A model for the transmission of dengueHui Wan

Institute of Mathematics, School of Mathematicsand Computer Sciences, Nanjing Normal University

Nanjing 210097, P.R.ChinaEmail: wanh2046@163.com

Jing-an CuiInstitute of Mathematics, School of Mathematics

and Computer Sciences, Nanjing Normal UniversityNanjing 210097, P.R.ChinaEmail: cuija@njnu.edu.cn

Abstract—In this paper, we formulate a dengue model toexplore the complex dynamics induced by antibody-dependentenhancement (ADE). The basic reproduction number R0 iscalculated. We analyze the regions of existence and stabilityfor the steady state solutions. Numerical simulation manifeststhat backward bifurcation may occur and there may be a stableendemic equilibrium even if the basic reproduction number isbelow 1, which implies that the basic reproduction number isnot enough to describe whether dengue will prevail or not. Weshould pay more attention to the initial states.

I. INTRODUCTION

Dengue has been known clinically for over 200 years, butthe etiology of the disease was not discovered until 1944 [10].The infective agent is the Dengue virus of the family Flaviviri-dae. The two recognized species of the vector transmittingdengue are Aedes aegypti and Aedes albopictus. The former,which is known as the main vector, is highly anthropophilic,thriving in crowded cities and biting primarily during the daywhile the latter is less anthropophilic and inhabits rural areas.Dengue is known to exhibit as many as four coexisting typesof serotype (strains) in a region. Once a person is infected andrecovered from one serotype, they confer life-long immunityfrom that serotype. However, the antibodies that the bodydevelops for the first serotype will not counteract a secondinfection by a different serotype. In fact, due to the nature ofthe disease, the antibodies developed from the first infectionform complexes with the second serotype so that the viruscan enter more cells, increasing viral production. This effect isknown as antibody-dependent enhancement (ADE). Therefore,the sequential infection increases the risk of more seriousdisease resulting in DHF in the region where different typesof serotype coexist.

Currently vector control is the available method for thedengue and DHF prevention and control but research ondengue vaccines for public health use is in process. Due toADE, an optimal vaccination must protect against the fourtypes of serotype simultaneously, or the vaccinations couldincrease transmission of the types of serotype not covered.In addition, epidemiological evidence suggests that denguehemorrhagic fever (DHF) and dengue shock syndrome (DSS)are associated with heterologous dengue infections occurringat an interval of a year or more. Treatment of DHF iscomplicated by the fact that it can progress from a non-specificviral syndrome to irreversible shock and death within a few

hours, so hospitalization is required. Therefore, the coexistenceof different types of serotype is particularly dangerous.

The need for models of dengue disease has reached apinnacle as the transmission of this mosquito-borne virushas increased dramatically. Some mathematical models havebeen proposed. With an evaluation of the impact of ultra-lowvolume (ULV) insecticide applications on dengue epidemics,a SEIRS model is studied in [8]. Feng et al. ([9]) studied asystem that models the population dynamics of an SIR vectortransmitted disease with two pathogen strains. They arguedthat the existence of competitive exclusion in this systemis product of the interplay between the host superinfectionprocess and frequency-dependent (vector to host) contact rates.Esteva and Varga ([4]) studied a one-serotype dengue modelwith a constant human population and variable vector popu-lation. With the assumption that the human population wassupposed to grow exponentially, the same authors proposedanother models facing only one type of serotype [5]. In[6], they considered the impact of vertical transmission andinterrupted feeding on the dynamics of the disease. Supposingboth population size of human and vector were constant, amulti-serotype of dengue was also studied by them ([7]). Theydiscussed conditions for the asymptotic stability of equilibria,supported by analytical and numerical methods and found thatcoexistence of both types of serotype was possible for a largerange of parameters. Recently, Billings and the others ([3])formulated a multi-serotype disease model which did not in-clude vector population to investigates the complex dynamicsinduced by ADE. In this paper, they derived approximationsof the ADE parameter needed to induce oscillations andanalyzed the associated bifurcations that separate the typesof oscillations and then investigated the stability of thesedynamics by adding stochastic perturbations to the model.Taking account of the effect of vaccination, a new multi-serotype disease model was studied in [2]. After describingthe model and its steady state solutions, they modified itto include vaccine campaigns and explored if there existsvaccination rates that can eradicate one or more strains of avirus with ADE. In addition, Shaw et al. ([11]) presented andanalyzed a dynamic compartmental model for multiple typesof serotype exhibiting ADE. Using center manifold techniques,they showed how the dynamics rapidly collapsed to a lowerdimensional system.

Our interest here is to explore the effect of ADE to

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the transmission dynamics of multi-serotype dengue virus.For the sake of mathematical tractability, only two types ofserotype is considered. Since the mosquito dynamics operateson a much faster time-scale than the human dynamics, themosquito population can be considered to be at equilibriumwith respect to changes in the human population. In this paper,we suppose that the human population is variable while thevector population is constant, which has not been consideredin the multi-serotype dengue model mentioned above.

II. FORMULATION OF THE MODEL

We formulate a compartmental model for the spread ofdengue fever in the human and mosquito population with thetotal population sizes at time t are given by Nh(t) and Nv(t),respectively.

For the human population: Sh is the number of individualssusceptible to serotype 1 and 2; Ihj is the number of primaryinfective human with serotype j; Rhj is the number of recov-ered human from serotype j, susceptible to serotype k; Ijk isthe number of secondary infective human with serotype k; Rh

is the number of human recovered from secondary infection ofeither types of serotype, and immune to both types of serotype.Λh > 0 is the human input (birth) rate. μh > 0 is the naturaldeath rates. ej ≥ 0 is the disease-induced death rate producedby serotype j. γj > 0 is the rate at which human hosts withserotype j recover. βhj > 0 is the proportion of bites on humanthat produce an infection of serotype j. Since viral productionis increased during a secondary infection due to ADE, weintroduce an ADE parameter, σk, to mimic primary infectionswith serotype j increase susceptibility to serotype k. In orderto study the effect of ADE, we are considering the situationof σk ≥ 1, where σk = 1 implies that there is no ADE whileσk > 1 implies ADE will increase the susceptibility of the hostto the second serotype. All the subscripts j and k mentionedin this paragraph satisfy j, k = 1, 2 and j �= k.

For the mosquito population: the three compartments rep-resent susceptible vectors Sv , infectious vectors Ivj withserotype j (j = 1, 2) respectively. Due to its short life span,we assume that once a mosquito is infected with one kindof serotype it never recover from the infection, and thereforeit cannot be reinfected with different types of serotype. Then,secondary infections may take place only in human. Our modelalso excludes the immature mosquitoes since they do notparticipate in the infection cycle and are, thus, in the waitingperiod, which limits the vector population growth. We assumethat, in a given period of time, the mosquito population isconstant and equal to Nv , with birth and death rate constantsequal to μv . βvj is the probability that a mosquito whichbites an infectious people with serotype j becomes infectious,j = 1, 2.

The biting rate b of mosquitoes is the average number ofbites per mosquito per day. This rate depends on a number offactors, in particular, climatic ones, but for simplicity in thispaper we assume b constant.

With the preceding assumptions and parameters, using stan-dard incidence rate, the two-serotype dengue model with ADE

is governed by the following equations:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dSh/dt = Λh − bβh1Iv1Sh/Nh − bβh2Iv2Sh/Nh − μhSh,

dIhj/dt = bβhjIvjSh/Nh − (μh + γj)Ihj ,

dRhj/dt = γjIhj − μhRhj − σkbβhkIvkRhj/Nh,

dIjk/dt = σkbβhkIvkRhj/Nh − (μh + γk + ek)Ijk,

dRh/dt = γ1I21 + γ2I12 − μhRh,

dSv/dt = μvNv − bβv1(Ih1 + I21)Sv/Nh

− bβv2(Ih2 + I12)Sv/Nh − μvSv,

dIvj/dt = bβvj(Ihj + Ikj)Sv/Nh − μvIvj ,

dNh/dt = Λh − μhNh − e1I21 − e2I12,(A.1)

where j, k = 1, 2, j �= k and Nh = Sh + Ih1 + Ih2 + I12 +I21 + Rh1 + Rh2 + Rh and Nv = Sv + Iv1 + Vv2 are totalnumber of human and mosquitoes respectively.

The first quadrant in the Sh Ih1 Ih2 Rh1 Rh2 I12 I21 Rh

Sv Iv1 Vv2 Nh space is positively invariant for system (A.1)since the vector field on the boundary does not point to theexterior. What’s more, since dNh/dt < 0 for Nh > Λh/μh

and Nv is constant, all trajectories in the first quadrant enteror stay inside the region

D+ = {(Sh, Ih1, Ih2, Rh1, Rh2, I12, I21, Nh, Iv1, Iv2) | Nh

� Λh/μh, Sv + Iv1 + Vv2 = Nv}.

The continuity of the right-hand side of (A.1) implies thatunique solution exists on a maximal interval. Since solutionsapproach, enter or stay in D+, they are eventually boundedand hence exist for t > 0. Therefore, the initial value problemfor system (A.1) is mathematically well posed and biologicallyreasonable since all variables remain nonnegative.

In order to reduce the number of parameters and simplifysystem (A.1), we normalize the human and mosquito vectorpopulation sh = Sh

Λh/µh, ihj = Ihj

Λh/µh, rhj = Rhj

Λh/µh, ijk =

Ijk

Λh/µh, rh = Rh

Λh/µh, nh = Nh

Λh/µh, sv = Sv

Nv, ivj = Ivj

Nvand

m = Nv

Λh/µh, where j, k = 1, 2, j �= k.

It follows from the relations rh = nh − sh − ih1 − ih2 −rh1 − rh2 − i12 − i21 and sv = 1− iv1 − iv2 that the originalmodel (A.1) can be reduced to the following nonlinear system

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dsh

dt= μh − mbβh1

nhiv1sh − mbβh2

nhiv2sh − μhsh,

dihj

dt=

mbβhj

nhivjsh − (γj + μh)ihj ,

drhj

dt= γjihj − μhrhj − σk

mbβhk

nhivkrhj

dijk

dt= σk

mbβhk

nhivkrhj − (μh + γk + ek)ijk,

dnh

dt= μh − μhnh − e1i21 − e2i12,

divj

dt=

bβvj(ihj + ikj)nh

(1 − iv1 − iv2) − μvivj ,

(A.2)

2

where j, k = 1, 2, j �= k. Note that all trajectories in the firstquadrant enter or stay inside the region

Ω = {(sh, ih1, rh1, iv1, ih2, rh2, iv2, i21, i12, nh) | 0 � sh,

0 � ihk, 0 � ijk, 0 � rhk, 0 � ivk,

sh + ih1 + ih2 + i12 + i21 + rh1 + rh2 � 1, iv1 + iv2 � 1,

j, k = 1, 2, j �= k}.III. THE BOUNDARY EQUILIBRIA

According to the concept of next generation matrix (Diek-mann et al., 1990 [1]) and reproduction number presented invan den Driessche and Watmough (2002) [12], by calculation,we can get the basic reproduction number

R0 = max{√

R1,√

R2}, (A.3)

where

Rk =mb2βhkβvk

μv(μh + γk), k = 1, 2. (A.4)

Thus, according to our analysis, for the boundary Equilibria,we have the following results:

Theorem III.1. For the model (A.1), the disease-free equi-librium E0 always exists; there also exists a boundary equi-librium Ek if and only if Rk > 1, k = 1, 2. E1 is the statewhere only serotype 1 is present and E2 is the state whereonly serotype 2 is present.

The stability of the disease-free equilibrium E0 is given bythe following theorem:

Theorem III.2. If R0 < 1 (R0 > 1), E0 is locally asymptot-ically stable (unstable).

Consider now the stability of the boundary equilibriaEk, k = 1, 2. Note that E1 and E2 only make biological sensewhen R1 > 1 and R2 > 1, respectively. For E1 and E2, wehave the following theorem:

Theorem III.3. E1 is locally asymptotically stable whenR1 > 1 and

R2 <R1

1 + ω2(R1 − 1). (A.5)

Similarly, E2 is locally asymptotically stable when R2 > 1and

R1 <R2

1 + ω1(R2 − 1), (A.6)

where

ωj =σjμvγk(μh + γj)

(μhbβvk + μvμh + μvγk)(μh + γj + ej), (A.7)

j, k = 1, 2, j �= k.

Remark III.4. In the case where R1 = R2, we find that theboundary equilibria, E1 and E2, are never stable. In addition,E1 and E2 cannot be stable simultaneously, which implies thatonly one serotype for which Rk is bigger will exist eventuallyif the two serotypes can not coexist when R0 > 1.

Noting that, using R1 and R2 as parameters, in the planeof R1R2,

Rj =Rk

1 + ωj(Rk − 1), (A.8)

(j, k = 1, 2, j �= k) is a hyperbola when ωk �= 1, otherwise, itis a straight line. It is easy to see that the stability regions ofE1 and E2 become smaller as ω1 and ω2 increase. Since ωj

will increase with the increasing of σj , we can conclude thatthe effect of antibody-dependent enhancement (ADE) reducethe possibility of the permanence of only one type of serotype,which is more dangerous for dengue because of the risk forDHF/DSS and the higher mortality rates and transmissibilityfor second infections.

IV. ENDEMIC EQUILIBRIA AND BACKWARD BIFURCATION

Due to the complexity of this system, it is not easy toanalyze the existence and stability of the endemic equilibriumE3.

0.01

0.02

0.03

0.04

ih1

0 2000 4000 6000 8000t

(a) The time course ofih1.

0.01

0.02

0.03

0.04

ih2

0 2000 4000 6000 8000t

(b) The time course ofih2.

Fig. 1. A stable endemic equilibrium where both types of serotype are presentwhen R0 < 1. The units of time is day and σ1 = σ2 = 50, μh = 0.0015,μv = 1/14, βh1 = 0.02, βh2 = 0.02, βv1 = βv2 = 0.15, m = 5,e1 = e2 = 0.04, γ1 = γ2 = 1/14, b = 0.5. Note that R0 ≈ 0.848 andω1 = ω2 ≈ 30.960.

Now, we consider the situation when R0 < 1. By TheoremIII.2, the DFE is always existing and local stable when R0 < 1.To begin exploring the dynamics numerically, we set μh =0.0015, μv = 1/14, βh1 = βh2 = 0.02, βv1 = βv2 = 0.15,m = 5, e1 = e2 = 0.04, γ1 = γ2 = 1/14, b = 0.5. Note thatR1 = R2 ≈ 0.848 < 1. For symmetry, ih1 = ih2, iv1 = iv2,rh1 = rh2, i21 = i12 at the endemic equilibrium, if it exists.Thus, we can evaluate the existence for a endemic equilibrium.

Interestingly, by calculation and numerical simulation, thereexists a threshold condition for the ADE parameter σj . Ifσj is big enough, for example, let σ1 = σ2 = 50, thereare two endemic equilibria. One is E01 where sh ≈ 0.801,ih1 ≈ 0.00204, rh1 ≈ 0.0135, i12 ≈ 0.00111, nh ≈ 0.941,iv1 ≈ 0.0035 and the other is E02 where sh ≈ 0.891,ih1 ≈ 0.00112, rh1 ≈ 0.0132, i12 ≈ 0.000535, nh ≈ 0.971,iv1 ≈ 0.00179. In addition, the former is a locally stableendemic equilibria (see Fig. 1), which implies the globalstability of E0 is impossible and backward bifurcation isinduced by ADE. The basic reproductive number itself is notenough to describe whether dengue will prevail or not in thissituation. We should pay more attention to the effect of ADEand the initial state of the disease.

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V. DISCUSSIONS

In this paper, we have formulated a model to study thedynamics of a two-serotype dengue with ADE. The basicreproduction number was produced. We analyzed the systemand found regions of stability for the steady state solutions.The effect of ADE was studied.

According to our analysis, ADE shrank the area of parame-ter space in which only one type of serotype was present andenlarged the area in which both types of serotype were present,which was not desirable because of the risk for DHF/DSSand the higher mortality rates and transmissibility for secondinfection. Numerical simulation manifested if the parameterof ADE was big enough, there existed a endemic equilibrium,which implied that the global stability of E0 was impossible.ADE induced complex dynamics.

By the definition of R0, it was independent to the ADEparameter. Nevertheless, due to ADE , the susceptibility wasincreased when a new kind of serotype entered a population inwhich some individuals have had previous dengue infections.Therefore, the basic reproduction number obtained by justfitting a time series may give an incorrect estimation of thetrue number of secondary cases produced by a primary casein a fully susceptible population.

Because of the effect of ADE, endemic equilibria mightexist even if R0 < 1 and there might be backward bifurcation.This was a new result which implied that the basic reproduc-tion number was not enough to describe whether dengue willprevail or not. We should pay more attention to the initialstates.

ACKNOWLEDGMENT

Research is supported by the National Natural ScienceFoundation of China (10771104).

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