Modeling the Impact of Climate Change on the Dynamics of ... · A deterministic SEIR model of rift...

Research ArticleModeling the Impact of Climate Change onthe Dynamics of Rift Valley Fever

Saul C. Mpeshe,1 Livingstone S. Luboobi,1,2 and Yaw Nkansah-Gyekye1

1 School of CoCSE, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania2Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda

Correspondence should be addressed to Saul C. Mpeshe; [email protected]

Received 30 August 2013; Revised 20 January 2014; Accepted 3 February 2014; Published 30 March 2014

Academic Editor: Gabriel Turinici

Copyright © 2014 Saul C. Mpeshe et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A deterministic SEIR model of rift valley fever (RVF) with climate change parameters was considered to compute the basicreproduction number R

0and investigate the impact of temperature and precipitation on R

0. To study the effect of model

parameters to R0, sensitivity and elasticity analysis of R

0were performed. When temperature and precipitation effects are not

considered,R0is more sensitive to the expected number of infectedAedes spp. due to one infected livestock andmore elastic to the

expected number of infected livestock due to one infectedAedes spp.When climatic data are used,R0is found to be more sensitive

and elastic to the expected number of infected eggs laid by Aedes spp. via transovarial transmission, followed by the expectednumber of infected livestock due to one infected Aedes spp. and the expected number of infected Aedes spp. due to one infectedlivestock for both regions Arusha and Dodoma. These results call for attention to parameters regarding incubation period, theadequate contact rate of Aedes spp. and livestock, the infective periods of livestock and Aedes spp., and the vertical transmission inAedes species.

1. Introduction

Rift valley fever (RVF) is a viral disease that primarily affectsanimals (such as sheep, horses, cattle, goats, camels, andbuffalos) and has the capacity to affect human beings. Riftvalley fever virus (RVFV) is a member of the Phlebovirusgenus family Bunyaviridae which has been isolated from atleast 40 mosquito species in the filed and other arthropods[1, 2]. RVFV infection can cause severe disease in bothanimals and humans, leading to high disease induced deathrate in livestock, long-term health effects in humans, andeconomic destruction of people [3, 4]. Currently, two typesof vaccine for animals exist: a live vaccine and inactivatedvaccine. However, the current live vaccine cannot be used forprevention, and prevention using the inactivated vaccine isdifficult to sustain in RVF affected countries for economicreasons [3, 5].

RVF can be transmitted through an initial aerosol releaseand subsequent transmission through the mosquito vector.RVFV can remain dormant in Aedes spp. mosquito eggs

in dry soil for years. During periods of heavy rainfall,larval habitats frequently become flooded, enabling the eggsto 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 ofinfected mosquitoes, mainly Aedes and Culex spp. whichcan acquire the virus from feeding on an infected animal[8–10]. The female Aedes spp. mosquito is also capable oftransmitting the virus directly to her offspring (verticaltransmission) via eggs leading to new generations of infectedmosquitoes hatching from eggs [11, 12].This is not the case forCulex spp. mosquito.

RVFV can be transmitted to humans through the han-dling of animal tissue during slaughtering or butchering,assisting with animal births, conducting veterinary proce-dures, or from the disposal of carcasses or fetuses. Humaninfections have also resulted from the bites of infectedmosquito vector, and by ingesting unpasteurized or uncookedmilk and meat of infected animals [8–10, 13]. Transmission

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014, Article ID 627586, 12 pageshttp://dx.doi.org/10.1155/2014/627586

http://dx.doi.org/10.1155/2014/627586

2 Computational and Mathematical Methods in Medicine

of RVFV by blood feeding flies is also possible. To dateno human-to-human transmission of RVF has been docu-mented [12].

RVF was first reported in Kenya (Africa) in 1931 [8], andit was primarily considered to be of sub-Saharan Africa untilSeptember, 2000, when RVF cases were confirmed in SaudiArabia and Yemen (outside Africa) [14]. The recent outbreakin East Africa is that of 2006-2007 where 684 cases and 155deathswere confirmed inKenya, and 264 cases and 109 deathsin Tanzania.There were outbreaks also in Somalia and Sudanin the same period [12].

RVF outbreaks in East Africa have been largely correlatedwith the unusual heavy rainfall associated with El Nino[15], which consequently flooded many Aedes spp. breedinghabitats. The hatching dynamics of Aedes spp. mosquitoes,the main reservoir of RVF in Africa, strongly depends onthe 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, secondaryvectors such as Culex spp. and other biting flies, which breedin semipermanent pools of water, become involved in thetransmission of the virus [16].

Global temperature change, on the other hand, wouldaffect the biology of the vectors, including feeding rate andegg production, and the length of the development cycle andthe extrinsic incubation period.Thismay result in high vectordensity, an increased vector capacity to transmit the virusand a higher transmission rate [16]. When temperature risesabove the biological maximum threshold for a species, it maydecrease the vector population. Sustained climate shifts maylead to changes in the RVF burden in endemic areas and newoutbreaks in areas of similar conditions. Thus, modeling theimpact of climate change in the dynamics of RVF and itsinterventions is important for understanding of the disease.

Mathematical epidemiological models have been devel-oped to assess the dynamics of RVF. Gaff et al. [17] proposeda theoretical model in a closed system which included twomosquito populations Aedes and Culex spp. and a livestockpopulation. Their proposed model was a system of ordinarydifferential equations developed to explain the behaviour ofthe RVF transmission. The result of the development processwas the production of a first-time model of this disease.The model was later modified by Gaff et al. [18] to assessthe relative effectiveness of RVF countermeasures such asvector adulticide, vector larvicide, livestock vaccination, andlivestock culling.

A theoretical model involving mosquito population, live-stock and human population has been developed to studythe dynamics of the disease using nonlinear differentialequations [19]. The results show that the disease prevalencein both human and livestock is more sensitive to livestockand human recruitment rates suggesting isolation of livestockfrom human as a viable measure during the outbreak. Theinitial transmission and disease prevalence were found tobe highly linked to mosquito population suggesting controlmeasures such as vector adulticides and larvicides to beapplied to reduce the mosquito population.

Fischer et al. [20] investigated the transmission poten-tial of RVFV among livestock in the Netherlands. Themodel included the effect of temperature on the bitingrate, mosquito population size, and the mortality of thevectors. The results show that high degree of vaccination andvector control strategy are needed to prevent RVF outbreaks.Other studies include that of Xue et al. [21] who developeda network-based metapopulation model approach to RVFepidemics to assess the disease spread in both time and spaceusing network theory, Xue et al. [22] who investigate thespread of RVFV when introduced in United States, Chitnis etal. [23] who developed a model to assess the effect of verticaltransmission in vector-borne disease with applications toRVF, and Niu et al. [24] who developed an epidemiologicalmodel of RVF with spatial dynamics to study the spatialeffects.

In this paper, we propose amodel that assesses the impactof climate change on the dynamics of RVF. The approach isbased on the previous model of RVF transmission byMpesheet al. [19] and modifications have been made to incorpo-rate vertical transmission and climate-driven parameters. Tosimplify the model, only temperature and precipitation areconsidered in this study. While Aedes spp. mosquito eggsare naturally infected by RVF virus via vertical transmission,this is not a case for Culex spp. mosquito and, therefore, weassume vertical transmission in our model only for Aedesspecies. To accommodate the impact of climate change weassert that temperature and precipitation can affect the layingand hatching of the eggs as well as the death rate, the effectivecontact rate, and the incubation period of the mosquitoes.When the epizootic is very high human can also be a sourceof infection for mosquitoes [25] and, therefore, also we assertin ourmodel the human-to-mosquito transmission when themosquitoes feed on an infected human.

2. Materials and Methods

2.1. Model Formulation. The model considers three pop-ulations: mosquitoes, livestock, and humans with disease-dependent death rate for livestock and humans.Themosquitopopulation is subdivided into two: Aedes species and Culexspecies. Due to vertical transmission in Aedes spp., weinclude both infected and uninfected eggs in the modelfor determining the effect of vertical transmission in theinitial transmission of RVF. The mode of transmission ofRVF virus from vector to host, host to host, and host tovector is shown in the model flowchart shown by Figure 1.The egg population of Aedes spp. consists of uninfectedeggs (𝑋

𝑎) and infected eggs (𝑌

𝑎). The population for adult

Aedes spp. consists of susceptible adults (𝑆𝑎), latently infected

adults (𝐸𝑎), and infectious adults (𝐼

𝑎). The egg population

of Culex spp. consists of uninfected eggs (𝑋𝑐) only and the

population for adult Culex 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 (𝑅𝑙). The human population consists

of susceptible humans (𝑆ℎ), latently infected humans (𝐸

ℎ),

Computational and Mathematical Methods in Medicine 3

Host 2(human)

Host 1(livestock)

Uninfected InfectedXa Ya

Sa Ea Ia

Sl El Il Rl

Sh Eh Ih Rh

Sc Ec Ic

Xc

bc bc

bc

𝜇c𝜇c 𝜇chc

𝜀c

𝜀ababa haha

𝜇a𝜇a𝜇a

𝜇h𝜇h𝜇h𝜇h

𝜙h

bh 𝜀h 𝛾h

𝛾l

𝜙l 𝜇l𝜇l𝜇l 𝜇l

bl 𝜀l𝜆al

𝜆cl 𝜆ah

𝜆lh𝜆lc 𝜆ha

𝜆la

𝜆hc𝜆ch

bafa

ba(1 − fa)

Adults Culex

Culex eggs

Adults Aedes

Aedes eggs

Figure 1: Flow diagram for the RVF model.

infectious humans (𝐼ℎ) and recovered humans (𝑅

ℎ). Table 1

shows the model parameters and their description as theyhave been used in this work. 𝑇 and 𝑃 represent temperatureand precipitation, respectively.

The epidemiology cycle of RVF presented by Balenghienet al. [26] and Chevalier et al. [27] is here applied to developthe flow diagram shown by Figure 1. The inclusion of thetransmission dynamics of RVF from Aedes spp. to humanand vice versa is due to the fact that some Aedes spp. suchasAedes vexans,Aedes aegpti,Aedes albopictus,Ae. ochraceus,Ae. mcintonshi, and Ae. dalzieli and many others numerouslyfeed on humans, and therefore has the capacity to causeinfection to human [26–31].

Using the parameters in Table 1 and the model flowdiagram in Figure 1, an SEIR model is derived on the basis ofthe explanations above using first-order nonlinear ordinarydifferential equations as follows:

AedesMosquito

𝑑𝑋

𝑎

𝑑𝑡

= 𝑏

𝑎(𝑇, 𝑃) (𝑁

𝑎− 𝑓

𝑎𝐼

𝑎) − ℎ

𝑎(𝑇, 𝑃)𝑋

𝑎,

(1a)

𝑑𝑌

𝑎

𝑑𝑡

= 𝑏

𝑎(𝑇, 𝑃) 𝑓

𝑎𝐼

𝑎− ℎ

𝑎(𝑇, 𝑃) 𝑌

𝑎,

(1b)

𝑑𝑆

𝑎

𝑑𝑡

= ℎ

𝑎 (𝑇, 𝑃)𝑋𝑎

− 𝜇

𝑎 (𝑇) 𝑆𝑎

− 𝜆

𝑙𝑎(𝑇)

𝐼

𝑙

𝑁

𝑙

𝑆

𝑎− 𝜆

ℎ𝑎(𝑇)

𝐼

ℎ

𝑁

ℎ

𝑆

𝑎,

(1c)

𝑑𝐸

𝑎

𝑑𝑡

= 𝜆

𝑙𝑎(𝑇)

𝐼

𝑙

𝑁

𝑙

𝑆

𝑎+ 𝜆

ℎ𝑎(𝑇)

𝐼

ℎ

𝑁

ℎ

𝑆

𝑎− (𝜀

𝑎(𝑇) + 𝜇

𝑎(𝑇)) 𝐸

𝑎,

(1d)

𝑑𝐼

𝑎

𝑑𝑡

= ℎ

𝑎(𝑇, 𝑃) 𝑌

𝑎+ 𝜀

𝑎(𝑇) 𝐸

𝑎− 𝜇

𝑎(𝑇) 𝐼

𝑎,

(1e)

𝑑𝑁

𝑎

𝑑𝑡

= ℎ

𝑎(𝑇, 𝑃) (𝑋

𝑎+ 𝑌

𝑎) − 𝜇

𝑎(𝑇)𝑁

𝑎.

(1f)

CulexMosquito𝑑𝑋

𝑐

𝑑𝑡

= 𝑏

𝑐(𝑇, 𝑃)𝑁

𝑐− ℎ

𝑐(𝑇, 𝑃)𝑋

𝑐,

(2a)

𝑑𝑆

𝑐

𝑑𝑡

= ℎ

𝑐(𝑇, 𝑃)𝑋

𝑐− 𝜇

𝑐(𝑇) 𝑆

𝑐

− 𝜆

𝑙𝑐 (𝑇)

𝐼l𝑁

𝑙

𝑆

𝑐− 𝜆

ℎ𝑐 (𝑇)

𝐼

ℎ

𝑁

ℎ

𝑆

𝑐,

(2b)

𝑑𝐸

𝑐

𝑑𝑡

= 𝜆

𝑙𝑐(𝑇)

𝐼

𝑙

𝑁

𝑙

𝑆

𝑐+ 𝜆

ℎ𝑐(𝑇)

𝐼

ℎ

𝑁

ℎ

𝑆

𝑐− (𝜀

𝑐(𝑇) + 𝜇

𝑐(𝑇)) 𝐸

𝑐,

(2c)

𝑑𝐼

𝑐

𝑑𝑡

= 𝜀

𝑐(𝑇) 𝐸

𝑐− 𝜇

𝑐(𝑇) 𝐼

𝑐,

(2d)

𝑑𝑁

𝑐

𝑑𝑡

= ℎ

𝑐 (𝑇, 𝑃)𝑋𝑐

− 𝜇

𝑐 (𝑇)𝑁𝑐

.(2e)

Livestock𝑑𝑆

𝑙

𝑑𝑡

= 𝑏

𝑙𝑁

𝑙− 𝜇

𝑙𝑆

𝑙− 𝜆

𝑎𝑙(𝑇)

𝐼

𝑎

𝑁

𝑎

𝑆

𝑙− 𝜆

𝑐𝑙(𝑇)

𝐼

𝑐

𝑁

𝑐

𝑆

𝑙, (3a)

𝑑𝐸

𝑙

𝑑𝑡

= 𝜆

𝑎𝑙(𝑇)

𝐼

𝑎

𝑁

𝑎

𝑆

𝑙+ 𝜆

𝑐𝑙(𝑇)

𝐼

𝑐

𝑁

𝑐

𝑆

𝑙− (𝜀

𝑙+ 𝜇

𝑙) 𝐸

𝑙, (3b)

𝑑𝐼

𝑙

𝑑𝑡

= 𝜀

𝑙𝐸

𝑙− (𝜇

𝑙+ 𝜙

𝑙+ 𝛾

𝑙) 𝐼

𝑙,

(3c)

𝑑𝑅

𝑙

𝑑𝑡

= 𝛾

𝑙𝐼

𝑙− 𝜇

𝑙𝑅

𝑙,

(3d)

𝑑𝑁

𝑙

𝑑𝑡

= (𝑏

𝑙− 𝜇

𝑙)𝑁

𝑙− 𝜙

𝑙𝐼

𝑙.

(3e)

Humans𝑑𝑆

ℎ

𝑑𝑡

= 𝑏

ℎ𝑁

ℎ− 𝜇

ℎ𝑆

ℎ− 𝜆

𝑙ℎ

𝐼

𝑙

𝑁

𝑙

𝑆

ℎ

− 𝜆

𝑎ℎ(𝑇)

𝐼

𝑎

𝑁

𝑎

𝑆

ℎ− 𝜆

𝑐ℎ(𝑇)

𝐼

𝑐

𝑁

𝑐

𝑆

ℎ,

(4a)

𝑑𝐸

ℎ

𝑑𝑡

= 𝜆

𝑙ℎ

𝐼

𝑙

𝑁

𝑙

𝑆

ℎ+ 𝜆

𝑎ℎ(𝑇)

𝐼

𝑎

𝑁

𝑎

𝑆

ℎ

+ 𝜆

𝑐ℎ(𝑇)

𝐼

𝑐

𝑁

𝑐

𝑆

ℎ− (𝜀

ℎ+ 𝜇

ℎ) 𝐸

ℎ,

(4b)

𝑑𝐼

ℎ

𝑑𝑡

= 𝜀

ℎ𝐸

ℎ− (𝜇

ℎ+ 𝜙

ℎ+ 𝛾

ℎ) 𝐼

ℎ,

(4c)

𝑑𝑅

ℎ

𝑑𝑡

= 𝛾

ℎ𝐼

ℎ− 𝜇

ℎ𝑅

ℎ,

(4d)

𝑑𝑁

ℎ

𝑑𝑡

= (𝑏

ℎ− 𝜇

ℎ)𝑁

ℎ− 𝜙

ℎ𝐼

ℎ.

(4e)

To test whether themodel is well posed epidemiologicallyand mathematically, we need to investigate the feasibility ofthe model solution. Since 𝑅

𝑙and 𝑅

ℎcan be determined when

𝑆

𝑙, 𝑆

ℎ, 𝐸

𝑙, 𝐸

ℎ, 𝐼

𝑙, and 𝐼

ℎare known, without loss of generality,

we omit the expression for 𝑑𝑅𝑙/𝑑𝑡 and 𝑑𝑅

ℎ/𝑑𝑡 and write the

system in compact form as

𝑑𝑋

𝑑𝑡

= 𝑀 (𝑥)𝑋 + 𝐹,(5)


Table 1: Parameters used in the model formulation and their description.

Parameter Description Dependent on climate change1/ℎ

𝑎(𝑇, 𝑃) Development time of Aedesmosquitoes Temperature and precipitation

1/ℎ

𝑐(𝑇, 𝑃) Development rate of Culexmosquitoes Temperature and precipitation

𝑏

𝑎(𝑇, 𝑃) Number of Aedes eggs laid per day Temperature and precipitation

𝑏

𝑐(𝑇, 𝑃) Number of Culex eggs laid per day Temperature and precipitation

𝑏

ℎDaily birth rate in humans Not considered

𝑏

𝑙Daily birth rate in livestock Not considered

1/𝜇

𝑎(𝑇) Lifespan of Aedesmosquitoes Temperature

1/𝜇

𝑐(𝑇) Lifespan of Culexmosquitoes Temperature

1/𝜇

ℎLifespan of humans Not considered

1/𝜇

𝑙Lifespan of livestock Not considered

1/𝜀

𝑎(𝑇) Incubation period of Aedesmosquitoes Temperature

1/𝜀

𝑐(𝑇) Incubation period of Culexmosquitoes Temperature

1/𝜀

ℎIncubation period of humans Not considered

1/𝜀

𝑙Incubation period of livestock Not considered

𝜙

𝑙Death rate of livestock due to disease Not considered

𝜙

ℎDeath rate of humans due to disease Not considered

1/𝛾

𝑙Infectious period in livestock Not considered

1/𝛾

ℎInfectious period in humans Not considered

𝜆al(𝑇) Adequate contact rate: Aedes to livestock Temperature𝜆cl(𝑇) Adequate contact rate: Culex to livestock Temperature𝜆la(𝑇) Adequate contact rate: livestock to Aedes Temperature𝜆lc(𝑇) Adequate contact rate: livestock to Culex Temperature𝜆ah(𝑇) Adequate contact rate: Aedes to humans Temperature𝜆ch(𝑇) Adequate contact rate: Culex to humans Temperature𝜆ha(𝑇) Adequate contact rate: humans to Aedes Temperature𝜆hc(𝑇) Adequate contact rate: humans to Culex Temperature𝜆lh Adequate contact rate: livestock to humans Not considered𝑓

𝑎Vertical transmission rate in Aedes Not considered

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, 𝐸ℎ,

𝐼

ℎ)

𝑇, 𝑀(𝑥) is a 15 by 15 matrix, and 𝐹 is a column matrix.Substituting 𝐼

𝑎= 𝑁

𝑎− (𝑆

𝑎+ 𝐸

𝑎) in 𝑑𝑋

𝑎/𝑑𝑡, we have

𝑑𝑋

𝑎

𝑑𝑡

= 𝑏

𝑎 (𝑇, 𝑃)𝑁𝑎

(1 − 𝑓

𝑎) + 𝑏

𝑎𝑓

𝑎(𝑆

𝑎+ 𝐸

𝑎) − ℎ

𝑎 (𝑇, 𝑃)𝑋𝑎

,

(6)

and, therefore,

𝑀(𝑥) =

[

[

𝑀

1(𝑥) 0 0

0 𝑀

2(𝑥) 0

0 0 𝑀

3(𝑥)

]

]

, (7)

where

𝑀

1(𝑥) =

[

[

[

[

[

[

−ℎ

𝑎(𝑇, 𝑃) 0 𝑏


𝑎𝑏


𝑎0

0 −ℎ

𝑎(𝑇, 𝑃) 0 0 𝑏


𝑎

ℎ

𝑎(𝑇, 𝑃) 0 − (𝜇

𝑎+ 𝐴) 0 0

0 0 𝐴 − (𝜀

𝑎 (𝑇, 𝑃) + 𝜇𝑎 (

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)

]

]

]

]

]

]

]

]

(8)


with

𝐴 = 𝜆

𝑙𝑎 (𝑇)

𝐼

𝑙

𝑁

𝑙

+ 𝜆

ℎ𝑎 (𝑇)

𝐼

ℎ

𝑁

ℎ

, (9a)

𝐵 = 𝜆

𝑙𝑐 (𝑇)

𝐼

𝑙

𝑁

𝑙

+ 𝜆

ℎ𝑐 (𝑇)

𝐼

ℎ

𝑁

ℎ

, (9b)

𝐶 = 𝜆

𝑎𝑙(𝑇)

𝐼

𝑎

𝑁

𝑎

+ 𝜆

𝑐𝑙(𝑇)

𝐼

𝑐

𝑁

𝑐

, (9c)

𝐷 = 𝜆

𝑙ℎ (𝑇)

𝐼

𝑙

𝑁

𝑙

+ 𝜆

𝑎ℎ (𝑇)

𝐼

𝑎

𝑁

𝑎

+ 𝜆

𝑐ℎ (𝑇)

𝐼

𝑐

𝑁

𝑐

, (9d)

𝐹 = (𝑏

𝑎(𝑇, 𝑃)𝑁𝑎 (1 − 𝑓

𝑎) , 0, 0, 0, 0, 𝑏

𝑐(𝑇, 𝑃)𝑁

𝑐,

0, 0, 0, 𝑏

𝑙𝑁

𝑙, 0, 0, 𝑏

ℎ𝑁

ℎ, 0, 0)

𝑇≥ 0.

(10)

Combining all together, the matrix 𝑀(𝑥) is a Metzlermatrix for all R15

+. Therefore, the model system is positively

invariant in R15+, and 𝐹 is Lipschitz continuous. Thus, the

feasible region for the model system is the set

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)

≥ 0 ∈ R15

+} .

(11)

That is, the solution remains in the feasible regionD if it startsin this region. Hence, it is sufficient to study the dynamics ofthe model inD.

2.2. Climate Driven Parameters. Several parameters relatedtomosquito vector, such as the hatching rate, vectormortalityand longevity, biting rate, and extrinsic incubation period,depend on the temperature and precipitation. Using theexisting studies and information from Aedes vexans, Aedesaegypti, Culex pipiens, and Culex quinquefasciatus [20, 32–34] which are potential vectors of RVF, we generalise thefollowing relations for Aedes and Culex spp. mosquitoes.

2.2.1. Hatching Rate or Mosquito Birth Rate, ℎ(𝑇, 𝑃). Thisis the number of eggs hatching into adult mosquitoes at acertain period of time which we also refer to as the mosquitobirth rate. It will depend on the the daily survival probability𝜌from eggs to adults and the duration𝑑 it takes to develop fromeggs to adults. The daily survival probability is assumed todepend independently on temperature, precipitation/rainfall,and prolonged period of desiccation. Thus,

𝜌 (𝑇, 𝑃,𝐷) = 𝜌 (𝑇) 𝜌 (𝑃) 𝜌 (𝐷) , (12)

where 𝜌(𝑇) is the daily survival probability of immaturity dueto temperature effect 𝑇; 𝜌(𝑃) is the daily survival probabilityof immaturity due to precipitation effect 𝑃; and 𝜌(𝐷) is thedaily survival probability of immaturity due to desiccationeffect 𝐷. The duration of maturation 𝑑 is assumed to dependon temperature. Therefore, the hatching rate is given by

ℎ (𝑇, 𝑃,𝐷) =

𝜌 (𝑇, 𝑃,𝐷)

𝑑 (𝑇)

. (13)

2.2.2. Survival Probability due to Temperature Effect 𝜌(𝑇).The daily survival probability 𝜌(𝑇) is affected by the durationof maturation 𝑑(𝑇) in exponential form, that is,

𝜌 (𝑇) = exp{− 1

𝑑 (𝑇)

} . (14)

Fitting the data from [34] we obtain that 1/𝑑(𝑇) = 𝛼

1𝑇

2+

𝛼

2𝑇 + 𝛼

3for Culex spp. and 1/𝑑(𝑇) = 𝛼

1𝑇

3+ 𝛼

2𝑇

2+ 𝛼

3𝑇 +

𝛼

4for Aedes spp., where 𝛼

1= 0.0095, 𝛼

2= −0.4684, 𝛼

3=

5.8343 for Culex spp. and 𝛼1= −0.0025, 𝛼

2= 0.2069, 𝛼

3=

−5.5285, 𝛼

4= 48.2951 for Aedes spp.

2.2.3. Survival Probability due to Precipitation Effect 𝜌(𝑃).Precipitation or rainfall is important in creating breeding sitesfor mosquitoes and causing massive hatching. But excessiverainfall increasesmortality of immature due to flushing effect.Since rainfall has two effects, that is, positive and negativeeffect, we use the idea from [35] and assume the daily survivalprobability of immaturity due to precipitation effect to be

𝜌 (𝑃) = (1 − exp {−𝛽1(𝑃 − 𝑃

1)}) (1 − exp {−𝛽

2(𝑃

2− 𝑃)}) ,

(15)

where 𝛽𝑖(𝑖 = 1, 2) are the sensitivity parameters; 𝑃

1is

the minimum amount of rainfall to support maturity; and𝑃

2is the maximum amount of rainfall which reduces their

survival. For computational purposes we set 𝜌(𝑃) = 0 for𝑃 < 𝑃

1and for 𝑃 > 𝑃

2.

2.2.4. Survival Probability due to Desiccation Effect 𝜌(𝐷).Lack of precipitation affects the development of the imma-ture. Following the approach by [32] we define the dailysurvival probability due desiccation as

𝜌 (𝐷

𝑡) =

exp (−𝜔𝐷𝑡)

𝑐 + exp (−𝜔𝐷𝑡)

, (16)

where𝐷 depends on precipitation 𝑃 and is defined as

𝐷

𝑡= {

𝐷

𝑡−1+ 1 𝑃

𝑡≤ 𝑃th

0 otherwise,(17)

where 𝑃th is the threshold precipitation; 𝐷𝑡is the number of

consecutive days up to time 𝑡 when the precipitation 𝑃𝑡was

below the threshold 𝑃th; 𝜔 is the sensitivity parameter; and𝑐 is the constant that ensures that 𝜌(𝐷

𝑡) is close to 1 at small

values of𝐷𝑡.

2.2.5. Daily Egg Laying Rate 𝑏(𝑇). The egg laying rate isassumed to depend on the moisture index. High moistureindex correlates with high egg laying rate [33]. To model thedaily egg laying rate we employ the equation derived by Gonget al. [33] that

𝑏 (𝑇, 𝑃) = Baseline Egg rate

+

𝐸max1 + exp {− (Moisture Index − 𝐸mean) /𝐸var}

,

(18)


where Baseline Egg rate is the baseline for fecundity, 𝐸max isthe maximum daily egg laying rate, 𝐸mean is the mean of dailyegg laying rate, and 𝐸var is the variance function.

To compute the moisture index, we apply Thornthwaite’smoisture index [36] that

Moisture Index (𝐼𝑚) = 100 (

𝑟

𝐸

0

− 1) , (19)

where 𝑟 is the precipitation rate, and𝐸0is the potential evapo-

transpiration. In absence of the potential evapotranspiration,Linacre’s method [37] can be applied. That is,

𝐸

0=

700𝑇

𝑚/ (100 − 𝐴) + 15 (𝑇 − 𝑇

𝑑)

(80 − 𝑇)

mmday−1, (20)

where, 𝑇𝑚= 𝑇 + 0.00ℎ with ℎ being the elevation (metres), 𝑇

is the mean temperature,𝐴 is the latitude (degrees), and 𝑇𝑑is

the mean dew-point.

2.2.6. Longevity ofMosquitoes 1/𝜇(𝑇). Different studies showthat the longevity of mature mosquitoes also depends on thetemperature. To model the longevity, equations deduced byFischer et al. [20] are applied. That is,

1

𝜇 (𝑇)

= 𝑎

0− 𝑎

1𝑇, (21)

where 𝑎0= 25.8, 𝑎

1= 0.45 for Aedes spp., and 𝑎

0= 69.1, 𝑎

1=

2.14 for Culex spp.

2.2.7. Extrinsic Incubation Period ofMosquitoes 1/𝜀(𝑇). Extr-insic incubation period is the time between a blood meal onan infections host and the first successful transmission fromvector to host during another blood meal. We also adapt theexpressions by Fischer et al. [20]. That is,

1

𝜀 (𝑇)

= 𝜀max − 𝜀slope𝑇, (22)

where 𝜀max = 18.9, 𝜀slope = 0.30 for Aedes spp., and 𝜀max =11.3, 𝜀slope = 0.30 for Culex spp.

2.2.8. Adequate Contact Rate 𝜆(𝑇). Adequate contact rate iscontact which is sufficient for transmission of the infectionfrom an infective to a susceptible. Thus, in this study

adequate contact rate

= biting rate × probability of transmission.(23)

The biting rate depends on temperature, and we assume alinear relationship as in Fischer et al. [20]. That is,

𝑎 (𝑇) = 𝑎slope (𝑇 − 𝑇min) , (24)

where 𝑎slope = 0.0173, 𝑇min = 9.60 for all mosquito species.Assume that the probability of transmission is independentto temperature, we have

𝜆

𝑎𝑙(𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑎𝑙, 𝜌

𝑎𝑙= 0.70, (25a)

𝜆

𝑐𝑙 (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑐𝑙, 𝜌

𝑐𝑙= 0.78, (25b)

𝜆

𝑙𝑎 (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑙𝑎, 𝜌

𝑙𝑎= 0.38, (25c)

𝜆

𝑙𝑐 (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑙𝑐, 𝜌

𝑙𝑐= 0.22, (25d)

𝜆

𝑎ℎ (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑎ℎ, 𝜌

𝑎ℎ= 0.01, (25e)

𝜆

ℎ𝑎 (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌ℎ𝑎, 𝜌

ℎ𝑎= 0.05, (25f)

𝜆

𝑐ℎ (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌𝑐ℎ, 𝜌

𝑐ℎ= 0.01, (25g)

𝜆

ℎ𝑐 (𝑇) = 𝑎slope (𝑇 − 𝑇min) 𝜌ℎ𝑐, 𝜌

ℎ𝑐= 0.015. (25h)

2.3. The Basic Reproduction Number. Thebasic reproductionnumberR

0is computed using themethod of next generation

matrix as outlined by [38]. Let 𝑘𝑖𝑗be the expected number of

the new cases of type 𝑖 caused by one infected individual oftype 𝑗, during the entire period of infectiousness. Define amatrix 𝐾 whose entries are 𝑘

𝑖𝑗, that is, 𝐾 = [𝑘

𝑖𝑗]. Then,R

0=

𝜌(𝐾), where 𝜌(𝐾) is spectral radius of 𝐾. For our model, wedefine four type-at-infection consisting of two vectors andtwo hosts, namely, Aedes spp. (type 1), Culex spp. (type 2),livestock (type 3), and humans (type 4). The resulting nextgeneration matrix is

𝐾 =

[

[

[

[

𝑘

11𝑘

12𝑘

13𝑘

14

𝑘

21𝑘

22𝑘

23𝑘

24

𝑘

31𝑘

32𝑘

33𝑘

34

𝑘

41𝑘

42𝑘

43𝑘

44

]

]

]

]

, (26)

where 𝑘11

is the expected number of infected eggs laid byAedes spp. via transovarial transmission, 𝑘

12is the expected

number of infected Aedes spp. due to one infected Culex, 𝑘21

is the expected number of infected Culex spp. due to oneinfected Aedes spp., 𝑘

13is the expected number of infected

Aedes spp. due to one infected livestock, 𝑘31

is the expectednumber of infected livestock due to one infected Aedes spp.,𝑘

14is the expected number of infected Aedes spp. due to

one infected human, 𝑘41

is the expected number of infectedhumans due to one infected Aedes spp., 𝑘

22is the expected

number of infected eggs laid by Culex spp. via transovarialtransmission, 𝑘

23is the expected number of infected Culex

spp. due to one infected livestock, 𝑘32is the expected number

of infected livestock due to one infected Culex spp., 𝑘24is the

expected number of infected Culex spp. due to one infectedhuman, 𝑘

42is the expected number of infected humans due

to one infected Culex spp., 𝑘33

is the expected number ofinfected livestock due to one infected livestock, 𝑘

34is the

expected number of infected livestock due to one infectedhuman, 𝑘

43is the expected number of infected humans due

to one infected livestock, and and 𝑘44is the expected number

of infected humans due to one infected human.Since there is no vertical transmission in Culex spp., then

𝑘

22= 0. The same applies for 𝑘

33and 𝑘

44. Also Aedes spp.


cannot infectCulex spp. and vice versa; therefore, 𝑘12= 𝑘

21=

0. Humans cannot infect livestock, so 𝑘34= 0. Hence, we have

𝐾 =

[

[

[

[

𝑘

110 𝑘

13𝑘

14

0 0 𝑘

23𝑘

24

𝑘

31𝑘

320 0

𝑘

41𝑘

42𝑘

430

]

]

]

]

. (27)

The entry 𝑘𝑖𝑗depends 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 periodof individual of type 𝑗. For example, 𝑘

13will depend on the

probability that livestock survives the incubation period, theadequate contact rate from livestock to Aedes spp., and theinfective period of livestock.We therefore derive the 𝑘

𝑖𝑗values

as follows:

𝑘

11=

𝑏


𝑎

𝜇

𝑎

, 𝑘

13= (

𝜀

𝑙

𝜀

𝑙+ 𝜇

𝑙

)(

𝜆

𝑙𝑎(𝑇)

𝜇

𝑙+ 𝜙

𝑙+ 𝛾

𝑙

) ,

(28a)

𝑘

14= (

𝜀

ℎ

𝜀

ℎ+ 𝜇

ℎ

)(

𝜆

ℎ𝑎(𝑇)

𝜇

ℎ+ 𝜙

ℎ+ 𝛾

ℎ

) ,

𝑘

23= (

𝜀

𝑙

𝜀

𝑙+ 𝜇

𝑙

)(

𝜆

𝑙𝑐(𝑇)

𝜇

𝑙+ 𝜙

𝑙+ 𝛾

𝑙

) ,

(28b)

𝑘

24= (

𝜀

ℎ

𝜀

ℎ+ 𝜇

ℎ

)(

𝜆

ℎ𝑐(𝑇)

𝜇

ℎ+ 𝜙

ℎ+ 𝛾

ℎ

) ,

𝑘

31= (

𝜀

𝑎(𝑇)

𝜀

𝑎 (𝑇) + 𝜇𝑎 (

𝑇)

)(

𝜆

𝑎𝑙(𝑇)

𝜇

𝑎 (𝑇)

) ,

(28c)

𝑘

32= (

𝜀

𝑐(𝑇)

𝜀

𝑐 (𝑇) + 𝜇𝑐 (

𝑇)

)(

𝜆

𝑐𝑙(𝑇)

𝜇

𝑐 (𝑇)

) ,

𝑘

41= (

𝜀

𝑎(𝑇)

𝜀

𝑎 (𝑇) + 𝜇𝑎 (

𝑇)

)(

𝜆

𝑎ℎ(𝑇)

𝜇

𝑎 (𝑇)

) ,

(28d)

𝑘

42= (

𝜀

𝑐(𝑇)

𝜀

𝑐 (𝑇) + 𝜇𝑐 (

𝑇)

)(

𝜆

𝑐ℎ(𝑇)

𝜇

𝑐 (𝑇)

) ,

𝑘

43= (

𝜀

𝑙

𝜀

𝑙+ 𝜇

𝑙

)(

𝜆

𝑙ℎ(𝑇)

𝜇

𝑙+ 𝜙

𝑙+ 𝛾

𝑙

) .

(28e)

2.4. Sensitivity and Elasticity Analyses of R0. Sensitivities

quantify how R0changes in response to the small shifts

in the value of a parameter, while elasticities quantify theproportional change in R

0in response to the proportional

change in a parameter. Both sensitivity and elasticity valuescan be used to judge which parameters are important tomeasure accurately and where variation in parameters willtranslate into variation inR

0.

Caswell [39] developed a way to quantify sensitivity andelasticity of the growth rate 𝜆 to changes in vital rates 𝑎

𝑖𝑗

where 𝑎𝑖𝑗are the entries of population matrix 𝐴. That is, the

sensitivity of the growth rate 𝜆 to changes in vital rates 𝑎𝑖𝑗is

given by

𝑠

𝑖𝑗=

𝜕𝜆

𝜕𝑎

𝑖𝑗

=

V𝑖𝑤

𝑗

⟨w, k⟩, (29)

wherew and v are the right and left eigenvectors, respectively,corresponding to the dominant eigenvalue 𝜆 of the matrix𝐴, and ⟨w, k⟩ is the dot product of w and v. In case 𝑎

𝑖𝑗is a

function of other lower-level parameters, then, the chain rulecan be applied to estimate the sensitivity of 𝜆 to changes inany model parameter 𝑝 as

𝑠 (𝑝) =

𝜕𝜆

𝜕𝑝

= ∑

𝑖𝑗

𝜕𝜆

𝜕𝑎

𝑖𝑗

𝜕𝑎

𝑖𝑗

𝜕𝑝

. (30)

The elasticity of the growth rate 𝜆 to changes in vital rates𝑎

𝑖𝑗, the entries of population matrix 𝐴, is given by

𝑒

𝑖𝑗=

𝜕 log 𝜆𝜕 log 𝑎

𝑖𝑗

=

𝑎

𝑖𝑗

𝜆

𝜕𝜆

𝜕𝑎

𝑖𝑗

. (31)

For 𝑎𝑖𝑗a function of other lower-level parameters 𝑝, the

elasticity is given by

𝑒 (𝑝) =

𝑝

𝜆

𝜕𝜆

𝜕𝑝

=

𝑝

𝜆

∑

𝑖𝑗

𝜕𝜆

𝜕𝑎

𝑖𝑗

𝜕𝑎

𝑖𝑗

𝜕𝑝

. (32)

The theory of sensitivity analysis developed for thematrixmodels by Caswell [39] can be extended to the diseasemodelsto study the sensitivity and elasticity of R

0to the changes

in the reproduction numbers 𝑘𝑖𝑗or the parameters defining

them.Thus, the sensitivity 𝑠𝑖𝑗of amatrix element 𝑘

𝑖𝑗is defined

as the change in the eigenvalue (R0) due to change in 𝑘

𝑖𝑗given

by

𝑠

𝑖𝑗=

𝜕R0

𝜕𝑘

𝑖𝑗

. (33)

For individual parameter, the sensitivity 𝑠(𝑝) is given by

𝑠 (𝑝) = ∑

𝑖𝑗

𝜕R0

𝜕𝑘

𝑖𝑗

𝜕𝑘

𝑖𝑗

𝜕𝑝

. (34)

The elasticity 𝑒𝑖𝑗of a matrix element 𝑘

𝑖𝑗is defined as

𝑒

𝑖𝑗=

𝑘

𝑖𝑗

R0

𝜕R0

𝜕𝑘

𝑖𝑗

. (35)

For individual parameters 𝑝, the elasticity is given by

𝑒 (𝑝) =

𝑝

R0

∑

𝑖𝑗

𝜕R0

𝜕𝑘

𝑖𝑗

𝜕𝑘

𝑖𝑗

𝜕𝑝

. (36)

In order to study the impact of climate change to climate-driven parameter in the distribution of R

0we use climate

data from two different regions in Tanzania, namely, Arushaand Dodoma for the 2006-2007 outbreak. According toWHO [12], RVF was reported in 10 out of the 21 regions ofTanzania where 12 cases were reported in Arusha region, 1 inDar es Salaam, 156 inDodoma, 4 in Iringa, 6 inManyara, 50 inMorogoro, 5 inMwanza, 5 in the Pwani, 24 in Singida, and 1 inTanga regions. From the data we find that Dodoma has morethan 50% of the total cases giving a justification for being acase of study, and Arusha is considered in this study becausethe first case was reported in January 2007 in this region.


Table 2: Parameters with their estimated lower and higher valueswithout considering impact of climate change.

Parameter low value high value Reference1/𝑏

𝑎100 200 Assumed

1/𝜇

𝑎3 days 60 days [17]

1/𝜇

𝑐3 days 60 days [17]

1/𝜇

ℎ40 yrs 60 yrs [19]

1/𝜇

𝑙1 yr 10 yrs [40]

1/𝜀

𝑎4 days 8 days [19]

1/𝜀

𝑐4 days 8 days [19]

1/𝜀

ℎ2 day 6 days [19]

1/𝜀

𝑙1 day 6 days [19]

𝜙

𝑙0.025 0.10 [19]

𝜙

ℎ0.01 0.10 [19]

𝑓

𝑎0.05 0.1 [18]

1/𝛾

𝑙1 day 5 days [19]

1/𝛾

ℎ4 days 7 days [19]

𝜆al 0.15 0.48 [18]𝜆

𝑐𝑙0.05 0.13 [18]

𝜆la 0.15 0.395 [18]𝜆lc 0.15 0.56 [18]𝜆ah 0.001 0.002 [22]𝜆ch 0.0005 0.001 [22]𝜆ha 0.001 0.0015 Assumed𝜆hc 0.0015 0.002 Assumed𝜆lh 0.001 0.002 [19]

3. Results and Discussion

In this section we first present the result for R0when the

parameters are assumed to be independent of climate change.Then, we will compute the numerical value for R

0when

climate change is considered to climate-driven parameters.Sensitivity and elasticity analysis results in both cases will bepresented. Table 2 shows the parameter values for low rangeand high range which are used to compute the numericalvalue for R

0when temperature and precipitation effects are

not considered.When we substitute the values in Table 2 to the expres-

sions of the elements of matrix𝐾 and computeR0, we obtain

that for low parameter values R0= 0.1941 and for high

parameter valuesR0= 6.8071.

When climate change parameters were evaluated usingthe climate variable the value of R

0change from 0.4747

to 14.2007 in Arusha with the highest value marked inNovember 2006 (= 14.2007) followed by December 2006(= 14.1530). The value of R

0dropped below 1 in January

2007 and February 2007, but it rose again in March, April,and May. Figure 2(a) shows the distribution ofR

0from July

2006 to June 2007 in Arusha region.In Dodoma, the highest R

0was marked February 2007

(= 12.7438) followed by January 2007 (= 12.7368) thenMarch 2007 (= 7.9899) and December 2006 (= 1.5088) asFigure 2(b) indicates.

Table 3: Sensitivity and elasticity ofR0for low and high parameter

values.

Parameter Sensitivity ElasticityLow parameter values

𝑘

110.37750 0.00292

𝑘

130.49913 0.37438

𝑘

140.00629 0.00012

𝑘

230.16513 0.12386

𝑘

240.00208 0.00006

𝑘

310.37679 0.37443

𝑘

320.37399 0.12388

𝑘

410.01118 0.00007

𝑘

420.01109 0.00003

𝑘

430.01497 0.00008

High parameter values𝑘

110.36274 0.00160

𝑘

131.87148 0.36107

𝑘

140.01149 0.00001

𝑘

230.50469 0.13804

𝑘

240.00310 0.00000

𝑘

310.09672 0.36107

𝑘

320.13653 0.13804

𝑘

410.00045 0.00001

𝑘

420.00063 0.00000

𝑘

430.00231 0.00000

Table 4: Sensitivity and elasticity of R0for Dodoma and Arusha

climate data.

Parameter Sensitivity ElasticityDodoma

𝑘

110.99874 0.99748

𝑘

130.12996 0.00126

𝑘

140.00020 0.00000

𝑘

230.00010 0.00000

𝑘

240.00000 0.00000

𝑘

310.00971 0.00126

𝑘

320.00001 0.00000

𝑘

410.00010 0.00000

𝑘

420.00000 0.00000

𝑘

430.00001 0.00000Arusha

𝑘

110.99921 0.99841

𝑘

130.10903 0.00079

𝑘

140.00016 0.00000

𝑘

230.00012 0.00000

𝑘

240.00000 0.00000

𝑘

310.00727 0.00079

𝑘

320.00000 0.00000

𝑘

410.00008 0.00000

𝑘

420.00000 0.00000

𝑘

430.00001 0.00000


Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

70123456789101112131415

Months

Basic

repr

oduc

tion

num

bers

(a) Distribution ofR0 for Arusha

Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

7

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Months

Basic

repr

oduc

tion

num

bers

(b) Distribution ofR0 for Dodoma

Figure 2: Distribution ofR0for climatic data in Arusha and Dodoma.

Prec

ipita

tion

Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

70 00

100

200

300

5

10

15

Months

Basic

repr

oduc

tion

num

bers

(a) R0 and precipitation for Arusha

Prec

ipita

tion

Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

70 00

100

200

300

5

10

15

Months

Basic

repr

oduc

tion

num

bers

(b) R0 and precipitation for Dodoma

Figure 3:R0and precipitation for climatic data in Arusha and Dodoma.

While it is clear thatR0increaseswith increase in rainfall,

it is not the case for temperature where we experience highR0for low temperatures. Figure 3 shows the plots forR

0and

precipitation over months, and Figure 4 shows the plots forR0and temperature over months.Table 3 shows the sensitivity and elasticity values ofR

0, to

both low and high parameter values. For both low and highparameter values, R

0is most sensitive to 𝑘

13, the expected

number of infected Aedes spp. Due to one infected livestock,and to and most elastic to 𝑘

31, the expected number of

infected livestock due to one infected Aedes spp. Table 3shows the sensitivity and elasticity values of R

0for low

and high parameter values, and Figure 5 shows the plots ofsensitivity and elasticity values plotted against the parameter𝑘

𝑖𝑗. The results suggest that attention should be given to

parameters regarding incubation period, the adequate con-tact rate, and the infective period of livestock and Aedes spp.

When climatic data are used, R0is found to be more

sensitive and elastic to 𝑘11, the expected number of infected

eggs laid byAedes spp. via transovarial transmission, followedby 𝑘13and 𝑘31for both regions Arusha and Dodoma. Table 4

shows the sensitivity and elasticity values ofR0for Dodoma

and Arusha climate data, and Figure 6 shows the plots ofsensitivity and elasticity against the parameters 𝑘

𝑖𝑗.The results

call for attention to parameters regarding incubation period,the adequate contact rate of Aedes spp. and livestock, theinfective periods of livestock and Aedes spp., and the verticaltransmission in Aedes spp.

4. Conclusion

A deterministic SEIR model of RVF has been presented tostudy the impact of climate change variables mainly tem-perature and precipitation. The model presented here is just

10 Computational and Mathematical Methods in MedicineTe

mpe

ratu

re

Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

7 00

20

25

30

10

20

Months

Basic

repr

oduc

tion

num

bers

(a) R0 and temperature for Arusha

Jul0

6

Aug0

6

Sep0

6

Oct

06

Nov

06

Dec

06

Jan0

7

Feb0

7

Mar

07

Apr0

7

May

07

Jun0

7 00

5

10

15

Months

Basic

repr

oduc

tion

num

bers

Tem

pera

ture

26

28

30

32

(b) R0 and temperature for Dodoma

Figure 4:R0temperature for climatic data in Arusha and Dodoma.

k13 k14 k23 k24 k31 k32 k41 k42 k430

0.05

0.15

0.25

0.35

0.45

Sens

itivi

ty

k110

Elas

ticity

0.5 0.5

0.4

0.3

0.2

0.1

Parameters, kij

(a) Sensitivity and elasticity ofR0 for low parameter values

k13 k14 k23 k24 k31 k32 k41 k42 k430

Sens

itivi

ty

k110

Elas

ticity

0.5

0.4

0.3

0.2

0.1

1.5

1

2

Parameters, kij

(b) Sensitivity and elasticity ofR0 for high parameter values

Figure 5: Sensitivity and elasticity ofR0plotted against the low and high parameters values.

k13 k14 k23 k24 k31 k32 k41 k42 k430

Sens

itivi

ty

k110

Elas

ticity

0.5

0.6 0.6

0.7

0.8 0.8

0.9

1 1

0.4 0.4

0.3

0.2 0.2

0.1

Parameters, kij

(a) Sensitivity and elasticity ofR0 for Arusha

k13 k14 k23 k24 k31 k32 k41 k42 k430

Sens

itivi

ty

k110

Elas

ticity

0.5

0.6 0.6

0.7

0.8 0.8

0.9

1 1

0.4 0.4

0.3

0.2 0.2

0.1

Parameters, kij

(b) Sensitivity and elasticity ofR0 for Dodoma

Figure 6: Sensitivity and elasticity ofR0plotted against the parameters 𝑘

𝑖𝑗for climatic data in Arusha and Dodoma.


a simple representation of the complex ecological situationinvolved in the epidemiology of RVF. The formulation ofthe model, computation of R

0, and sensitivity and elasticity

analyses of R0are based on the assumptions made to build

themodel aswell as the chosen parameter values. Real climatedata from Dodoma and Arusha where outbreak occured in2006-2007 have been used to study the distribution ofR

0in

thewhole period of the outbreak.Though the current analysispresented in this work may not be exhaustible, it remains,however, an important step toward the study of the impactof climate change on the dynamics of RVF.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to thank the following institutions forsupport: TanzaniaMeteorological Agency (TMA), Universityof Iringa, and Nelson Mandela African Institution of Scienceand Technology (NM-AIST). However, the conclusionsmadeso far from this study are not influenced by any institution.

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