627586-2014-03-30.xml*