multiseason transmission for rift valley fever in north america · rift valley fever’s success at...

Multiseason transmission for Rift Valley fever in NorthAmericaRachelle E. Mirona, Gaël A. Giordanoa, Alison D. Kealeyb, and Robert J. Smith?c

aDepartment of Mathematics, The University of Ottawa, ON, Canada; bDepartment of CommunityHealth and Epidemiology, Queen’s University, Kingston, ON, Canada; cDepartment of Mathematics andFaculty of Medicine, The University of Ottawa, ON, Canada

ABSTRACTRift Valley fever is a vector-borne disease, primarly found inWest Africa, that is transmitted to humans and domestic live-stock. Its similarities to the West Nile virus suggest that estab-lishment in the developed world may be possible. Rift Valleyfever has the potential to invade North America, where seasonsplay a role in disease persistence. The values for the basicreproductive number show that, in order to eradicate thedisease, the survival time of mosquitoes must decrease below8.67 days. Mechanisms such as aggressive spraying thatdecreases the mosquito population can contain an outbreak.Otherwise, Rift Valley fever is likely to establish itself as arecurring seasonal outbreak. Rift Valley fever poses a potentialthreat to North America that would require aggressive inter-ventions in order to prevent a recurring seasonal outbreak.

KEYWORDSImpulsive differentialequations; mathematicalmodel; Rift Valley fever;seasons; spraying

1. Introduction

The emergence of West Nile virus in North America has drawn attention tothe possibility for other mosquito-borne pathogens to expand their naturalranges (Morse, 1995; Patz et al., 2005; Favier et al., 2006; Moutailler et al.,2008; Turell, Dohm et al., 2008). The West Nile virus first appeared in theUnited States in 1999, causing acute illness in 62 individuals, 7 of whom died(Roche, 2002). Outbreaks of encephalitis caused by the West Nile virus haveoccurred in the late summer and early autumn months yearly in New YorkCity since 1999 (Karpati et al., 2004). By the end of 2002, West Nile virusactivity had been reported in all but four continental U.S. states, with morethan 3,500 human cases reported (Mostashari et al., 2003).

Of the many potential mosquito-borne pathogens able to invade NorthAmerica like the West Nile virus, one particular concern is Rift Valley fever,an arthropod-borne viral zoonosis, which has seen an expansion in its geo-graphic range and virulence since its formal identification in Kenya’s Rift

CONTACT Robert J. Smith? [email protected] Department of Mathematics and Faculty of Medicine,The University of Ottawa, 585 King Edward Avenue, Ottawa, ON K1N 6N5, Canada.The question mark in “Smith?” is part of the author’s name.

MATHEMATICAL POPULATION STUDIES2016, VOL. 23, NO. 2, 71–94http://dx.doi.org/10.1080/08898480.2013.836426

© 2016 Taylor & Francis

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Valley in 1931 (House et al., 1992; Favier et al., 2006; Moutailler et al., 2008;Turell, Dohm et al., 2008). Incidence of Rift Valley fever is believed to have aparticularly strong connection to climate variation and has been linked with ElNiño-southern oscillation events or incidents of localized heavy rainfall (Houseet al., 1992; Traoré-Lamizana et al., 2001; Porphyre et al., 2005; Despommierset al., 2007; Anyamba et al., 2009). Prior to 1977, incidents of Rift Valley feverhad primarily been concentrated in sub-Saharan Africa and were mainly aconcern of the agricultural industry. Infection was fatal in young domesticruminants and initiated abortion in pregnant animals, while cases were mild inhumans (Meegan, 1980; House et al., 1992; Moutailler et al., 2008; Turell,Dohm et al., 2008). In 1977, Rift Valley fever’s potential to spread beyond itsendemic range in sub-Saharan Africa was confirmed when it appeared withoutapparent precedent in Egypt. Unlike previously documented outbreaks, humaninfection exceeded 200,000 cases, 600 of which were fatal (Meegan, 1980;House et al., 1992; Traoré-Lamizana et al., 2001; Favier et al., 2006; Gaffet al., 2007). Since then, Rift Valley fever has become endemic in previouslyunexposed areas of Western and Eastern Africa. It reached the ArabianPeninsula in the early 21st century (Gerdes, 2004). Rift Valley fever’s successat establishing its endemicity in novel environments, with a wide variety ofarthropods capable of acting as its vectors, constitutes a serious threat (Houseet al., 1992; Turell, Dohm et al., 2008; Turell, Linthicum et al., 2008).

Rift Valley fever is an arthropod-borne viral zoonosis: it is transmittedbetween humans and animals by an arthropod vector (House et al., 1992;Meegan, 1980; Gerdes, 2004). Rift Valley fever can infect a fairly wide rangeof vertebrate hosts and has an even wider range of insect vectors: more than40 species of mosquitoes from 8 different genera have been isolated in thefield (Meegan, 1980; House et al., 1992; Gerdes, 2004; Turell, Dohm et al.,2008; Turell, Linthicum et al., 2008). Rift Valley fever’s success at establishingits endemicity in novel environments is due in part to its flexibility in bothhosts and vectors (Balkhy and Memish, 2003). Mosquitoes belonging to theAedes species are an important vector since they may vertically transmitinfection to their young through transovarial transmission, enabling themaintenance of Rift Valley fever during inter-epizootic periods (Houseet al., 1992; Traoré-Lamizana et al., 2001; Bowman et al., 2005). Culex speciesand those belonging to Eretmapodites are believed to be strongly involvedwith epizootic outbreaks (House et al., 1992; Traoré-Lamizana et al., 2001;Bowman et al., 2005). Both the Aedes and Culex species are known totransmit the West Nile virus. Rift Valley fever can also be transmittedthrough aerosol exposure from handling infected carcasses. Laboratory work-ers, veterinarians, and individuals involved with meat processing are parti-cularly vulnerable to this form of infection (House et al., 1992; Traoré-Lamizana et al., 2001; Moutailler et al., 2008; Turell, Dohm et al., 2008;Turell, Linthicum et al., 2008).

72 R. E. MIRON ET AL.D

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Rift Valley fever has a very wide range of viable vertebrate hosts, as mightbe expected, considering the diversity of its vectors (Balkhy and Memish,2003). Rift Valley fever’s most significant host species are domestic rumi-nants such as sheep and cattle (House et al., 1992; Balkhy and Memish, 2003;Turell, Dohm et al., 2008; Turell, Linthicum et al., 2008). Newborn lambs andgoats are most susceptible to disease, followed by calves and sheep. Moderatedisease occurs in adult cattle, sheep, goats, humans, water buffalo, and rats.Humans and ruminants are capable of developing enough viremia to infectmosquitoes. Camels, horses, pigs, cats, dogs, guinea pigs, rabbits, hedgehogs,and monkeys are susceptible but do not necessarily display clinical disease.Birds, with the exception of pigeons and chickens, and reptiles are believed tobe resistant to infection (House et al., 1992; Gerdes, 2004). Balkhy andMemish (2003) showed that fatality in adult cattle and sheep can be ashigh as 30%, while young animals may experience 100% fatality. Gerdes(2004) showed that the disease is most significant in young animals, parti-cularly lambs and goats, which may experience 70–100% mortality. Newbornanimals experience 100% fatality, and most epidemics are defined by highincidences of abortion (Gerdes, 2004). Prior to the Egyptian outbreak in1977, the disease in humans had mostly been asymptomatic or mild, oftenmanifesting as influenza-like symptoms. Complications such as encephalitismay occur, and fewer than 1% of cases may present hemorrhagic fever(House et al., 1992; Balkhy and Memish, 2003; Gerdes, 2004; Turell, Dohmet al., 2008).

In order for a vector-borne emerging infectious disease to become endemic in anew location, there must be enough susceptible hosts and vectors (House et al.,1992). Due to Rift Valley fever’s large range of viable hosts and vectors, itspotential to establish itself is high compared to other vector-borne diseases. Gaffet al. (2007) modeled Rift Valley fever transmission using ordinary differentialequations for two populations of mosquito species — those that can transmitvertically and those that cannot — and one population of domestic livestockanimals with disease-dependent mortality. They found the stability of the disease-free equilibrium and identified parameters affecting the stability. They showedthat, for any given contact rate, there is a low level of endemic prevalence, whichimplies that the disease could persist if introduced into an isolated system.

Bicout and Sabatier (2004) and Zell (2004) demonstrated that the incidence ofmany vector-borne infectious diseases shows seasonality, and extreme weatherevents are often accompanied by additional outbreaks. Favier et al. (2006) assessedthe possibility of endemicity without wild animals providing a permanent virusreservoir. Using a deterministic model, endemicity without a permanent virusreservoir is impossible in a single site except when there is a strictly periodicrainfall pattern, but it is possible when there are herd movements and sufficientinter-site variability in rainfall, which drives mosquito emergence.

MATHEMATICAL POPULATION STUDIES 73D

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2. The model

We use a compartmental model derived from Gaff et al. (2007) with threepopulations: the human, domestic livestock, and mosquito populations(Figure 1). The model has four different classes: the S classes are susceptibleindividuals, the E classes are infected but non-infectious individuals, the I classesare infectious individuals, and the R classes are recovered and immune indivi-duals. When initially infected, the individual is non-infectious for a while; onlyindividuals in class I can infect the susceptible populations. Once recovered fromthe disease, the individual has lifelong immunity.

The human population is described by

S0H t! " # ΛH $ dHSH t! " $ βLHSH t! "IL t! "NL t! " $ βMHSH t! "IM t! "

E0H t! " # βLH

SH t! "IL t! "NL t! " % βMHSH t! "IM t! " $ dHEH t! " $ εHEH t! "

I0H t! " # εHEH t! " $ dHIH t! " $ μHIH t! " $ γHIH t! "R0

H t! " # γHIH t! " $ dHRH t! ";

8>>><

>>>:(1)

where

NL t! " # SL t! " % EL t! " % IL t! " % RL t! ":

The domestic livestock population is described by

S0L t! " # ΛL $ dLSL t! " $ βMLSL t! "IM t! "E0L t! " # βMLSL t! "IM t! " $ dLEL t! " $ εLEL t! "I0L t! " # εLEL t! " $ dLIL t! " $ μLIL t! " $ γLIL t! "R0

L t! " # !LIL t! " $ dLRL t! ":

8>><

>>:(2)

SH

EH

IH

RH

H

dH

dH

dH

dH

µH

!MH

!LH

"H

#H

SL

EL

IL

RL

L

dL

dL

dL

dL

µL

!ML

"L

#L

SM

EM

IM

M

dM

dM

dM

!LM

!HM

"M

Figure 1. The model. The index “H” refers to human, “L” to livestock, and “M” to mosquitoes. SH,SL, and SM are the susceptible populations, EH, EL, and EM are the exposed populations, IH, IL, andIM are the infected populations, and RH and RL are the recovered populations (mosquitoes do notrecover from the disease). Other parameters are listed in Table 2.


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The Aedes mosquito population is described by

S0M t! " # ΛM $ dMSM t! " $ βLMSM t! "IL t! " $ βHMSM t! "IH t! "E0M t! " # βLMSM t! "IL t! " % βHMSM t! "IH t! " $ dMEM t! " $ εMEM t! "I0M t! " # εMEM t! " $ dMIM t! ":

8<

: (3)

Each population has infection rate βj j # H; L;M! ". The natural death ratesare dj, the disease-induced death rates μj, the rates at which a noninfectiousindividual becomes infectious εj, and the recovery rates !j. The mosquitopopulation contains no R class, because mosquitoes never clear the infectiononce they are infected. A summary of the parameters and units are given inTable 1.

The birth rates in the system are constant sources of susceptible indi-viduals Λi for i # H; L;M; in the absence of disease, the susceptiblepopulations converge to the disease-free equilibrium at an exponentialrate. We also include fetal death caused by Rift Valley fever infection inthe domestic livestock population, which is not included in Gaff et al.(2007). Human and livestock populations are unlikely to be well-mixed(meaning that each infected individual has equal chance of infecting everysusceptible individual), so we use standard incidence for transmission. Allother terms, including the interactions between mosquitoes and hosts, aremass-action because the mosquito and host populations are assumed to bewell-mixed. The spread of Rift Valley fever is much more likely betweenmosquitoes and hosts than between hosts because the interaction betweenhost and mosquito populations is higher in a given area than host to hostinteraction.

Table 1. Definition of parameters.Parameter Units DefinitionΛH population ! days!1 Human birth rateΛL population ! days!1 Immigration of livestockΛM population ! days!1 Mosquito birth rate1/dH days Human survival time1/dL days Livestock survival time1/dM days Mosquito survival timeμH days!1 Disease death rate for humansμL days!1 Disease death rate for livestockβMH (days ! population)!1 Infection rate from mosquitoes to humansβLH (days ! population)!1 Infection rate from livestock to humansβML (days ! population)!1 Infection rate from mosquitoes to livestockβLM (days ! population)!1 Infection rate from livestock to mosquitoesβHM (days ! population)!1 Infection rate from humans to mosquitoes1=εH days Human incubation time1=εL days Livestock incubation time1=εM days Mosquito incubation time1=!H days Human recovery time1=!L days Livestock recovery time


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3. Asymptotic behavior: One-season model

The model has two equilibria: the disease-free equilibrium and the ende-mic equilibrium (developed in the Appendix). Several solutions could existfor the endemic equilibrium, resulting in a backward bifurcation. If this isthe case, then lowering the basic reproductive number below 1 may nolonger be sufficient for control (Li et al., 2011). This presents a seriouscomplication when a disease is already endemic because the outcomedepends on initial conditions. In the case of Rift Valley fever, we onlyconsider sufficiently small perturbations away from the disease-free equili-brium because the disease is present in North America. This contrasts withthe situation in Africa, where the disease is already established, so morerigorous control efforts would be required.

The Jacobian matrix is computed at the disease-free equilibrium(expressed in the Appendix). For fixed parameters and state variables, thestability changes as the mosquito death rate dM varies. For the values used inTable 2, the Routh–Hurwitz conditions are satisfied for the characteristicequation

f α! " # α6 % c1α5 % c2α4 % c3α3 % c4α2 % c5α% c6; (4)

when 0:15< dM < 0:68. The disease-free equilibrium is locally stable in thisregion.

Table 2. Range of parameters.Parameter Sample Value Range Units ReferencesΛH 1000 ! dH 1000 ! dH people

days!1(Bowman et al., 2005)

ΛL 0.1 0.01–0.5 livestockdays!1

Assumed

ΛM 10000 ! (1/14) 200–1000 mosquitoesdays!1

(Gaff et al., 2007)

1/dH 80 ! 365 70–90 days (Bowman et al., 2005)1/dL 20 ! 365 10–25 days (Gaff et al., 2007)1/dM varies 3–60 days (Gaff et al., 2007)μH 0.01 0.01–0.1 days!1 (Directors of Health Promotion and

Education, 2013)μL 1/25 1/10–1/40 days!1 (Gaff et al., 2007)βMH 0.2762/1000 0.2762/10000–

0.2762/100days!1 (Bowman et al., 2005)

βLH 10!5 10!5–10!3 days!1 (Xue et al., 2012)βML βMH βMH days!1 (Gaff et al., 2007)βLM βML=8 βML=8 days!1 (Gaff et al., 2007)βHM βMH=10 βMH=10 days!1 (Xue et al., 2012)1=εH 5 4–6 days (Bowman et al., 2005), (CDC, 2003)1=εL 2 1–3 days (Gaff et al., 2007), (CDC, 2003)1=εM 6 4–8 days (Gaff et al., 2007)1=γH 5 2–7 days (Bowman et al., 2005), (CDC, 2003)1=γL 3 1–5 days (Gaff et al., 2007)


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We calculated R0, the average number of secondary infections that anysingle infected individual will cause by using the next-generation method(van den Driessche and Watmough, 2002). The solution (developed in theAppendix) to the characteristic polynomial gives a cubic equation with onereal solution and two complex conjugates. The value for R0 is the largestmodulus of our eigenvalues (Greenhalgh, 1996).

4. Multiple-season model

4.1. Impulse conditions

In most of northern America, the mosquito population drops during thewinter months. This fact is taken into account though impulsive differentialequations (Lakshmikantham et al., 1989; Bainov and Simeonov, 1989, 1993,1995). At impulse times tk:

Δx # x t%k! "

$ x t$k! "

# f tk; x t$k! "! "

; (5)

where f(t, x) maps the solution before the impulse, x t$k! "

, to x t%k! "

. We resetthe mosquito, human, and domestic livestock populations at the beginning ofeach summer.

We consider two seasons: the summer season, when the mosquitoes willinfect hosts, and the winter season, when the mosquitoes die from the coldweather. During the winter, there is no vector to spread the disease; thedomestic livestock and human populations increase, as abortions no longeroccur.

The mosquitoes decrease to zero with the cold but reappear at the beginning ofsummer. The Aedesmosquito has very few offspring surviving through the winter(Dohm et al., 2002). We assume that all adult mosquitoes die during winter.

At time tk,

ΔSH # r1S$H % p1 E$H % I$H % R$H

! "

ΔEH # $E$HΔIH # $I$HΔRH # E$H % I$H $ p1 E$H % I$H % R$

H

! ";

8>><

>>:(6)

where SH, EH, IH and RH are the susceptible, exposed, infected, and recoveredhuman populations. The susceptible population increases by a fraction r1. Becausethe infection period is short, by the end of winter, the total infected population hasrecovered so that IH # 0. A fraction p1 of the recovered offspring is susceptible.

At time tk,

ΔSL # r2S$L % c% p2 E$L % I$L % R$L

! "

ΔEL # $E$LΔIL # $I$LΔRL # E$L % I$L $ p2 E$L % I$L % R$

L

! ";

8>><

>>:(7)


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where SL, EL, IL, and RL are the susceptible, exposed, infected, and recovereddomestic livestock populations, respectively. Because infection causes deathin livestock, the susceptible livestock increases by a fixed amount c, reflectingthe purchase of livestock for breeding.

The opposite happens with the mosquito population. By the end of winterseason 1, few mosquitoes remain. At time tk,

ΔSM # $r3S$M % p3I$MΔEM # $E$MΔIM # $ r3 % p3! "I$M;

8<

: (8)

where SM and IM are the susceptible and infected mosquito populations; r3 isthe total decrease in the mosquito population; and p3 is the proportion ofmosquitoes who are born with no infection from an original infected mos-quito. The total number of exposed mosquitoes who have offspring is zero,because a mosquito only stays in the exposed class for a short period of time.

The basic reproductive number for the one-season model has similarproperties to that of the extended model. In the one-season model, R0 largerthan 1 causes an outbreak of Rift Valley fever in the human and livestockpopulations. The dynamics in the first season of the extended model are thesame for subsequent seasons. The reset initial conditions for the secondseason do not change the equilibrium points for this season and, becauseR0 is greater than 1, we have an outbreak in the second season.

5. Simulations

5.1. One-season model

We calculated the effects on the human and domestic livestock populationsby having infected mosquitoes enter North America. The initial site ofinfection is a small, fictitious coastal city. All the values used in the simula-tions are shown in Table 1. Each farmer buys one cow every 10 days, soΛL # 0:1. We take βHM 10 times smaller than βMH because the infection ratefrom human to mosquito is smaller than from mosquito to human. Theinfection rate from livestock to mosquito is likewise smaller than mosquito tolivestock. Mosquitoes infect humans and livestock at the same rate.

Using the values in Table 2, Figure 2 shows the effect of decreasing thesurvival time of mosquitoes. As the survival time of mosquitoes is decreased(due to mosquito spraying), R0 is reduced below 1.

For Figure 3, we choose the mosquito survival time to be such that R0 < 1,which caused no outbreaks within the year and led to eradication of thedisease. For Figure 4, we choose the mosquito survival time to be such thatR0 > 1, which caused a disease outbreak at the beginning of the year, but thedisease was still eradicated after the outbreak.


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4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

mosquito survival time (days)

Basic reproductive number

Figure 2. The effects of mosquito survival time on the basic reproductive number, R0. R0 dropsbelow 1 if the mosquito survival time is sufficiently small. All other parameters are set to theirmedian values listed in Table 2.

0 50 100 150 200 250 300 3500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

time (days)

Mosquito population(c)

infectedexposed

susceptibles

0 50 100 150 200 250 300 35010 15

10 10

10 5

100

105

time (days)

Human population(a)

exposed

infected

susceptibles

recovered

0 50 100 150 200 250 300 35010 10

10 5

100

105

time (days)

Domestic livestock population(b)

susceptibles

recovered

exposed

infected

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

time (days)

infected

exposed

(d) Mosquito population

Figure 3. The behavior of (a) the human population, (b) the domestic livestock population, and(c) the mosquito population when the mosquito survival time is such that R0 < 1 for one season(assuming winter does not change the dynamics of populations). The solid lines represent thesusceptible populations, the dotted lines the exposed populations, the dashed lines the infectedpopulations, and the dash-dot lines are the recovered populations. (d) The initial behavior of themosquito population when R0 < 1.


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For R0 < 1, the mosquito population in Figure 3c decreases. Becausemosquitoes do not recover from the disease, and the mosquito survivaltime is such that R0 < 1, we take the death rate to be large enough so thatthe mosquitoes do not have enough time to infect the human and domesticlivestock populations in order to have an outbreak. Because infectionbetween humans and livestock is nonexistent, it suggests that the disease-free equilibrium is stable and Rift Valley fever does not become endemic.

For a small value of dM resulting in R0 > 1, the disease causes a decrease inthe susceptible human and domestic livestock populations, which follows anincrease in the recovered human and domestic livestock populations, becauseboth populations have lifelong immunity (Figure 4a and 4b). The infectiondoes not completely eradicate the virus, which suggests that the endemicequilibrium is stable.

5.2. Multiple-season model

We allow 2% of the second generation of susceptible mosquitoes to survivethe winter. Meanwhile, the susceptible human population increases by 10%,

0 50 100 150 200 250 300 3500

5000

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15000

time (days)


susceptibles

infected

exposed

0 50 100 150 200 250 300 3500

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time (days)

Human population(a)

susceptibles

recovered

infected

exposed

0 50 100 150 200 250 300 3500

100

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600

700

800

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1000

time (days)


recovered

susceptibles

infected

exposed

Figure 4. The behavior of (a) the human population, (b) the domestic livestock population and(c) the mosquito population when the mosquito survival time is such that R0 > 1 for one season(assuming winter does not change the dynamics of populations). The solid lines represent thesusceptible populations, the dotted lines the exposed populations, the dashed lines the infectedpopulations, and the dash-dot lines are the recovered populations.


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the infected population decreases to zero, and 10% of the recovered popula-tion becomes susceptible. The domestic livestock population grows at thesame rate as the human population. The susceptible population increases by40% during the winter months, so 40% of the recovered population becomessusceptible. The domestic livestock increases by 10% to account for thebuying of livestock due to deaths in the previous season.

Figure 5 shows the effects of choosing the mosquito survival time to besuch that R0 < 1, which causes no outbreaks within 5 years and leads toeradication of the disease. Figure 6 shows the effects of choosing the mos-quito survival time to be such that R0 > 1, which causes a disease outbreakevery year in the human and domestic livestock populations.

Figure 5c shows the effects of impulses on the mosquito population bydecreasing 98% of all the susceptible mosquitoes during the winter seasonswhen R0 < 1. For both the human and domestic livestock populations, thedisease-free equilibrium is stable (Figures 5a and 5b). The only difference is achange in the susceptible populations due to the discontinuous increase inpopulations during the winter months.

0 200 400 600 800 10000

2000

4000

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10000

time (days)


susceptible

exposed infected

0 200 400 600 800 10000

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time (days)

Human population(a)

recovered infected exposed

susceptible

0 200 400 600 800 10000

500

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time (days)


susceptible

recoveredinfected exposed

Figure 5. The behavior of (a) the human population, (b) the domestic livestock population, and(c) the mosquito population when the mosquito survival time is such that R0 < 1 for multipleseasons. The solid lines represent the susceptible populations, the dotted lines the exposedpopulations, the dashed lines the infected populations, and the dash-dot lines the recoveredpopulations. (d) The initial behavior of the mosquito population when R0 < 1.


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For R0 > 1, outbreaks occur every year. The first outbreak is larger thansubsequent years, but the infected populations remain endemic. Infectionpersists in the mosquito population even though 98% of the infected areremoved from the model due to winter (Figure 6c).

5.3. Sensitivity

Latin Hypercube Sampling is a statistical sampling that allows for an efficientanalysis of parameter variations across simultaneous uncertainty ranges ineach parameter; partial rank correlation coefficients rank the coefficients bythe degree of influence each has on the outcome, regardless of whether thatinfluence increases or decreases the effect. Latin Hypercube Sampling is mostefficient if the outcome variable is a monotonic function of each of the inputparameters (Blower and Dowlatabadi, 1994). Stein (1985) showed that, formany simulations, Latin Hypercube Sampling is the most efficient design,even if the outcome variable is not monotonic. Figure 7a shows the partialrank correlation coefficient sensitivity for all parameters for 1000 runs. R0 ismost sensitive to the mosquito death rate dM, as Figure 7b shows.

0 200 400 600 800 10000

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1000

time (days)

Human population

infected exposedsusceptible

recovered

(a)

0 200 400 600 800 10000

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400

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1000

time (days)

(b) Domestic livestock population

recovered

infectedexposed

susceptible

0 200 400 600 800 10000

5000

10000

15000

time (days)

(c) Mosquito population

susceptible

infected

exposed

Figure 6. The behavior of (a) the human population, (b) the domestic livestock population, and(c) the mosquito population when the mosquito survival time is such that R0 > 1 for multipleseasons. The solid lines represent the susceptible populations, the dotted lines the exposedpopulations, the dashed lines the infected populations, and the dash-dot lines the recoveredpopulations.


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6. Conclusion

An outbreak of Rift Valley fever in North America is theoretically possible:only a very small number of infected mosquitoes is needed to establish thevirus in North America. As seen with West Nile virus, a disease can establishitself with a mosquito as a vector, and humans and livestock as hosts. Thevirulence of West Nile virus in North America demonstrates the potential ofother mosquito-borne pathogens.

We showed that reducing the survival time of mosquitoes below approxi-mately 8.67 days would be sufficient to control an imported outbreak of RiftValley fever. This could be achieved through spraying insecticides.Insecticide control of mosquitoes has been effective in reducing malaria

0 0.2 0.4 0.6 0.8 16

4

2

0

2

4

6

8

mosquito death rate (dM

)

ln(basic reproductive number)(b)

1 0.8 0.6 0.4 0.2 0 0.2 0.4

disease death rate (livestock)rate of becoming infectious (human)

infection rate (livestock to human)rate of becoming infectious (mosquito)

natural death rate (livestock)rate of becoming infectious (livestock)

human birth ratenatural death rate (human)

disease death rate (human)livestock birth rate

infection rate (livestock to mosquito)recovery rate (livestock)

mosquito birth rateinfection rate (mosquito to livestock)

recovery rate (human)infection rate (mosquito to human)infection rate (human to mosquito)

natural death rate (mosquito)

(a)

0.21290.1908

0.0511

0.02280.0231

0.14320.1458

0.10880.0983

0.8267

0.00520.0140

0.04880.0497

0.05650.0728

0.1263

0.1528

Figure 7. (a) Partial rank correlation coefficient sensitivity analysis on R0 for all parameters. (b)The effect of the disease death rate dM on R0 using Monte Carlo simulations with parametersdrawn using Latin Hypercube Sampling.


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worldwide between the 1940s and 1960s (Trigg and Kondrachine, 1998;Mabaso et al., 2004; Macintyre et al., 2006; Al-Arydah and Smith?, 2011).Female mosquitoes live between two weeks and a month, depending onwarmth and moisture (CDC, 2012), so reducing this duration to less than8.67 days results in disease eradication.

Themultiple seasonmodel shows the same results. If Rift Valley fever can enterNorth America, an outbreak is likely to occur unless it can be controlled quickly.Furthermore, if Rift Valley fever invades North America once, it is likely to breakout again each year. The sufficient condition given in Section A.2 of the Appendixto have eigenvalues with negative real part (ensuring that the disease-free equili-brium is stable) is dM > 0:15, where dM is themosquito death rate. This threshold isdifferent from the one found using R0 dM > 0:115! ". The difference is due to thecondition used in Section A.2 of the Appendix, which is sufficient but notnecessary in order to have all eigenvalues negative. A sufficient and necessarythreshold could be calculated from the Routh–Hurwitz conditions if fewer para-meters were involved.

While the behavior of the system is consistent with expectations, mostparameters have little effect on R0. Density-independence and mass-actionterms limit the model to small populations. We used initial values corre-sponding to a small coastal population in North America instead of the entireNorth American population. We ignored the presence of cities.

Our results suggest that countries in the developed world face ongoingthreats of disease introduction. As global travel increases the movement ofhumans, livestock, and vectors around the world, the potential for newoutbreaks is heightened. We must be ready to face these challenges.

Acknowledgments

The authors are grateful to Jing Li for technical discussions.

Funding

RM is funded by an Ontario Graduate Scholarship. Robert J. Smith? is supported by anNSERC Discovery Grant, an Early Researcher Award, and funding from MITACS.

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A. Appendix

A.1. Equilibria

The model has two equilibria. The disease-free equilibrium is

S&H; E&H; I

&H;R

&H; S

&L;E

&L; I

&L;R

&L; S

&M;E

&M; I

&M

! "# ΛH

dH; 0; 0; 0; ΛL

dL; 0; 0; 0; ΛM

dM; 0; 0

# $. The endemic

equilibrium is

S&H; E&H; I

&H;R

&H; S

&L;E

&L; I

&L;R

&L; S

&M;E

&M; I

&M

! "; (9)

where I&M is the solution to

ρI3M % ηI2M % δIM % ω # 0 (10)


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with

!1 #βLMεLβMLΛL

dL % μL % γL! "

dL % εL! "(11)

!2 #βHMεLΛL

dH % μH % γH! "

dH % εH! "(12)

and

ρ # dM dM % εM! " !1βMHβML % β2MLdMβMH % !2βMHβ2ML

! "(13)

η # d2M dM % εM! "βML !1 % βML % 2dLβMH

! "

% dM dM % εM! "βML !1!2 % !2βMHdL % !1dH! "

% !21dM dM % εM! "$ βMLεLΛL $ dLdM dM % εM! "! "

!1βMH % !2βMHβML

! "(14)

δ # d2MdL dM % εM! " !1 % 2βML % dLβMH

! "

$ βMLεLΛL $ dLdM dM % εM! "! "

!1!2 % !2βMHdL % !1dH! "

$ βMHdLεMΛM !1 % !2βML

! "$ !21εMΛM

(15)

ω # d2Md2L dM % εM! " $ !1dHdLεMΛM $ !2βMHd

2LεMΛM $ !1!2dLεMΛM; (16)

where

S&H # ΛH

dH % βMHI&M % βLHI

&L

(17)

E&H # βMHS&HI

&M % βLHS

&HI

&L=N

&L

dH % εH(18)

I&H # εHE&HdH % μH % !H

(19)

R&H # !HI

&H

dH(20)

S&L #ΛL

dL % βMLI&M

(21)

E&L #βMLS

&LI

&M

dL % εL(22)

I&L # εLE&LdL % μL % γL

(23)

R&L #

γLI&L

dL(24)

S&M # ΛM

dM % βHMI&M % βLMI

&L

(25)

E&M #βHMS

&MI

&H % βLMS

&MI

&L

dM % εM(26)

and where I&M is the solution to Eq. (10).


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A.2. Stability of the disease-free equilibrium

The Jacobian matrix evaluated at the disease-free equilibrium is JDFE # J 1! "DFEjJ

2! "DFE

# $, where

J 1! "DFE #

$dH 0 0 0 0 00 $dH $ εH 0 0 0 00 εH $dH $ μH $ γH 0 0 00 0 γH $dH 0 00 0 0 0 $dL 00 0 0 0 0 $dL $ εL0 0 0 0 0 εL0 0 0 0 0 00 0 $βHMS

&M 0 0 0

0 0 βHMS&M 0 0 0

0 0 0 0 0 0

0

BBBBBBBBBBBBBBBB@

1

CCCCCCCCCCCCCCCCA

(27)

J 2! "DFE #

$βLHS&H=S

&L 0 0 0 $βMHS

&H

βLHS&H=S

&L 0 0 0 βMHS

&H

0 0 0 0 00 0 0 0 00 0 0 0 $βMLS

&L

0 0 0 0 βMLS&L

$dL $ μL $ γL 0 0 0 0γL $dL 0 0 0

$βLMS&M 0 $dM 0 0

βLMS&M 0 0 $dM $ εM 0

0 0 0 εM $dM

0

BBBBBBBBBBBBBBBB@

1

CCCCCCCCCCCCCCCCA

: (28)

The matrix has the characteristic equation

0 # det JDFE S&H;E&H; I

&H;R

&H; S

&L;E

&L; I

&L;R

&L; S

&M; E

&M; I

&M

! "$ αI11

! "(29)

# $dH $ α! "2 $dL $ α! "2 $dM $ α! "f α! "; (30)

where f α! " is the determinant of (M1|M2) where

M1 #

$dH $ εH $ α 0 0εH $dH $ μH $ γH $ α 00 0 $dL $ εL $ α0 0 εL0 βHM

ΛMdM

00 0 0

0

BBBBBB@

1

CCCCCCA(31)

M2 #

βLHΛHdLdHΛL

0 βMHΛHdH

0 0 00 0 βML

ΛLdL

$dL $ μL $ γL $ α 0 0βLM

ΛMdM

$dM $ εM $ α 00 εM $dM $ α

0

BBBBBB@

1

CCCCCCA(32)

and S&H , S&L, and S&M are the disease-free equilibrium values. Solving for f !α", we have

f α! " # α6 % c1α5 % c2α4 % c3α3 % c4α2 % c5α% c6; (33)


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where

c1 # 2dH % μH % γH % εH % 2dL % εL % μL % γL (34)

c2 # dH % εH % γH! "

dH % εH! " % 2dH % εH % μH % γH! "

2dL % εL % μL % γL! "

% 2dH % εH % μH % γH! "

2dM % εM! " % dL % εL! " dL % μL % γL! "

% 2dL % εL % μL % γL! "

2dM % εM! " % dM dM % εM! "(35)

c3 # dH % μH % γH! "

dH % εH! " 2dL % εL % μL % γL! "

% dH % μH % γH! "

dH % εH! " 2dM % εM! "% dL % εL! " dL % μL % γL

! "2dH % μH % εH % γH! "

% 2dL % μL % εL % γL! "

2dM % εM! " 2dH % μH % εH % γH! "

% dM dM % εM! " 2dH % μH % εH % γH! "

% dL % εL! " 2dM % εM! " dL % μL % γL! "

% dM dM % εM! " 2dL % εL % μL % γL! "

(36)

c4 # dH % μH % γH! "

dH % εH! " dL % εL! " dL % μL % γL! "

% dH % εH! " dH % μH % γH! "

2dM % εM! " 2dL % εL % μL % γL! "

% dM dM % εM! " dH % εH! " dH % μH % γH! "

% dL % εL! " 2dM % εM! " dL % μL % γL! "

2dH % μH % εH % γH! "

% dM dM % εM! " 2dH % μH % εH % γH! "

2dL % μL % εL % γL! "

% dM dM % εM! " dL % εL! " dL % μL % γL! "

$ εHεMβHMβMHΛM

dM

ΛH

dH

$ εLεMβLMβMLΛL

dL

ΛM

dM

(37)

c5 # dH % μH % γH! "

dH % εH! " dL % εL! " 2dM % εM! " dL % μL % γL! "

% dM dM % εM! " dH % μH % γH! "

dH % εH! " 2dL % εL % μL % γL! "

% dM dM % εM! " dL % μL % γL! "

dL % εL! " 2dH % εH % μH % γH! "

$ εHεMβHMβMHΛM

dM

ΛH

dH2dL % εL % μL % γL! "

$ εLεMβLMβMLΛL

dL

ΛM

dM2dH % εH % μH % γH! "

(38)

and

c6 # dm dM % εM! " dH % μH % γH! "

dH % εH! " dL % μL % γL! "

dL % εL! "

$ εLεMβLMβMLΛL

dL

ΛM

dMdH % μH % γH! "

dH % εH! "

$ εHεMβMHβHMΛH

dH

ΛM

dMdL % μL % γL! "

dL % εL! "

$ εHεLεMβLHβHLβHMΛM

dM

ΛH

dH:

(39)

Necessary and sufficient conditions for all the zeros of f(α) to have negative real parts arethat c4, c5, and c6 be bigger than zero (because c1, c2, and c3 are strictly positive), and

a1a2a3 % a1a5 > a33 % a21a4 (40)

a1a6a2 2a1a5 $ a23! "

% a21a6 a4a3 $ a1a6! " % a1a5 a4a5 $ 3a3a6! " % a6a33% a25a1a4 > a

21a5a

24 % a5 a3a4 $ a5a2! " a3 $ a1a2! ":

(41)


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If these Routh–Hurwitz conditions are satisfied, the disease-free equilibrium is stable(Truccolo et al., 2003; Allen, 2006). For the values used in Table 2, c4, c5, and c6 varydepending on parameter values and the state variables S&H , S

&L, and S&M. By fixing the state

variables, the values for dM (mosquito death rate) can be varied in order to change c4, c5,and c6 from positive to negative. The stability changes depending on the mosquito survivaltime. Numerically, c4, c5, and c6 are positive when dM > 0:15 (Figure A1), and the twoinequalities are satisfied when 0< dM < 0:68. The disease-free equilibrium is stable if0:15< dM < 0:68.

A.3. Basic reproductive number

The model has six infected populations, EH, IH, EL, IL, EM, and IM. The vector F representingnew infections and the vector V representing transfers between compartments are given by

F #

βLHS&HI

&L

N&L% βMHS

&HI

&M

0βMLS

&LI

&M

0βLMS

&MI

&L % βHMS

&MI

&H

0

0

BBBBBB@

1

CCCCCCA(42)

V #

dH % εH! "E&HdH % μH % !H! "

I&H $ εHE&HdL % εL! "E&L

dL % μL % !L! "

I&L $ εLE&LdM % εM! "E&M

dMI&M $ εME&M

0

BBBBBB@

1

CCCCCCA: (43)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410 7

10 6

10 5

10 4

10 3

10 2

10 1

100

death rate of mosquitoes (days 1)

Coefficients of f(!)

c4

c6

c5

Figure A1. Sign change for certain parameters in f(!). When c6 becomes negative (at around0.15), both c4 and c5 are still positive. The values of c4 and c5 only become negative after c6 hasalready fallen below zero.


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The matrices F and V are

F #

0 0 0 βLHS&H=S

&L 0 βMHS

&H

0 0 0 0 0 00 0 0 0 0 βMLS

&L

0 0 0 0 0 00 βHMS

&M 0 βLMS

&M 0 0

0 0 0 0 0 0

0

BBBBBB@

1

CCCCCCA(44)

V #

dH % εH 0 0 0 0 0$εH dH % μH % !H 0 0 0 00 0 dL % εL 0 0 00 0 $εL dL % μL % γL 0 00 0 0 0 dM % εM 00 0 0 0 $εM dM

0

BBBBBB@

1

CCCCCCA; (45)

where S&H , S&L, and S&M are the disease-free equilibrium values. Next, FV$1 # FV$1

1! " jFV$12! "

# $

was calculated, and the eigenvalue with the largest modulus is the value of R0. We have

FV$11! " #

0 0 βLHS&HεL

S&L dL%μL%!L! " dL%εL! "0 0 00 0 00 0 0

βHMS&MεHdH%εH! " dH%μH%!H! "

βHMS&MdH%μH%!H

βLMS&MεLdL%εL! " dL%μL%!L! "

0 0 0

0

BBBBBBB@

1

CCCCCCCA

(46)

FV$12! " #

βLHS&H

S&L dL%μL%!L! "βMHS

&HεM

dM dM%εM! "βMHS

&H

dM

0 0 00 βMLS

&LεM

dM dM%εM! "βMLS

&L

dM0 0 0

βLMS&MdL%μL%!L

0 00 0 0

0

BBBBBBB@

1

CCCCCCCA

: (47)

The characteristic polynomial is

det FV$1 $ αI6! "

# α3g α! "; (48)

where g α! " is the determinant of

$α βLHS&HεL

S&L dL%μL%!L! " dL%εL! "βMHS

&HεM

dM dM%εM! "

0 $α βMLS&LεM

dM dM%εM! "βHMS&MεH

dH%εH! " dH%μH%!H! "βLMS&MεL

dL%εL! " dL%μL%!L! " $α

0

BBB@

1

CCCA: (49)

Solving for α, we have:

$ α3 % Aα% B # 0; (50)

where

A # βLMS&MεL

dL % εL! " dL % μL % γL! " βMLS

&LεM

dM dM % εM! "

%βHMS

&MεH

dH % εH! " dH % μH % γH! " βMHS

&HεM

dM dM % εM! "

(51)


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B #βHMS

&MεH

dH % εH! " dH % μH % γH! " βLHS

&HεL

dL % μL % γL! "

dL % εL! "βMLεM

dM dM % εM! " : (52)

Because A and B are positive, the equation α3 # Aα% B has a real solution (Figure A2).Eq. (50) has solutions

α1 #16!13 % 2A!$

13 (53)

α2 #16

$ 12% i

312

2

!

!13 % 2A $ 1

2$ i

312

2

!

!$13 (54)

α3 #16

$ 12$ i

312

2

!

!13 % 2A $ 1

2% i

312

2

!

!$13; (55)

where

! # 108B% 12 81B2 $ 12A3! "12: (56)

The value for R0 is the largest modulus of the eigenvalues αi (Greenhalgh, 1996).

Case 1. If 81B2 $ 12A3 > 0, with M # 16!

13 and N # 2A!$

13, then

α1 # M % N (57)

α2;3 # $ 12

M % N! " ' 312

2i M $ N! " (58)

and the modulus of each eigenvalue is

α1j j # M % N! "2! "1

2 (59)

!3 !2 !1 0 1 2 3 4 5!400

!200

0

200

400

600

800

!

y = !3

y = A! + B

y

Figure A2. Intersection of the line y # Aα% B with the cubic polynomial y # α3 when A and Bare positive.


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# M2 % 2MN % N2! "12 (60)

and

α2;3%% %% # 1

4M % N! "2 % 3

4M $ N! "2

& '12

(61)

# 12

4N2 $ 4MN % 4M2! "12 (62)

# M2 $ 2MN % N2! "12; (63)

where α1j j> α2;3%% %%. R 1! "

0 # α1j j, where R 1! "0 is the basic reproductive number if 81B2 $ 12A3 is

positive.Case 2. If 81B2 $ 12A3 < 0 then α1j j # α2;3

%% %%. R 2! "0 # α1j j, where R 2! "

0 is the basic repro-

ductive number if 81B2 $ 12A3 is negative. We use the fact that α1j j # α1!α1! "12 and polar

coordinates to find the modulus. Because r # 108B! "2 % 122j81B2 $ 12A3j! "1

2 andcos θ # 108B

r ,

α1 #16r13 exp

iθ3

& '% 2Ar$

13 exp $ iθ

3

& '(64)

!α1 #16r13 exp $ iθ

3

& '% 2Ar$

13 exp i

θ3

& '(65)

and so

R 2! "0 # α1j j (66)

# 136

r13 % A

3cos

2θ3% 4A2r$

23: (67)

We have

R0 # α1j j: (68)

A.4. Effects of reducing winter to an impulse

The seasonal changes occurring after an introduction of Rift Valley fever to North Americareduce the dynamics of the human, livestock, and mosquito populations in the winter months toa single impulse. We compare this seasonal-jumpmodel to a continuous ODEmodel for both thesummer and winter months with impulses only at the end of each season.

In a continuous model for the winter with SM # EM # IM # ΛM # 0 at the end of winter,the susceptible, exposed, and infected mosquitoes tend to zero exponentially fast. Figure A3shows the dynamics of the continuous winter season for the susceptible mosquito population.

At the end of summer, the susceptible mosquito population is at S$M. Throughout thewinter, the mosquito population decreases exponentially. At the end of the winter season,SendsM # S$M exp $rT! ", where T is the duration of winter. A proportion p3 of offspring of theinfected mosquitoes hatch at the end of winter, which increases the total population ofmosquitoes before the next summer season to S%M # S$M exp $rT! " % p3IM. Ifr3 # 1$ exp $rT! ", the two models give the same results for the mosquito population. Thesame results apply to the human and livestock populations.


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Winter Starts Winter Ends

Susceptible mosquito population (SM

)

p3

SM

SM+

SM

exp( rt)

SMend

Figure A3. Comparison between full-time model and seasonal-jump model. The curve SM #S$M exp $rt! " reflects the dynamics of the susceptible mosquito population during the wintermonths. At the end of the winter SendM

! ", the mosquito population increases by p3. S$M is the

value before the impulse for the seasonal-jump model and S%M is the value after the impulse.


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multiseason transmission for rift valley fever in north america · rift valley fever’s success at...

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