Mathematical model of Zika virus with vertical transmission

Accepted Manuscript Mathematical model of Zika virus with vertical transmission F.B. Agusto, S. Bewick, W.F. Fagan PII:

S2468-0427(16)30048-3

DOI:

10.1016/j.idm.2017.05.003

Reference:

IDM 27

To appear in:

Infectious Disease Modelling

Received Date: 28 December 2016 Revised Date:

19 May 2017

Accepted Date: 20 May 2017

Please cite this article as: Agusto F.B., Bewick S. & Fagan W.F., Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling (2017), doi: 10.1016/j.idm.2017.05.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Mathematical Model of Zika Virus with Vertical Transmission F.B. Agusto†∗, S. Bewick‡ and W.F. Fagan‡ †

Department of Ecology & Evolutionary Biology, University of Kansas, Lawrence KS, 66045 ‡

Department of Biology, University of Maryland, College Park, MD 20742

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Abstract

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May 20, 2017

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Zika is a flavivirus transmitted to humans through either the bites of infected Aedes mosquitoes or sexual transmission. Zika has been linked to congenital anomalies such as microcephaly. In this paper, we analyze a new system of ordinary differential equations which incorporates human vertical transmission of Zika virus, the birth of babies with microcephaly and asymptomatically infected individuals. The Zika model is locally and globally asymptotically stable when the basic reproduction number is less than unity. Our model shows that asymptomatic individuals amplify the disease burden in the community, and the most important parameters for ZIKV spread are the death rate of mosquitoes, the mosquito biting rate, the mosquito recruitment rate, and the transmission per contact to mosquitoes and to adult humans. Scenario exploration indicates that personal-protection is a more effective control strategy than mosquito-reduction strategy. It also shows that delaying conception reduces the number of microcephaly cases, although this does little to prevent Zika transmission in the broader community. However, by coupling aggressive vector control and personal protection use, it is possible to reduce both microcephaly and Zika transmission.

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2000 Mathematics Subject Classifications: 92B05, 93A30, 93C15. Keywords: Zika virus; vertical transmission; microcephaly; stability; control

1

Introduction

Zika virus (ZIKV) is a mosquito-borne disease transmitted to humans through the bites of infected Aedes mosquitoes, including Aedes aegypti, Aedes africanus, Aedes apicoargenteus, Aedes furcifer, Aedes hensilli, Aedes luteocephalus and Aedes vitattus. First identified in a rhesus macaque population in 1947 in the Zika forest of Uganda, ZIKV is from the Spondweni serocomplex of the Flaviviridae family of viruses. Historically, ZIKV was thought to cause mild symptoms in humans, including headaches, maculopapular rash, fever, malaise, conjunctivitis, and arthralgia, occurring three to twelve days after the bite from an infected mosquito. Recently, however, there have been reported increases in congenital anomalies (such as microcephaly), Guillain-Barre syndrome, and other neurological and autoimmune disorders in regions where ZIKV has been newly introduced ∗

Email: [email protected]

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ACCEPTED MANUSCRIPT [9, 54]. Many researchers believe that ZIKV is responsible for these increases, suggesting that ZIKV is a more serious disease than initially realized.

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In December 2015, the European Centre for Disease Prevention and Control issued a comprehensive update on the possible association between ZIKV, congenital microcephaly and GuillainBarre [21]. Most evidence, however, is correlative. In Brazil, for example, 2,782 cases of microcephaly were reported in the year following ZIKV introduction, as compared with 147 cases and 167 cases in the two years prior to ZIKV arrival [41]. Retrospective analysis of data from French Polynesia similarly uncovered an unusual number of babies born with neural defects during the height of the ZIKV outbreak [52]. Over this same period, French Polynesia also saw a spike in Guillain-Barre syndrome [22, 35], as well as increases in a range of other neurologic conditions including meningitis, meningoencephalitis, and myelitis [47]. More recently, a series of Latin American countries, including Brazil, Colombia, and Venezuela have observed similar upticks in the incidence of Guillain-Barre [56], consistent with the proposed relationship between this disorder and ZIKV infection.

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In addition to correlative support, several clinical and lab-based findings hint at potential mechanisms to explain the link between ZIKV and neural complications [31]. In 1952, for example, Dick et al. [15] demonstrated ZIKV tropism to the brain in intraperitoneally infected mice. Expanding on this finding, Bell, and colleagues [4] later showed that both neurons and glia could be infected by ZIKV. More recently, a number of studies, have demonstrated evidence of intrauterine infection with ZIKV [36], including infection of the fetal brain [27, 42]. This latter finding, in particular, provides a direct path from maternal ZIKV infection to microcephaly – a rare neurological condition in which an infant’s brain develops abnormally in the womb or does not grow as it should after birth [28]. Ultimately, microcephaly results in an infant’s head size being significantly smaller than the heads of other children of the same age and sex [28]. Although microcephaly can range from mild to severe, cases currently associated with the ZIKV outbreak in Brazil are notable for the level of damage observed in the brains of affected infants [13, 47]. Furthermore, congenital Zika usually come with a wide spectrum of clinical features [13].

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In this paper, we develop and analyze a mathematical model for ZIKV. Our focus is multifold. First, we consider overall ZIKV transmission in the adult population. Second, we consider ZIKV transmission to infants, either directly by mosquitoes or else prior to birth through vertical transmission from the mother. Infant ZIKV cases may be particularly severe because central nervous system (CNS) infections in young children can cause long-term damage to the developing brain [8]. Finally, we consider microcephaly rates, which we assume occur as a result of vertical transmission of ZIKV to the fetus during the early stages of pregnancy. The paper is organized as follows. The model is formulated in Section 2 and we investigate the theoretical properties of the Zika model with mother-to-child vertical transmission in Section 3. In Section 4, we assess the impact of the asymptomatic classes and identify key parameters with the most impact on disease burden in Section 5. We conduct numerical exploration of three control strategies in Section 6. The study results are discussed in Section 7.

2

Model Formulation

We model the transmission dynamics of ZIKV using a compartmental framework. We consider two human populations consisting of adults and newly born babies as well as the vector population. 2

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The population of newly born babies consists of susceptible (SB (t)), exposed (EB (t)), asymptomatic (AB (t)), symptomatic newly born without microcephaly, newly born with microcephaly (IBM (t)) and recovered newly born babies (RB (t)). The total population of adults, NW (t), at time t is split into mutually exclusive sub-populations of individuals who are susceptible (SW (t)), exposed (EW (t)), asymptomatic (AW (t)), symptomatic (IW (t)), adult with microcephaly (IW M (t)) and recovered adults (RW (t)). The population of the mosquitoes include the susceptible (SV (t)), exposed (EV (t)) and infected mosquitoes (IV (t)). The total population for each group is given as:

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NB (t) = SB (t) + EB (t) + AB (t) + IB (t) + IBM (t) + RB (t), NW (t) = SW (t) + EW (t) + AW (t) + IW (t) + IW M (t) + RW (t), NV (t) = SV (t) + EV (t) + IV (t).

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and total human population is NH (t) = NB + NW . Equations representing the mathematical model are given below. The flow diagram of the model is depicted in Figure 1, and the associated state variables and parameters are described in Table 1.

(2.1)

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SB′ (t) = πB − qA πB AW (t) − qI πB IW (t) − qR πB RW (t) −λB (IV , NB )SB (t) − (α + µB )SB (t) ′ EB (t) = λB (IV , NB )SB (t) − (α + σB + µB )EB (t) A′B (t) = qA πB AW (t) + (1 − p)σB EB (t) − (α + γB + µB )AB (t) IB′ (t) = qI πB IW (t) + pσB EB (t) − (α + γB + µB )IB (t) ′ IBM (t) = rqR πB RW (t) − (α + µB )IBM (t) ′ RB (t) = (1 − r)qR πB RW (t) + γB AB (t) + γB IB (t) − (α + µB )RB (t) ′ SW (t) = αSB (t) − λW (IV , NW )SW (t) − µW SW (t) ′ EW (t) = λW (IV , NW )SW (t) − (σW + µW )EW (t) A′W (t) = (1 − p)σW EW (t) − (γW + µW )AW (t) ′ IW (t) = pσW EW (t) − (γW + µW )IW (t) ′ IW M (t) = αIBM (t) − µW IW M (t) ′ RW (t) = αRB (t) + γW AW (t) + γW IW (t) − µW RW (t) SV′ (t) = πV − λV (AB , IB , AW , IW , NB , NW )SV (t) − µV SV (t) EV′ (t) = λV (AB , IB , AW , IW , NB , NW )SV (t) − (µV + σV )EV (t) IV′ (t) = σV EV (t) − µV IV (t).

where,

βW bV IV ηβB bV IV , λB (IV , NB ) = , NW NB   IW + ρW AW + η(IB + ρB AB ) λV (AB , IB , AW , IW , NB , NW ) = βV bV , NW + ηNB λW (IV , NW ) =

are the disease forces of infection rates, and all other parameters are as defined in Table 1. In particular, βW , βB and βV are the transmission probability per contact in adults, newly born babies and mosquitoes, bV is the mosquito biting rate, ρW and ρB are modification parameters modeling the infectivity of the asymptomatic babies and adults. The parameter η is a modification parameter that indicates that babies’ exposure rate is different from that of adults. For instance, 3

ACCEPTED MANUSCRIPT they may be protected from mosquito bites, making they less likely to get the infections, on the other hand, they may receive more mosquito bites if left unprotected; we assume that η > 0. We assume that the infection in the asymptomatic individuals might not be high enough to infect the susceptible mosquitoes or is the same level as for the infectious individuals, in which case the modification parameters are taken as 0 ≤ ρB , ρW ≤ 1.

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Zika virus is passed prenatally from a pregnant woman to her unborn fetus [32]. For example, during the 2015 Zika outbreak in Brazil, Zika virus RNA was found in the amniotic fluid of two women whose fetuses were determined via prenatal ultrasound to have microcephaly [44]. Depending on timing of infection in the womb, newborn babies can also be infected from birth [5]. Thus, we assume that some babies are born with infected with the virus. The parameters qA , qI , qR represent fractions of newly born babies who are infected due to vertical transmission. So that the fraction πB (1 − qA AW − qI IW − qR RW ) are babies born healthy by infected and recovered mothers and the remaining fraction qA πB AW , qI πB IW , qR πB RW are born infected.

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Despite the fact that there is sufficient evidence to conclude that intrauterine Zika virus infection is a cause of microcephaly [32], not all newly born babies are born with microcephaly, although they may have other congenital abnormalities [13]. We assume that some babies are born recovered from the virus. Thus, the parameter r correspond to the fraction of the qR πB RW recovered babies born by recovered mothers, while the remaining portion (1 − r)qR πB RW are newly born babies who have microcephaly. The parameter p represent the fraction of adults and newly born babies who are asymptomatic and the remaining fraction (1 − p) are adults and newly born babies who are infectious. The parameter α denotes the maturation rate. Microcephalic individuals experience profound developmental delay [10]; although their lifespan is not known, they live for a short period due to severe neurologic impairments (some have been known to live up to 9 years) [10]. As a result, we assume that microcephalic adults do not reproduce.

Figure 1: Flow diagram of the Zika transmission model.

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Description

SB (t), SW (t) EB (t), EW (t) AB (t), AW (t) IB (t), IW (t) IBM (t), IW M (t) RB (t), RW (t) SV (t) EV (t) IV (t)

Susceptible newly born babies and adults Exposed newly born babies and adults Asymptomatic newly born babies and adults Symptomatic newly born without microcephaly and adults Microcephalic newly born babies and adults Recovered newly born babies and adults Susceptible female mosquitoes Exposed female mosquitoes Infected female mosquitoes

Parameter

Description

πB p 1−p α r, qA , qI , qR 1−r η βW , βB ρW , ρB σW , σB γW , γB µW , µB πV βV bV σV µV

Birth rate newly born babies Fraction of adults and newly born babies who are asymptomatic Remaining fraction of adults and newly born babies who are infectious Maturation rate Fractions of newly born babies who are infected and have microcephaly Remaining fraction of newly born babies who have microcephaly Modification parameter Transmission probability per contact of adults and newly born babies Infectivity modification parameters in asymptomatic adults and newly born babies Progression rate of exposed adults and newly born babies Recovery rate of asymptomatic and symptomatic adults and newly born babies Natural death rate of adults and newly born babies Recruitment rate of mosquitoes Transmission probability per contact of susceptible mosquitoes Mosquito biting rate Progression rate of exposed mosquitoes Natural death rate of mosquitoes

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Variable

2.1

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Table 1: Description of the state variables and parameters of the Zika model (2.1).

Basic properties

We shall now explore the basic dynamical features of model (2.1). Since the model (2.1) describes both human and mosquito populations during a Zika epidemic, it will only be epidemiologically meaningful if all state variables are non-negative for t ≥ 0. That is, its solution with positive initial data will remain positive for all time (t > 0). Lemma 1. Let the initial data F (0) ≥ 0, where F (t) = (SB , EB , AB , IB , IBM , SW , EW , AW , IW , RW , IW M , RB , SV , EV , IV ). Then the solutions F (t) of model (2.1) are non-negative for all time t > 0. Furthermore lim sup NH (t) ≤ t→∞

πB , µH

lim sup NV (t) = t→∞

where µH = min{µB , µW }. 5

πV , µV

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The proof of Lemma 1 is given in Appendix A. Invariant regions

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Model (2.1) will be analyzed in a biologically-feasible region as follows. Consider the feasible region 3 Γ = ΓH × ΓV ⊂ R12 + × R+ with,

ΓV

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ΓH

 = SB (t), EB (t), AB (t), IB (t), IBM (t), RB (t), SW (t), EW (t), AW (t), IW (t), IW M (t), RW (t)  πB : NH (t) = , , µH   πV . = SV (t), EV (t), IV (t) : NV (t) = µV

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3 Lemma 2. The region = ΓH × ΓV ⊂ R12 + × R+ is positively invariant for the basic model (2.1) 15 with non-negative initial conditions in R+

The proof of Lemma 2 is given in Appendix B.

Analysis of the Model

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In the next section the conditions for the stability of the disease-free equilibrium of model (2.1) are explored.

Model (2.1) with endogenous reactivation and exogenous reinfection is now analyzed to gain insight into its dynamical features.

3.1

Local Stability of the Disease-Free Equilibrium

The disease free equilibrium (DFE) of model (2.1), which is obtained by setting the right hand sides of the model equations to zero is given by: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ E0 = (SB∗ , EB∗ , A∗B , IB∗ , IBM , RB , SW , EW , A∗W , IW , IW M , RW , SV , EV , IV )   πB απB πV = , 0, 0, 0, 0, 0, , 0, 0, 0, 0, 0, , 0, 0 . α + µB µW (α + µB ) µV

(3.1)

The local asymptotic stability of E0 can be established using the next generation operator method ∗ ∗ ∗ ∗ on the system (2.1). Taking the infected compartments (EB∗ , A∗B , IB∗ , IBM , EW , A∗W , IW , IM W, 6

ACCEPTED MANUSCRIPT EV∗ , IV∗ ) at the DFE and using the notation in [51], the Jacobian matrices F and V for the new infection terms and the remaining transfer terms are respectively given by,

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       V =       

0 0 0 0 0 0 0 0

ηρB βV bV SV∗ ∗ +ηS ∗ SW B

ηβV bV SV∗ ∗ +ηS ∗ SW B

0

0

k2 −(1 − p)σB −pσB 0 0 0 0 0 0 0

0 k3 0 0 0 0 0 0 0 0

0 0 k4 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

ρW βV bV SV∗ ∗ +ηS ∗ SW B

βV bV SV∗ ∗ +ηS ∗ SW B

0

0

0 0 0 0 0 0 k5 0 0 k8 0 −(1 − p)σW 0 −pσW −α 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

∗ ηβB bV SB ∗ NB

0 0 0

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0 0 0 0 0 0 0 0

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and

         F =        

0 0 0 0 0 0 0 0 0 0

0

−qA πB 0 0 0 k9 0 0 0 0

0 0 −qI πB 0 0 0 k10 0 0 0

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0 0 0 0 0 0 0 k12 0 0

∗ βW bV SW ∗ NW

0 0 0 0 0 0 0 0 k14 −σV

0 0 0 0 0

0 0 0 0 0 0 0 0 0 µV

                  

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       ,       

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where k1 = α + µB , k2 = α + σB + µB , k3 = α + γB + µB , k4 = α + γB + µB , k5 = α + µB , k6 = α + µB , k7 = µW , k8 = σW + µW , k9 = γW + µW , k10 = γW + µW , k11 = µW , k12 = µW , k13 = µV , k14 = µV + σV .

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It follows that the basic reproduction number of model (2.1), denoted by R0 , is given by; R0 = ρ(F V −1 ) =

RV (RW + RB )

(3.2)

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where

p

RV

=

RB =

RW

=

SV∗ βV bV σV , k14 µV η 2 βB bV σB [ρB (1 − p)k4 + pk3 ] ∗ + ηS ∗ ) k2 k3 k4 (SW B   (1 − p)(k3 ρW + ηρB qA πB ) p(k4 + ηqI πB ) βW bV σW + . ∗ + ηS ∗ ) k8 (SW k3 k9 k4 k10 B

The following result is established using Theorem 2 in [51]. Lemma 3. The DFE of model (2.1), given by E0 , is locally asymptotically stable (LAS) if R0 < 1, and unstable if R0 > 1. The epidemiological quantity, R0 gives the average number of ZIKV cases generated by a typical infected individual introduced into an entirely susceptible human population [3, 16, 23, 51]. Furthermore, the expression RB is the number of secondary infections in newly born babies by one 7

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introduced infectious mosquito, while the expression RW is the number of secondary infections in adults by one infectious mosquito. The expression RW consists of infections from newly born babies due to vertical transmission (mother-to-child infection) and infections due to horizontal transmissions from adults and infections from infants that have matured into adults. Lastly, the expression RV is the number of secondary infections in mosquitoes resulting from a newly introduced infectious adult woman and newly born baby. ZIKV can be adequately controlled in the community with adults and newly born babies if the threshold quantity (R0 ) can be reduced to (and maintained at) a value less than unity (i.e. R0 < 1).

3.2

Global Asymptotic Stability of the Disease-Free Equilibrium

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For Zika elimination to be independent of the initial sizes of the sub-populations of the model, the global asymptotic stability of the DFE must be established. This is what we consider next. Consider the feasible region

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∗ Γ1 = {X ∈ Γ : SB ≤ SB∗ , SW ≤ SW , SV ≤ SV∗ },

where , X = RW , SB , EB , AB , IB , IBM , SW , EW , AW , IW , IW M , RB , SV , EV , IV . Lemma 4. The region Γ1 is positively invariant for model (2.1). The proof of Lemma 4 is given in Appendix C.

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Theorem 1. The DFE, E0 , of model (2.1), is globally asymptotically stable (GAS) in Γ1 whenever R0 ≤ 1.

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The proof of Theorem 1 is given in Appendix D. The above result shows that ZIKV will be eliminated from the community if the threshold quantity R0 can be brought to a value less than unity.

Assessing the Impact of the Asymptomatic Classes

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In this section we shall explore the impact of the asymptomatically infected individuals since an estimated 80% of Zika infections do not show symptoms [12, 18, 37], and when infections lead to illness, the symptoms are usually mild. Note, that we have elected to work with the square of the reproduction number, so our results remain tractable. The conclusion is not altered if the actual expression for the reproduction number is used. Thus, differentiating the square of the basic reproduction number, R20 , given in (3.2), partially with respect to the asymptomatic modification parameters ρB and ρW , gives σV βV b2V SV∗ η(1 − p)(ησB k8 k9 + qA πB σW βW k2 ) ∂R20 = > 0, ∗ ∂ρB k14 µV k2 k3 k8 k9 (SW + ηSB∗ ) and

∂R20 σV βV b2V SV∗ (1 − p)βW σW = > 0. ∂ρW k14 µV k8 k9 (SW + ηSB∗ ) 8

ACCEPTED MANUSCRIPT This, implies, the square of basic reproduction number, R20 , is an increasing function of the parameters ρB and ρW . Thus, the disease burden in the community will increase as the infectivity of the asymptomatic individuals increases.

lim

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Furthermore, if we take the limit of R20 , as ρB → 1 and ρW → 1 (meaning, that infectivity of the asymptomatic individuals is the same as that of the infectious individuals), we have R0 = σV βV b2V SV∗ {σW βW k2 k3 k9 (k4 + ηqI πB )p

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+σW βW k2 k4 k10 (k3 + ηqA πB )(1 − p) +η 2 σB βB k8 k9 k10 [pk3 + (1 − p)k4 ]} ∗ /[k14 µV k2 k3 k4 k8 k9 k10 (SW + ηSB∗ )] > 0.

Thus, as the infectivity of the asymptomatic individuals increases, the disease burden increases, thereby increasing the number of Zika infected individuals in the community.

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Figure 2 shows a contour plot of the reproduction number R0 , as a function of the asymptomatic modification parameters ρB and ρW . This figure indicates that Zika burden in the community gets amplified as the level of infectivity of the asymptomatic individuals increases toward that of the infectious individuals. 1

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Figure 2: Contour plot of the reproduction number (R0 ) of the Zika model (2.1) as a function of the asymptomatic modification parameters ρB and ρW . Parameter values used are as given in Table 3. The impact of the asymptomatic classes is further assessed by simulating the model (2.1) using various values for the modification parameters ρB and ρW and the same parameters in Table 1. The results obtained, depicted in Figure 3, show, as expected, that the cumulative number of new cases generated by infectious mosquitoes to susceptible humans increases with increasing values of ρB and ρW (see Figure 3(a)). Similarly, the cumulative number of new cases generated by infectious humans to susceptible mosquitoes increases with increasing values of the modification parameters (see Figure 3(b)). Notice that 3(a) and 3(b) differ in the direction of spread, since one is infection from mosquito to host and the other is from host to mosquito. 9

14000

ρB,ρW = 0.0

12000

ρ ,ρ B

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= 0.25

ρB,ρW = 0.50

10000

ρB,ρW = 0.75 ρ ,ρ

8000

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= 1.0

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150 Time (days)

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14000

12000

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ρB,ρW = 0.0 ρ ,ρ B

8000

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= 0.75

ρ ,ρ

= 1.0

B

6000

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4000

2000

0 0

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ρB,ρW = 0.50

50

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150 Time (days)

200

W W

250

(b)

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Nnumber of new cases generated by infectious humans

Number of new cases generated by mosquitoes

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Figure 3: Simulations of the Zika model (2.1) with different values of ρB = ρW = 0, 0.25, 0.5, 0.75, 1.0

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(a). Cumulative number of new cases generated by infectious mosquitoes transmitting to susceptible humans. (b). Cumulative number of new cases generated by infectious humans transmitting to susceptible mosquitoes. Parameter values used are as given in Table 3.

Next, we evaluate the contributions of the asymptomatic and infectious individuals to the disease burden in the community. We observed from Table 2, that the percentage contributed by asymptomatic humans goes up as pW and pB go up. 0.25 20.5% 79.5%

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ρW , ρB Asymptomatic Infectious

0.50 34.1% 65.9%

0.75 43.7% 56.3%

1.0 50.9% 49.1%

Sensitivity Analysis

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Table 2: Contribution of the asymptomatic and infectious individuals to the mosquitoes’ cumulative infections with various values of ρB and ρW .

The outputs of deterministic models are governed by the model input parameters, which may exhibit some uncertainty in their determination or selection. We employed a global sensitivity analysis to assess the impact of uncertainty and the sensitivity of the outcomes of the numerical simulations to variations in each parameter of the model (2.1) using Latin Hypercube Sampling (LHS) and partial rank correlation coefficients (PRCC). LHS is a stratified sampling without replacement technique which allows for an efficient analysis of parameter variations across simultaneous uncertainty ranges in each parameter [7, 26, 29, 43]. PRCC measures the strength of the relationship between the model outcome and the parameters, stating the degree of the effect that each parameter has on the outcome [7, 26, 29, 43]. Thus, sensitivity analysis determines the parameters with the most significant impact on the outcome of the numerical simulations of the model [7, 26, 30]. To generate the LHS matrices, we assume that all the model parameters are uniformly distributed. Then a total of 1,000 simulations of the models per LHS run were carried out, using the ranges and baseline values tabulated in Table 3 (with the basic reproduction number, R0 , as the response function). From Figure 4 it follows that the parameters that have 10

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the most influence on Zika transmission dynamics are the death rate of the mosquitoes (µV ), the mosquito biting rate (bV ), mosquito recruitment rate (πV ), the transmission probability per contact to mosquitoes (βV ) and to adult humans (βW ), and the adult recovery rate (γW ). Identification of these key parameters is important to the formulation of effective control strategies for combating the spread of disease. In particular, the results of this sensitivity analysis suggest that a strategy that reduces the transmission probability per contact to mosquitoes or to adult humans (i.e., reduces βV or βW respectively), will adequately reduce the spread of ZIKV in the community. Furthermore, a strategy that reduces the mosquito recruitment rate or the mosquito biting rate (i.e., reduces πV or bV , respectively) or else increases the death rate of the mosquitoes (i.e., increases µV ) will be effective in curtailing the spread of ZIKV in the community.

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Figure 4: PRCC values for the Zika model (2.1), using as response functions the reproduction number

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R0 . Parameter values (baseline) and ranges used are given in Table 3.

Assessment of Control Strategies

Motivated by the sensitivity analysis in the previous section, we now explore some of the key model parameters to determine their effectiveness as targets for control strategies. In particular, we consider mosquito recruitment rate (πV ), which can be modified with larvicides or through effective management of mosquito breeding sites. We also consider the mosquito death rate, (µV ), which can be modified with adulticides, and the mosquito biting rates, (bV ), which can be modified using repellants or behavioral avoidance (e.g, remaining in buildings with screened windows and air-conditioning). We then consider a combined strategy where all three management strategies are employed simultaneously. Finally, we consider delayed pregnancy, because this is a recommendation put forward by several Latin American and Caribbean governments as a means of reducing microcephaly.

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Range

1 16×365

1 18×365

1 7.5 1 8.5

1 70×365 1 15×365

0.33 1 7.5 1 8.5

1 18.60×365

500 0.33 0.5 1 3.5 1 21

0.1 - 1 0.1 - 1 0.10 - 0.75 1 - 31 12 1 1 14 3 1 1 - 68×365 76×365 1 1 - 12×365 18×365 0.001 - 0.54 1 - 31 12 1 - 31 14 1 1 - 22.32×365 14.88×365 50 - 5000 0.10 - 0.75 0.33 - 1.0 1 - 12 6 1 - 81 42

References Assumed Assumed Assumed [25, 34, 38] [6] [6] [25] [53] [20, 19, 25, 39, 50] [6] [6] [55] [6] [25, 34, 38] [40, 48, 25] [17, 19, 33, 45] [46, 48, 49]

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Values

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Parameter α ρB , ρW p, r, qA , qI , qR βW σW γW µW πB βB σB γB µB πV βV bV σV µV

Table 3: Parameter values of model (2.1)

Epidemiological Consequences of Mosquito-Reduction Strategies

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6.1

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For our analysis, moderate control levels correspond to the baseline parameter values used in our sensitivity analysis (Table 3, column 2) whereas low and high control levels correspond to the extreme parameter values (Table 3, column 3). The parameter values and initial conditions used in these simulations are theoretical in the sense that they are similar to comparable parameters for other mosquito-transmitted diseases, but are not specific to ZIKV, which is only poorly studied. The goal is thus to illustrate the control strategies proposed in this paper for relatively broad, generic parameter ranges. These predictions should then be retested as more information becomes available that is specific to ZIKV.

We consider the following control levels for this strategy: 1. Low mosquito-reduction strategy: πV = 500/day, µV =

1 /day; 21

2. Moderate mosquito-reduction strategy: πV = 250/day, µV =

1 /day; 14

3. High mosquito-reduction strategy: πV = 125/day, µV = 81 /day. This strategy combines the larviciding and adulticiding strategies discussed in Appendices E and F.

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Figure 5: Simulation of the Zika model (2.1) for various control levels of the mosquito reduction control strategy. (a). The cumulative number of new Zika cases in adults. (b). The cumulative number of new Zika cases in newly born babies. (c). The cumulative number of new cases of newly born babies with microcephaly. Parameter values used are as given in Table 3.

Figure 5 shows for each of the three control levels, the cumulative number of new infections in adults, new infections in newly born babies and newly born babies with microcephaly. Comparing the three control levels in Table 4 shows a decrease in the cumulative number of new cases with increasing control. Humans Adults newly born babies newly born with microcephaly

Low-Control 2.1 × 106 7.7 × 105 10.0 × 103

Moderate-Control 5.8 × 105 1.4 × 105 3.9 × 103

High-Control 1.2 × 105 2.8 × 104 2.0 × 103

Table 4: Simulation results of the Zika model (2.1) using the mosquito reduction control strategy.

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6.2

Epidemiological Consequences of the Personal Protection Strategy

Personal protection reduces mosquito biting rates (bV ). Similar to mosquito control, we consider three different levels of protection:

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1. Low-effectiveness personal-protection strategy: bV = 0.5/day; 2. Moderate-effectiveness personal-protection strategy: bV = 0.250/day; 3. High-effectiveness personal-protection strategy: bV = 0.125/day.

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Figure 6: Simulation of the Zika model (2.1) for various control levels of the personal-protection strategy. (a). The cumulative number of new Zika cases in adults. (b). The cumulative number of new Zika cases in newly born babies. (c). The cumulative number of new cases of newly born babies with microcephaly. Parameter values used are as given in Table 3.

Figure 6 shows the cumulative number of new infections in adults, new infections in newly born babies and newly born babies with microcephaly for each of the three protection levels. The high 14

ACCEPTED MANUSCRIPT effectiveness personal-protection strategy lead to a considerable reduction in the number of new cases compared to the moderate-effectiveness level (see Table 5) at the same time period. The low-effectiveness level performed the poorest producing the most number of new cases. Low Control 2.1 × 106 7.7 × 105 10.0 × 103

Moderate Control 5.4 × 105 1.2 × 105 3.4 × 103

High Control 9.5 × 104 2.1 × 104 1.9 × 103

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Humans Adults newly born babies newly born with microcephaly

6.3

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Table 5: Simulation results of the cumulative number of new cases for the Zika model (2.1) using the personalprotection strategy.

Combined Mosquito Control and Personal Protection Strategy

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The combined strategy (where both the mosquito reduction and personal protection strategies are implemented simultaneously) was assessed for the following three control levels: 1. Low-control strategy: πV = 500/day, µV =

1 /day, 21

2. Moderate-control strategy: πV = 250/day, µV =

bV = 0.5/day;

1 /day, bV 14

= 0.25/day;

3. High-control strategy: πV = 125/day, µV = 81 /day, bV = 0.125/day.

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Figure 7 shows the cumulative number of new infections in adults, new infections in newly born babies, and babies with microcephaly for each control strategy. A comparison of the three control levels in Table 6 shows that higher levels of combined control are more effective for preventing new ZIKV cases. Low Control 2.1 × 106 7.7 × 105 10.0 × 103

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Humans Adults newly born babies newly born with microcephaly

Moderate Control 1.2 × 105 2.6 × 104 2.0 × 103

High Control 2.2 × 104 4.9 × 103 1.6 × 103

Table 6: Simulation results of the cumulative number of new cases for the Zika model (2.1) using the combined control strategy.

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Figure 7: Simulation of the Zika model (2.1) for various control levels of the combined control strategy.

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(a). The cumulative number of new Zika cases in adults. (b). The cumulative number of new Zika cases in newly born babies. (c). The cumulative number of new cases of newly born babies with microcephaly. Parameter values used are as given in Table 3.

A comparison across control strategies (larviciding, adulticiding, mosquito-reduction, personalprotection, and the combined strategy) in each group (see Table 7) shows, as expected, that the combined strategy is more effective than the other strategies implemented separately. Indeed, combining strategies results in anywhere from a 43% reduction to a 94% reduction as compared to single control strategies. With respect to single control strategies, personal protection is more effective than mosquito-reduction for reducing ZIKV and also for preventing microcephaly in newborns.

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Humans Adults newly born babies newly born with microcephaly

Larviciding Control 7.5 × 105 1.8 × 105 4.4 × 103

Adulticiding Control 2.6 × 105 5.9 × 104 2.7 × 103

Mosquito Reduction 1.2 × 105 2.6 × 104 2.0 × 103

Personal Protection 9.5 × 104 2.1 × 104 1.9 × 103

Combined Control 2.2 × 104 4.9 × 103 1.6 × 103

6.4

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Table 7: Comparison of the cumulative number of new cases for the high-control levels of the various control strategies for the Zika model (2.1)

Delayed Pregnancy

1. No women delay : πB =

1 15×365

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In light of the warnings issued by the Brazilian, Colombian, El Salvadorian, and Jamaican governments for reproductive women to delay conceiving [2, 14], we explore the impact that this will have on ZIKV transmission and the number of babies born with microcephaly. To consider delayed pregnancy, we adjusted the human birth rate πB . As above, we consider three levels of delayed pregnancy: day−1 ;

1 )/2 day−1 ; 2. Some women delay: πB = ( 15×365

1 )/10 day−1 . 3. Many women delay: πB = ( 15×365

EP

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As expected, for all three scenarios, we observed a negligible difference in the cumulative number of new cases of ZIKV infections among adults and newly born babies. Indeed, there was even a small increase in the number of infected infants when pregnancy was delayed. However, there was a significant impact on the cumulative number of babies born with microcephaly, with the highly delayed rate producing the least number of babies with microcephaly (see Figure 8). This is further demonstrated in Table 8, where we compared the three delayed conception rates levels.

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Humans Adults newly born babies newly born with microcephaly

None-delayed 2.1 × 106 7.7 × 105 10.0 × 103

Some-delayed 2.1 × 106 7.8 × 105 5.0 × 103

Many-delayed 2.1 × 106 7.8 × 105 499

Table 8: Simulation results of the cumulative number of new cases for the Zika model (2.1) with various rates of delaying conception.

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Figure 8: Simulation of the Zika model (2.1) showing the cumulative number of new cases in newly born babies with microcephaly for various rates of delaying conception. Parameter values used are as given in Table 3.

Humans

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As a final scenario, we implement delayed conception simultaneously with the combined mosquito control and personal protection strategies (see Section 6.3). Again, we consider low, moderate and high levels of mosquito control, personal-protection, and pregnancy delays. Unlike the findings with delayed pregnancy alone (see above), delayed pregnancy in combination with other control strategies result in an appreciable benefit on ZIKV transmission in adult and new-born populations (see Table 9). However, when large numbers of women delay conception, and this is combined with high levels of mosquito control and personal protection, there is a dramatic reduction in babies born with microcephaly (see Table 10). Thus, delayed pregnancy, particularly when combined with mosquito control and personal protection, appears to be beneficial for reducing microcephaly rates in regions with ongoing ZIKV outbreaks.

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Adults newly born babies newly born with microcephaly

None-delayed & Low-Control 2.1 × 106 7.7 × 105 10.0 × 103

Some-delayed & Moderate-Control 1.2 × 105 2.6 × 104 1.0 × 103

Many-delayed & High-Control 2.2 × 104 4.8 × 103 78.2

Table 9: Simulation results of the cumulative number of new cases for the Zika model (2.1) using various rates of delaying conception, combined mosquito control and personal protection strategies.

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Figure 9: Simulation of the Zika model (2.1) showing the cumulative number of new cases in newly born

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babies with microcephaly for various levels of delayed pregnancy. Parameter values used are as given in Table 3.

Humans

Adults newly born babies newly born with microcephaly

Combined Control 2.2 × 104 4.9 × 103 1.6 × 103

Delayed Pregnancy 2.1 × 106 7.8 × 105 500

Delayed Pregnancy + Combined Control 2.2 × 104 4.8 × 103 78.2

Table 10: Comparison of the combined control strategies (involving mosquito reduction and personal protection), and delayed pregnancy for the Zika model (2.1).

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7

Discussion and Conclusion

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In this paper, we develop a new deterministic model to study the transmission dynamics of Zika virus. Our model incorporates mother-to-child transmission as well as the development of microcephaly in newly born babies. The analysis shows that the disease-free equilibrium of the model is locally and globally asymptotically stable whenever the associated reproduction number (R0 ), is less than unity and unstable otherwise. Sensitivity analysis further identifies parameters with the strongest impact on model outcome (i.e., the basic reproduction number). These are the mosquito biting rate, the transmission probability per contact to mosquitoes and to human adults, the mosquito recruitment rate, and the mosquito death rate. Identification of these key parameters is vital to the formulation of effective ZIKV control strategies.

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Based on our analysis, we consider various control strategies aimed at reducing mosquito biting rates, mosquito recruitment rates, and mosquito death rates to examine if these strategies will be effective in curtailing ZIKV spread in the community. We lack methods for reducing viral transmission probabilities; however, these would also be effective targets for control if management strategies existed. Specifically, we implement a mosquito-reduction strategy, a personal-protection strategy, and a combined strategy each with three different control levels (low, moderate, high). Our results show that the cumulative number of new ZIKV cases generally decreases with increasing control as does the number of cases of microcephaly. As expected, the combined strategy is most effective across the board; this is followed by the personal protection strategy.

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Some countries in the Western Hemisphere currently affected by ZIKV (including Brazil, El Salvador, Colombia and Jamaica [2, 14]) have issued warnings against women of reproductive age becoming pregnant. Using our model to numerically explore the effect of delayed pregnancy on disease transmission and microcephaly, we find that this strategy is highly effective for reducing the number of microcephaly cases, but does not impact levels of ZIKV transmission among either infants or adults. Coupling delayed pregnancy with mosquito control and personal protection, however, results in a considerable reduction in both ZIKV transmission and microcephaly. Thus, it appears that attacking ZIKV from all fronts, including aggressive mosquito control, strong adherence to repellant use, and delayed pregnancy, provides the best solution for ZIKV management, at least over the near term. The issue of near-term versus long-term ZIKV management strategies is an important point that warrants further discussion. The model presented in the current paper is a highly detailed study of short-term ZIKV transmission. In particular, we restrict our analysis to a 9-month window beginning at the start of a ZIKV outbreak. Consequently, any babies born from women who are or who have ever been infected by ZIKV are at high-risk for microcephaly during this window. Over longer periods, however, ZIKV infection could occur prior to pregnancy. Consequently, a history of infection (i.e., recovery adults) in mothers would not necessarily indicate high-risk pregnancy. Quite the opposite - women who acquire infection before becoming pregnant may actually be protected during pregnancy. Thus, long-term predictions may differ substantially from predictions of the current model, and this is something that we explore in a separate paper [6]. In this paper, we formulated and analyzed a system of ordinary differential equations for transmission dynamics of ZIKV. Some of our theoretical and epidemiological findings are summarized below: 20

ACCEPTED MANUSCRIPT (i). Model (2.1) is locally and globally asymptotically stable when R0 < 1 and unstable when R0 > 1; (ii). Disease burden in the community increases infectivity of the asymptomatic individuals. Similarly, the cumulative infections to mosquitoes from the asymptomatic increases as their infectivity level increases.

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(iii). Sensitivity analysis shows that the most important parameters for ZIKV spread are the death rate of the mosquitoes (µV ), the mosquito biting rate (bV ), mosquito recruitment rate (πV ), the transmission probability per contact to mosquitoes and to adult humans (βV and βW respectively) and the adult recovery rate (γW );

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(iv). Numerical simulations using mosquito control indicate that personal protection is a better and more effective strategy than the mosquito-reduction in reducing the disease burden in the population. As expected, a combined strategy is the most effective for reducing the ZIKV disease burden in the community.

Acknowledgments

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(v). Additional numerical simulations suggest that delaying conception reduces the number of cases of microcephaly, although it does little to prevent ZIKV transmission in the broader community. Coupled with aggressive mosquito control level and personal-protection; however, it is possible to both reduce microcephaly and prevent ZIKV transmission.

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References

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This research was supported by the Strategic Environmental Research and Development Program under grant RC-2639. One of the authors, FBA, had additional support from University of Kansas under grant NFGRF-2302100 and WFF, had further support from the US National Science Foundation under grant DMS-1225917.

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[42] E.J. Rubin, M.F. Greene, and L.R. Baden. Zika virus and microcephaly. New England Journal of Medicine, 374(10):984–985, 2016.

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[43] M.A. Sanchez and S.M. Blower. Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example. American Journal of Epidemiology, 145(12):1127–1137, 1997. [44] L. Schuler-Faccini, E.M. Ribeiro, I.M.L. Feitosa, D.D.G. Horovitz, D.P. Cavalcanti, A. Pessoa, M.J.R. Doriqui, J.I. Neri, J.M. de Pina Neto, H.Y.C. Wanderley, M.Cernach, A.S. El-Husny, M.V.S. Pone, C.L.C. Serao, and M.T.V. Sanseverino. Possible association between zika virus infection and microcephaly–brazil, 2015. MMWR. Morbidity and mortality weekly report, 65, 2016. [45] M.R. Sebastian, R. Lodha, and S.K. Kabra. Chikungunya infection in children. Indian Journal of Pediatrics, 76(2):185–189, 2009. [46] P.M. Sheppard, W.M. Macdonald, R.J. Tonn, and Grabs. B. The dynamics of an adult population of aedes aegypti in relation to dengue haemorrhagic fever in bangkok. Journal of Animal Ecology, 38:661–701, 1969. [47] J. Talan. Epidemiologists are tracking possible links between zika virus, microcephaly, and guillain–barr´e syndrome. Neurology Today, 16(4):1–18, 2016. 24

ACCEPTED MANUSCRIPT [48] M. Trpis and W. Haussermann. Dispersal and other population parameters of aedes aegypti in an african village and their possible significance in epidemiology of vector-borne diseases. American Journal of Tropical Medicine and Hygiene, 35:1263–1279, 1986.

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[49] M. Trpis, W. Haussermann, and G.B. Craig. Estimates of population size, dispersal, and longevity of domestic aedes aegypti by mark-release-recapture in the village of shauri moyo in eastern kenya. Journal of Medical Entomology, 32:27–33, 1995. [50] M.J. Turell, J.R. Beaman, and R.F. Tammariello. Susceptibility of selected strains of aedes aegypti and aedes albopictus (diptera: Culicidae) to chikungunya virus. Journal of Medical Entomology, 29(1):49–53, 1992.

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[51] P. van den Driessche and J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 2002.

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[52] G. Vogel. A race to explain brazil’s spike in birth defects. Science, 351(6269):110–111., 2016. [53] Wikipedia. List of sovereign states and dependent territories by birth rate. 2015. [54] World Health Organization. Neurological syndrome, congenital malformations, and zika virus infection. implications for public health in the americas. Epidemiological Alert. http: // www. paho. org/ hq/ index. php? option= com_ docman& task= doc_ view& Itemid= 270& gid= 32405& lang= en ., 2015.

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[55] World Health Organization. Microcephaly - brazil: Disease outbreak news. http: // www. who. int/ csr/ don/ archive/ disease/ microcephaly/ en/ , 2016. [56] World Health Organization. Zika situation report. http: // www. who. int/ emergencies/ zika-virus/ situation-report/ who-zika-report-12-02-2016. pdf? ua= 1 , 2016.

Appendix A: Proof of Lemma 1

EP

A

AC C

Lemma 1: Let the initial data F (0) ≥ 0, where F (t) = (SB , EB , AB , IB , IBM , SW , EW , AW , IW , RW , IW M , RB , SV , EV , IV ). Then the solutions F (t) of model (2.1) are non-negative for all time t > 0. Furthermore lim sup NH (t) = t→∞

πB , µH

lim sup NW (t) = t→∞

πV , µV

where µH = min{µB , µW }. with, NB (t) = SB (t) + EB (t) + AB (t) + IB (t) + IBM (t) + RB (t), NW (t) = SW (t) + EW (t) + AW (t) + IW (t) + IW M (t) + RW (t), NV (t) = SV (t) + EV (t) + IV (t).

25

ACCEPTED MANUSCRIPT Proof. Let t1 = sup{t > 0 : F (t) > 0}. Thus, t1 > 0. Then it follows from the first equation of the Zika model (2.1) that

which can be re-written as  Z SB (t)exp

tf

λB (IV (u), NH (u))du + (α + µB )t

0



Z

=

tf 0



πB − qA πB AW (u)

 + −qI πB IW (u) − qR πB RW (u)  Z tf λB (IV (u), NH (u))du + (α + µB )t . × exp

SC

d dt

RI PT

dSB = πB [1 − qA AW (t) − qI IW (t) − qR RW (t)] − λB (IV , NB )SB (t) − (α + µB )SB (t). dt

0

SB (t1 )exp

Z

M AN U

So that,



t1

λB (IV (u), NH (u))du + µW t − SB (0) =

Hence,

0



πB − qA πB AW (t)

 −qI πB IW (t) − qR πB RW (t) Z tf  × exp λB (IV (u), NH (u))du + (α + µB )t . 0

  Z t1 λB (IV (u), NH (u))du − (α + µB )t1 = SB (0)exp − 0  Z tf  πB − qA πB AW (u) − qI πB IW (u) − qR πB RW (u) = 0   Z t1 λB (IV (u), NH (u))du − (α + µB )t1 . ×exp −

AC C

EP

SB (t1 )

tf

TE D

0

Z

0

> 0.

It can similarly be shown that F > 0 for all t > 0. For the second part of the proof, it should be noted that 0 < SB (t) ≤ NH (t), 0 < EB (t) ≤ NH (t), 0 < AB (t) ≤ NH (t), 0 < IB (t) ≤ NH (t), 0 < IBM (t) ≤ NH (t), 0 < RB (t) ≤ NH (t), 0 < SW (t) ≤ NH (t), 0 < EW (t) ≤ NH (t), 0 < AW (t) ≤ NH (t), 0 < IW (t) ≤ NH (t), 0 < IW M (t) ≤ NH (t), 0 < RW (t) ≤ NH (t), 0 < SV (t) ≤ NV (t), 0 < EV (t) ≤ NV (t), 0 < IV (t) ≤ NV (t). Adding the components of the model (2.1) gives dNH (t) = πB − µB NB (t) − µW NW (t) dt dNV (t) = πV − µV NV (t). dt

26

ACCEPTED MANUSCRIPT Thus, dNH (t) = πB − µH NH (t), dt dNV (t) = πV − µV NV (t). dt

RI PT

where µH = min{µB , µW }. Hence,

πB , t→∞ µH t→∞ πV = lim inf NV (t) ≤ lim sup NV (t) = , t→∞ µV t→∞

= lim inf NH (t) = lim sup NH (t) ≤

SC

πB µH πV µV

B

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as required.

Appendix B: Proof of Lemma 2

3 Lemma 2: The region = ΓH × ΓV ⊂ R12 + × R+ is positively invariant for the basic model (2.1) with non-negative initial conditions in R15 +.

TE D

Proof. The following steps are followed to establish the positive invariance of Γ (i.e., solutions in Γ remain in Γ for all t > 0). The rate of change of the total population is obtained by adding the components of the model (2.1) to give

Thus,

AC C

EP

dNH (t) = πB − µB NB (t) − µW NW (t) dt dNV (t) = πV − µV NV (t). dt

dNH (t) = πB − µH NH (t), dt dNV (t) = πV − µV NV (t). dt

where µH = min{µB , µW }. A standard comparison theorem [24] can then be used to show that NH (t) ≤ NH (0)e−µH t + µπHB (1 − e−µH t ), in particular, NH (t) ≤ µπHB , if NH (0) ≤ µπBB . And lastly, NV (t) ≤ NV (0)e−µV t + µπVV (1 − e−µV t ), in particular, NV (t) ≤ µπVV , if NV (0) ≤ µπVV . Thus, the region Γ is positively invariant. Hence, it is sufficient to consider the dynamics of the flow generated by (2.1) in Γ. In this region, the model is epidemiologically and mathematically well-posed [23]. Thus, every solution of the model (2.1) with initial conditions in Γ remains in Γ for all t > 0. Therefore, the Γ-limit sets of the system (2.1) are contained in Γ. This result is summarized below. 27

ACCEPTED MANUSCRIPT

C

Appendix C: Proof of Lemma 4

Lemma 4: The region Γ1 is positively invariant for model (2.1). Proof. It follows from the first equation of Zika model (2.1), (where SB∗ =

πB ), so that α + µB

RI PT

ηβB bV SB IV dSB (t) = πB − qA πB AW (t) − qI πB IW (t) − qR πB RW (t) − − (α + µB )SB (t) dt NH ≤ πB − (α + µB )SB (t) πB ≤ (α + µB ) − SB (t) (α + µB )



SC



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= (α + µB )[SB∗ − SB (t)]. Thus,

(C-1)

SB (t) ≤ SB∗ − [SB∗ − SB (0)]e−(α+µB )t . Thus, if SB (0) ≤ SB∗ for all t ≥ 0, then SB (t) ≤ SB∗ for all t ≥ 0. ∗ Similarly, it follows from the sixth equation of the Zika model (2.1) (where SW =

TE D

that

απB ), (α + µB )µW

dSW (t) βW bV SW IV = αSB − − µW SW (t), dt NH

AC C

EP

≤ αSB − µW SW (t), ≤ µW



 αSB∗ − SW (t) , µW

= µW



απB − SW (t) µW (α + µB )

(C-2) 

∗ = µW [SW − SW (t)].

Hence,

∗ ∗ SW (t) ≤ SW − [SW − SW (0)]e−µW t . ∗ ∗ Thus, if SW (0) ≤ SW for all t ≥ 0, then SW (t) ≤ SW for all t ≥ 0.

Finally, it follows from the twelfth equation of the Zika model (2.1), that

28

ACCEPTED MANUSCRIPT   dSV (t) AW + IW + η(AB + IB ) SM − µM SM (t), = πV − βV bV dt NH ≤ πV − µV SV (t), (C-3)

 πV − SV (t) , µV

RI PT

≤ µV



= µV [SV∗ − SV (t)]. Thus,

∗ Hence, if NM =

πV µV

SC

SV (t) ≤ SV∗ − [SV∗ − SV (0)]e−µV t .

and SV (0) ≤ SV∗ for all t ≥ 0, then SV (t) ≤ SV∗ for all t ≥ 0.

D

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Thus, in summary, it has been shown that the region Γ1 is positively invariant and attracts all solutions in R14 + for the Zika model (2.1).

Appendix D: Proof of Theorem 1

Theorem 1: The DFE, E0 , of model (2.1), is globally asymptotically stable (GAS) in Γ1 whenever R0 ≤ 1.

TE D

Proof. To prove the global stability of the disease-free equilibrium, we will following the approach in [1]. Let X = (SW , RW , SB , RB , SV ) and Z = (EB , AB , IB , IBM , EW , AW , IW , IW M , EV , IV ) and group the system (2.1) into

AC C

EP

dX dt

= F (X, 0)

(D-1)

dZ = G(X, Z), dt

where F (X, 0) is the right hand side of S˙W , R˙W , S˙B , R˙B and S˙V with EB = AB = IB = IBM = EW = AW = IW = IW M = EV = IV = 0 and G(X, Z) is the right hand side of ˙ , E˙B , A˙B , I˙B , IBM ˙ , E˙V and I˙V . E˙W , A˙W , IW Next, consider the reduced system:

dX dt

= F (X, 0) given as:

29

ACCEPTED MANUSCRIPT

= πB − qR πB RW (t) − (α + µB )SB B(t)

dRB dt

= (1 − r)qR πB RW (t) − (α + µB )RB (t)

dSW dt

= αSB (t) − µW SW (t)

dRW dt

= αRB (t) − µW RW (t)

dSV dt

RI PT

dSB dt

Let

SC

= πV − µV SV (t).

(D-2)



 πB απB πV X = = , 0, . , 0, α + µB µW (α + µB ) µV be an equilibrium of the reduced system (D-2), we show that X ∗ is a globally stable equilibrium in Γ1 . ∗ ∗ ∗ (SB∗ , RB , SW , RW , SV∗ )

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∗

To do this, solve the first and second equations of (D-2), this gives Z

SB (t) =

t

(α+µB )u

e

0 t

TE D

RB (t) =

Z

 πB [1 − qR RW (u)]du + SB (0) e−(α+µB )t (α+µB )u

(1 − r)qR πW RW (u)e

0

(D-3)

 du + RB (0) e−(α+µB )t .

EP

Integrating SB (t) in (D-3) gives:

AC C

πB [1 − e(α+µB )t ] SB (t) = − (α + µB )

Z

t (α+µB )u

e 0

 πB qR RW (u)du + SB (0) e−(α+µB )t .

(D-4)

Taking the limit of SB (t) in (D-4) and RB (t) in (D-3) as t → ∞, we have lim SB (t) =

t=→∞

πB and lim RB (t) = 0. t→∞ (α + µB )

Next, solving the  of (D-2) SW (t) and  Z Rt W (t) gives  Z t third and fourth equations µW z −µW t µB z SW (t) = αSB (z)e dz + SW (0) e and RW (t) = RB (z)e dz + RW (0) e−µW t . 0

0

Integrating SW (t) and RW (t) and substituting SB (t) in (D-4) and RB (t) in (D-3) gives     Z t πB (1 − ek1 u ) k1 z −k1 u k7 t e πB qrRW (z)dz + SB (0) e e dt + SW (0) e−k7 t , α − SW (t) = k1 0 0 Z t  Z s    k6 u −k6 u k12 s RW (t) = RqπW RW (u)e du + RB (0) e e α dt + RW (0) e−k12 t , Z

t

0

0

30

ACCEPTED MANUSCRIPT where, k1 = α + µB , k6 = α + µB , k7 = µW , k12 = µW . Taking the limit as t → ∞, t=→∞

απB and lim RB (t) = 0. t→∞ µW (α + µB )

Similarly, solving for SV (t) in (D-2) gives SV (t) = as t → ∞.

πV µV

RI PT

lim SW (t) =

+ e−µV t [SV (0) − µπVV ] which converges to

πV µV

,

SC

These asymptotic dynamics are independent of initial conditions in Γ. Hence, the convergence of solutions of (D-2) is global in Γ1 . Next, following [11], we require G(X, Z) to satisfy the two stated conditions: (i). G(X, 0) = 0 and

0 −k3 0 0 0 0 0 0 ηρB ΨV 0

EP

       ∗ DZ G(X , 0) =        

−k2 (1 − p)σB pσB 0 0 0 0 0 0 0

∗ SB ∗ , NB

ΨW =

AC C

where, ΨB = βB bV

0 0 −k4 0 0 0 0 0 ηΨV 0

0 0 0 −k5 0 0 0 α 0 0

TE D



M AN U

b b (ii). G(X, Z) = DZ G(X ∗ , 0)Z − G(X, Z), G(X, Z) ≥ 0,   πB απB πV ∗ where (X , 0) = , 0, , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 and DZ G(X ∗ , 0) is the , 0, α + µB µW (α + µB ) µV Jacobian of G(X, Z) taken with respect to (EB , AB , IB , IBM , EW , AW , IW , IW M , EV , IV ) and evaluated at (X ∗ , 0), which is an M-matrix (the diagonal elements are nonnegative). Thus,

∗ βW bV SW ∗ NW



, ΨV =

       b G(X, Z) =         ηρB ΦV AW

0 0 0 0 −k8 (1 − p)σW pσW 0 0 0

βV bV SV∗ ∗ NH

0 qπW 0 0 0 −k9 0 0 ρW Ψ V 0

0 0 qπW 0 0 0 −k10 0 ΨV 0

0 0 0 0 0 0 0 −k11 0 0

0 0 0 0 0 0 0 0 −k14 σV

ηΨB 0 0 0 ΨW 0 0 0 0 −µV



       ,       

and

ΦB IV 0 0 0 ΦW IV 0 0 0 + ηΦV IW + ρW ΦV AB + ΦV IB 0

               

      ∗ ∗ ∗ ∗ βW bV SW βV bV SV∗ S B NB S W NW S V NH where, ΦB = 1 − NB S ∗ , ΦW = N ∗ 1 − NW S ∗ and ΦV = N ∗ 1 − NH S ∗ . B W W H V πB πV απB ∗ ∗ ∗ ∗ ∗ ∗ , S = , NH = SW + ηSB and SV = . We have in Γ1 Furthermore, SW = µW (α + µB ) B α + µB µV ∗ that, SW ≤ SW , SB ≤ SB∗ , and SV ≤ SV∗ . Therefore, it follows that NH ≤ NH∗ . ∗ βB bV SB ∗ NB

31

ACCEPTED MANUSCRIPT   ∗ SW NW > 0, Hence, if the human population is at equilibrium level, we have that 1 − ∗ SW NW     ∗ SV NH∗ NB SB b > 0; thus, G(X, Z) ≥ 0. 1 − S ∗ NB > 0 and 1 − B NH SV∗ Thus,   0 0 0 0 0 0 0 0

ηΦV IW 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ρW ΦV AB 0 0

0 0 0 0 0 0 0 0 ΦV IB 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

ΦB IV 0 0 0 ΦW IV 0 0 0 0 0

              

RI PT

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ηρB ΦV AW 0 0

SC

       G(X, Z) =        

E

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Therefore, the disease-free equilibrium is globally asymptotically stable by the theorem in [11] (page 246).

Mosquito-Larviciding Strategy

We consider the following three control levels:

1. Low mosquito-larviciding strategy: πV = 500/day;

TE D

2. Moderate mosquito-larviciding strategy: πV = 250/day; 3. High mosquito-larviciding strategy: πV = 125/day.

EP

Figure 10 shows the cumulative number of new infections in adults, new infections in newly born babies and newly born babies with microcephaly for each of the three control levels. Table 11 shows a clear decrease in the cumulative number of new cases with increasing control when comparing the three control levels.

AC C

Humans Adults newly born babies newly born with microcephaly

Low-Control 2.1 × 106 7.7 × 105 10.0 × 103

Moderate-Control 1.5 × 106 4.4 × 105 7.1 × 103

High-Control 7.5 × 105 1.8 × 105 4.4 × 103

Table 11: Simulation results of the cumulative number of new cases days for the Zika model (2.1) using mosquitolarviciding strategy.

32

ACCEPTED MANUSCRIPT

4000

8000 7000 Low Mosquito−Larviciding Moderate Mosquito−Larviciding High Mosquito−Larviciding

6000 5000 4000 3000 2000 1000 0 0

50

100

150

200

3500

Low Mosquito−Larviciding Moderate Mosquito−Larviciding High Mosquito−Larviciding

3000

2500

2000

RI PT

Newly−born babies cumulative new cases

Adults cumulative new cases

9000

1500

1000

500

0 0

250

50

100

150

200

250

Time (days)

SC

Time (days)

80 70 60 50 40 30 20

Low Mosquito−Larviciding Moderate Mosquito−Larviciding High Mosquito−Larviciding

(b)

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90

10

TE D

Newly−born babies with microcephaly cumulative new cases

(a)

0 0

50

100

150

200

250

Time (days)

(c)

EP

Figure 10: Simulation of the Zika model (2.1) for various control levels of the larviciding strategy. (a).

F

AC C

The cumulative number of new cases in adults (b). The cumulative number of new cases in newly born babies. (c). The cumulative number of new cases of newly born babies with microcephaly. Parameter values used are as given in Table 3.

Mosquito-Adulticiding Strategy

1. Low mosquito-adulticiding strategy: µV =

1 /day; 21

2. Moderate mosquito-adulticiding strategy: µV =

1 /day; 14

3. High mosquito-adulticiding strategy: µV = 81 /day. The cumulative number of new infections in adults, new infections in newly born babies and newly born babies with microcephaly for each of the three control levels is depicted in Figure 11. A comparison of the three control levels in Table 12 shows a decrease in the cumulative number of new cases with increasing control.

33

ACCEPTED MANUSCRIPT

4000

8000 7000 6000 5000

Low Mosquito−Adulticiding Moderate Mosquito−Adulticiding High Mosquito−Adulticiding

4000 3000 2000 1000 0 0

50

100

150

200

3500

Low Mosquito−Adulticiding Moderate Mosquito−Adulticiding High Mosquito−Adulticiding

3000

2500

2000

RI PT

Newly−born babies cumulative new cases

Adults cumulative new cases

9000

1500

1000

500

0 0

250

50

100

150

200

250

Time (days)

SC

Time (days)

80 70 60 50 40 30 20

Low Mosquito−Adulticiding Moderate Mosquito−Adulticiding High Mosquito−Adulticiding

(b)

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90

10

TE D

Newly−born babies with microcephaly cumulative new cases

(a)

0 0

50

100

150

200

250

Time (days)

(c)

EP

Figure 11: Simulation of the Zika model (2.1) for various control levels of the adulticiding strategy. (a).

AC C

The cumulative number of new cases in adults (b). The cumulative number of new cases in newly born babies. (c). The cumulative number of new cases of newly born babies with microcephaly. Parameter values used are as given in Table 3.

Humans Adults newly born babies newly born with microcephaly

Low-Control 2.1 × 106 7.7 × 105 10.0 × 103

Moderate-Control 1.4 × 106 4.0 × 105 6.9 × 103

High-Control 2.6 × 105 5.8 × 104 2.7 × 103

Table 12: Simulation results of the cumulative number of new cases for the Zika model (2.1) using mosquitoadulticiding strategy.

34