Accepted Manuscript. Backward Bifurcations in Dengue Transmission Dynamics. S.M. Garba, A.B. Gumel, M.R. Abu Bakar

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1 Accepted Manuscript Backward Bifurcations in Dengue Transmission Dynamics S.M. Garba, A.B. Gumel, M.R. Abu Bakar PII: S (08) DOI: /j.mbs Reference: MBS 6860 To appear in: Mathematical Biosciences Received Date: 15 September 2007 Revised Date: 24 April 2008 Accepted Date: 1 May 2008 Please cite this article as: S.M. Garba, A.B. Gumel, M.R. Abu Bakar, Backward Bifurcations in Dengue Transmission Dynamics, Mathematical Biosciences (2008), doi: /j.mbs 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.

2 Backward Bifurcations in Dengue Transmission Dynamics S.M. Garba,, A.B. Gumel,1 and M.R. Abu Bakar Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada Department of Mathematics, Universiti Putra Malaysia, Serdang, Selangor, Malaysia Abstract A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutuallyexclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number (R 0 ) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R 0 less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease. Keywords: dengue, mosquitoes, equilibria, stability, bifurcation, vaccine. 1 Corresponding author. gumelab@cc.umanitoba.ca 1

3 Introduction Dengue, a mosquito-transmitted disease caused by any of four closely-related virus serotypes (DEN-1-4) of the genus Flavivirus, is endemic in at least 100 countries in Africa, the Americas, the Eastern Mediterranean and subtropical regions of the world [18, 44], inhibited by over 2.5 billion people. Dengue ranks second to malaria amongst deadly mosquito-borne diseases, each year claiming about 100 million infections and 20,000 deaths globally [44]. Classical dengue fever causes relatively mild morbidity and mortality, and sufferers recover within one to two weeks after the onset of fever [31]. However, some individuals develop dengue hemorrhagic fever (DHF) or dengue shock syndrome (DSS)[33], where the severity of the disease is drastically increased (with mortality ranging from 5-15% [31, 36]. Figures from the World Health Organization show that hundreds of thousands of cases of DHF are recorded annually [6, 7, 49]. Dengue is transmitted to humans through mosquito bites. Female mosquitoes (of the genus Aedes (Stegomyia), mainly the Aedes aegypti [4]) acquire infection by taking a blood meal from an infected human (in the viremic phase of illness). These infected mosquitoes pass the disease to susceptible humans. Individuals who recover from one serotype become permanently immune to it, but may become partially-immune or temporarily-immune (or both) to the other serotypes [49]. Unfortunately, there is still no specific effective treatment for dengue. Fluid replacement therapy is used if an early clinical diagnosis is made [13, 21]. Although there is no effective and safe vaccine for dengue at the moment, a number of candidate vaccines (including live attenuated mono- and tetra-valent formulation, inactivated whole virus vaccines, and recombinant subunit vaccines) are undergoing various phases of clinical trials [6, 7, 13, 14, 22, 35, 37, 42, 45, 49, 50]. However, it is believed that any future dengue vaccine would not be able to offer perfect protection against all serotypes. Thus, any future dengue vaccine is expected to be imperfect [14]. It is instructive, therefore, to assess the potential impact of such a vaccine in a community. A number of mathematical models have been developed in the literature to gain insights into the transmission dynamics of dengue in a community (see, for instance, [18, 19, 20, 24, 25, 26, 27, 29, 30, 36, 45, 46, 52] ). While Feng and Velasco-Hernandez [29] investigated the competitive exclusion principle in a two-strain dengue model, Chowell et al. [18] estimated the basic reproduction number of dengue using spatial epidemic data. Tewa et al. [46] established global asymptotic stability of the equilibria of a single-strain dengue model, and Struchiner et al. [45] gave a detailed discussion on current research issues in modelling mosquito-borne diseases. The aforementioned studies typically subdivide the total population into mutually-exclusive compartments of susceptible and infected individuals and vectors. This (compartmental modelling) approach is followed in the current study. This paper extends previous studies by incorporating the dynamics of individuals and vectors who are exposed to the disease 2

4 (that is, individuals and vectors that are infected but not yet infectious are assumed capable of transmitting the disease) and an imperfect dengue vaccine. The paper is organized as follows. A basic single strain dengue model, which incorporates the dynamics of exposed humans and vectors, is formulated and rigorously analysed in Section 2. The model is then extended to include an imperfect vaccine in Section 3, and its detailed analysis provided. In both Sections 2 and 3, the associated mass action models are also analysed Model Formulation The model assumes a homogenous mixing of the human and vector (mosquito) populations, so that each mosquito bite has equal chance of transmitting the virus to susceptible human in the population (or acquiring infection from an infected human). The total human population at time t, denoted by N H (t), is sub-divided into four mutually-exclusive sub-populations of susceptible humans S H (t), exposed humans E H (t), infectious humans I H (t) and recovered humans R H (t), so that N H = N H (t) = S H (t) + E H (t) + I H (t) + R H (t). Similarly, the total vector population at time t, denoted by N V (t), is split into susceptible mosquitoes S V (t), exposed mosquitoes E V (t), infectious mosquitoes I V (t), so that N V = N V (t) = S V (t) + E V (t) + I V (t). The susceptible human population is generated via recruitment of humans (by birth or immigration) into the community (at a constant rate, Π H ). This population is decreased following infection, which can be acquired via effective contact with an exposed or infectious vector (at a rate λ H ; the force of infection of humans) given by λ H = C V H(N H, N V ) N V (η V E V + I V ), (1) where the modification parameter 0 < η V < 1 accounts for the assumed reduction in transmissibility of exposed mosquitoes relative to infectious mosquitoes. It is worth emphasizing that, unlike many of the published modelling studies on dengue transmission dynamics, the current study assumes that exposed vectors can transmit dengue disease to humans (that is, η V > 0). This is in line with some recent clinical studies (see, for instance, [28, 51, 53]). 2.1 Incidence Functions In this section, the functional forms of the incidence functions associated with the transmission dynamics of dengue disease will be derived. The derivation is based on the basic fact that for mosquito-borne diseases (such as dengue), the total number of bites made by mosquitoes must equal the total number of bites received by humans (see also [9]). 3

5 Since mosquitoes bite both susceptible and infected humans, it is assumed that the average number of mosquito bites received by humans depends on the total sizes of the populations of mosquitoes and humans in the community. It is assumed that each susceptible mosquito bites an infected human at an average biting rate, b S, and the human hosts are always sufficient in abundance; so that it is reasonable to assume that the biting rate, b S, is constant. Let, C HV = ρ HV b S, be the rate at which mosquitoes acquire infection from infected humans (exposed or infectious), where ρ HV is the transmission probability from an infected human to a susceptible mosquito and b S is the biting rate per susceptible mosquito, so that C HV is a constant. Similarly, let C V H (N H, N V ) = ρ V H b I, be the rate at which humans acquire infection from infected mosquitoes (exposed or infectious), where ρ V H is the transmission probability from an infected mosquito to a susceptible human and b I is the average biting rate per infected mosquito. Thus, for the number of bites to be conserved, the following equation must hold, C HV N V = C V H (N H, N V )N H, (2) 109 so that, 110 Using (3) in (1) gives, N V = C V H(N H, N V ) C HV N H. (3) λ H = C HV N H (η V E V + I V ). (4) Similarly, it can be shown that the force of infection of mosquitoes (denoted by λ V ) is given by, λ V = C HV N H (η H E H + I H ), (5) where the modification parameter 0 η H < 1 accounts for the relative infectiousness of exposed humans in relation to infectious humans. Here, too, it is assumed that susceptible mosquitoes can acquire dengue infection from exposed humans (see also [19]). 4

6 Model equations The basic model for the transmission dynamics of dengue is given by the following deterministic system of nonlinear differential equations ds H = Π H λ H S H µ H S H, dt de H = λ H S H (σ H + µ H )E H, dt di H = σ H E H (τ H + µ H + δ H )I H, dt dr H = τ H I H µ H R H, (6) dt ds V = Π V λ V S V µ V S V, dt de V = λ V S V (σ V + µ V )E V, dt di V = σ V E V (µ V + δ V )I V, dt where Π H is the recruitment of humans into the population (assumed susceptible), λ H is the infection rate of susceptible humans (which results following effective contact with exposed or infectious mosquitoes) and µ H is the natural death rate of humans. Exposed humans develop clinical symptoms of dengue disease, and move to the infectious class (I H ), at a rate σ H. Infectious humans recover (and move into the R H class) at a rate τ H and suffer disease-induced death at a rate δ H. It is assumed that recovered individuals acquire lifelong immunity against re-infection (so that they do not acquire dengue infection again). The susceptible vector population is generated by birth at a rate Π V. This population is reduced by infection, following effective contact with exposed or infectious humans, at the rate λ V and natural death at a rate µ V. Exposed vectors develop symptoms of disease (and move to the I V class) at a rate σ V. Infectious vectors die due to disease at a rate δ V. The basic model (6) is an extension of some dengue transmission models published in the literature, such as those in [18, 24, 25, 26, 46], by incorporating the dynamics of, and transmission by, exposed humans and mosquitoes. Since the above (basic) model monitors human and vector populations, all the associated parameters and state variables are non-negative. The parameters and variables of the models are described in Table 1, and a flow diagram of the model is depicted in Figure 1. The above derivation adopts a standard incidence formulation, where the contact rate is assumed to be constant, unlike in the case of the mass action formulation, where the contact rate depends on the size of the total host population (see, for instance, [34, 43] for detailed derivation of these incidence functions). A crucial question 5

7 is which of the two incidence functions, standard or mass action incidence, is more suitable for modeling infectious diseases in general, and dengue disease in particular? Hethcote [34] showed that formulating the incidence function using the mass action law implicitly assumes that the contact rate increases linearly with the population size and cited studies that suggest that this implicit assumption is not realistic. Although, as noted by Hethcote [34], data ([1, 2]) strongly suggests that standard incidence is more suited for modelling human diseases than mass action incidence, the choice of one formulation over the other really depends on the disease being modelled and, in some cases, the need for mathematical (analytical) tractability [43]. In the context of dengue disease, the vast majority of the models published over the last few decades, such as those in [5, 18, 20, 26, 29, 41, 52], adopt a standard incidence formulation. It should be mentioned, however, that although the models in [24, 25, 27] were initially formulated using standard incidence, the constant human population assumption made in those studies (perhaps to achieve mathematical tractability in the analysis) reduces the models therein to (essentially) mass action models. Furthermore, standard incidence formulation seems to be more generally favoured in modelling other vector-borne diseases, such as malaria ([15, 16, 17]) and West Nile virus ([9, 32, 40, 48]). Based on all these, it seems plausible that standard incidence formulation should be used in modelling dengue transmission dynamics. Consequently, this formulation is adopted in the current study. 2.3 Basic Properties In this section, the basic dynamical features of the model (6) will be explored. We claim the following Lemma 1 The closed set { } D = (S H, E H, I H, R H, S V, E V, I V ) R 7 + : S H + E H + I H + R H Π H µh ; S V + E V + I V Π V µ V, is positively-invariant and attracting with respect to the basic model (6). Proof. Adding the first four equations and the last three equations in the model (6), respectively, gives: 169 and, dn H dt = Π H µ H N H δ H I H, (7) dn V dt = Π V µ V N V δ V I V. (8) 6

8 Since dn H /dt Π H µ H N H and dn V /dt Π V µ V N V, it follows that dn H /dt < 0 and dn V /dt < 0 if N H (t) > Π H µh and N V (t) > Π V µ V, respectively. Thus, a standard comparison theorem (see [38]) can be used to show that N H (t) N H (0)e µ H(t) + Π H µh [1 e µ H(t) ] and N V (t) N V (0)e µ V (t) + Π V µ V [1 e µ V (t) ]. In particular, N H (t) Π H /µ H and N V (t) Π V /µ V if N H (0) Π H /µ H and N V (0) Π V /µ V, respectively. Thus, D is positively-invariant. Further, if N H (t) > Π H /µ H and N V (t) > Π V /µ V, then either the solution enters D in finite time, or N H (t) approaches Π H /µ H and N V (t) approaches Π V /µ V, and the infected variables E H, I H, E V and I V approach zero. Hence, D is attracting (i.e., all solutions in R 7 + eventually enter D). Thus, in D, the basic model (6) is well-posed epidemiologically and mathematically [34]. Hence, it is sufficient to study the dynamics of the basic model in D. 2.4 Existence and Stability of Equilibria Disease-free equilibrium (DFE) The basic model (6) has a DFE given by, E 0 = (S H, E H, I H, R H, S V, E V, I V ) = ( ΠH, 0, 0, 0, Π ) V, 0, 0. (9) µ H µ V Following [47], the linear stability of E 0 can be established using the next generation operator method on system (6). The matrices, F (for the new infection terms) and V (of the transition terms) are given, respectively, by F = where, C HV S V N H 0 0 C HV η V C HV Q C η HV SV H 0 0 N, V = σ H Q Q H 3 0, σ V Q Q 1 = σ H + µ H, Q 2 = τ H + µ H + δ H, Q 3 = σ V + µ V and Q 4 = µ V + δ V. It follows then that the basic reproduction number, denoted by R 0, is given by R 0 = ρ(f V 1 ) = C 2 HV Π V µ H (η H Q 2 + σ H )(η V Q 4 + σ V ) Π H µ V Q 1 Q 2 Q 3 Q 4, where ρ is the spectral radius (dominant eigenvalue in magnitude) of the next generation matrix F V 1. Hence, using Theorem 2 of [47], we have established the following result: 7

9 Lemma 2 The disease free equilibrium, E 0, of the model (6), is locally asymptotically stable (LAS) if R 0 < 1, and unstable if R 0 > 1. The threshold quantity R 0 is the basic reproduction number of the disease [1, 2, 34]. It represents the average number of secondary cases that one infected case can generate if introduced into a completely susceptible population. It can be interpreted as follows. Susceptible mosquitoes can acquire infection following effective contact with either an exposed human (E H ) or an infectious human (I H ). The number of vector infections generated by an exposed human (near the DFE) is given by the product of the infection rate of exposed humans (C HV η H /NH = C HV η H µ H /Π H ) and the average duration in the exposed (E H ) class (1/Q 1 ). Furthermore, the number of vector infections generated by an infectious human (near the DFE) is given by the product of the infection rate of infectious humans (C HV /NH = C HV µ H /Π H ), the probability that an exposed human survives the exposed stage and move to the infectious stage (σ H /Q 1 ) and the average duration in the infectious stage (1/Q 2 ). Thus, the average number of new mosquito infections generated by infected humans (exposed or infectious) is given by (noting that SV = Π V /µ V ) ( CHV η H µ H Π H Q 1 + C ) HV µ H σ H S (η H Q 2 + σ H )Π V µ H V = C HV. (10) Π H Q 1 Q 2 Π H µ V Q 1 Q 2 Similarly, susceptible humans acquire infection following effective contact with either an exposed (E V ) or infectious mosquito (I V ). The number of human infections generated by an exposed mosquito is the product of the infection rate of exposed mosquito (C HV η V /NH = C HV η V µ H /Π H ) and the average duration in the exposed class (1/Q 3 ). The number of infections generated by an infectious mosquito is the product of the infection rate of infectious mosquitoes (C HV /NH = C HV µ H /Π H ), the probability that an exposed mosquito survives the exposed class and move to the infectious stage (σ V /Q 3 ) and the average duration in the infectious stage (1/Q 4 ). Thus, the average number of new human infections generated by an infected mosquito (exposed or infectious) is given by (noting that SH = Π H/µ H ) ( CHV η V µ H Π H Q 3 + C HV µ H σ V Π H Q 3 Q 4 ) S H = C HV (η V Q 4 + σ V ) Q 3 Q 4. (11) The geometric mean of (10) and (11) gives the basic reproduction number, R 0 (interpretation for R 0 for dengue disease is also given in [18, 25]). The epidemiological implication of Lemma 2 is that, in general, when R 0 is less than unity, a small influx of infected mosquitoes into the community would not generate large outbreaks, and the disease dies out in time (since the DFE is LAS). However, we show in the next subsection that the disease may still persist even when R 0 < 1. 8

10 Endemic equilibrium and backward bifurcation In order to find endemic equilibria of the basic model (6) (that is, equilibria where at 230 least one of the infected components of the model (6) is non-zero), the following steps are taken. Let E 1 = (SH, H, I H, R H, S V, E V, I V ) represents any arbitrary endemic 232 equilibrium of the model (6). Further, let λ H = C HV (η NH V EV + IV ) and λ V = C HV (η NH H EH + IH ), (12) be the forces of infection of humans and vectors at steady state, respectively. Solving the equations in (6) at steady state gives 235 S H = R H = I V = Π H λ H + µ, EH = H τ H σ H λ H Π H Q 1 Q 2 µ H (λ H + µ H), S σ V λ V Π V Q 3 Q 4 (λ V + µ V ). λ H Π H Q 1 (λ H + µ H), I H = V = Π V λ V + µ V σ H λ H Π H Q 1 Q 2 (λ H + µ H), λ V Π V Q 3 (λ V + µ V ),, E V = Substituting (13) in (12), and simplifying, gives, respectively, (13) 236 and, λ H = Q 3 Q 4 Π H (λ V Q 1 Q 2 µ H C HV λ V Π V (η V Q 4 + σ V )(λ H + µ H) + µ V )[Q 2 µ H (Q 1 + λ H ) + σ Hλ H (µ H + τ H )], (14) λ V = µ H λ H C HV (η H Q 2 + σ H ) Q 2 µ H (Q 1 + λ H ) + σ Hλ H (µ H + τ H ). (15) By substituting (15) in (14), it can be shown that the non-zero equilibria of the model satisfy the following quadratic (in terms of λ H ) 239 where, a 0 (λ H ) 2 + b 0 λ H + c 0 = 0, (16) a 0 = Π H µ H Q 3 Q 4 [Q 2 µ H + σ H (µ H + τ H )][C HV (η H Q 2 + σ H ) + µ V (Q 2 + σ H + τ H )], b 0 = µ H Π H Q 1 Q 2 Q 3 Q 4 {C HV µ H (Q 2 η H + σ H ) + µ V [2(µ H Q 2 + µ H σ H + σ H τ H ) Q 1 Q 2 R 2 0]}, c 0 = Q 2 1Q 2 2Q 3 Q 4 µ 2 H µ V Π H (1 R 2 0). 9

11 Thus, the positive endemic equilibria of the basic model (6) are obtained by solving for λ H from the quadratic (16) and substituting the results (positive values of λ H ) into 242 the expressions in (13). Clearly, the coefficient a 0, of (16), is always positive, and c is positive (negative) if R 0 is less than (greater than) unity, respectively. Thus, the 244 following result is established Theorem 1 The dengue model (6) has: (i) a unique endemic equilibrium if c 0 < 0 R 0 > 1; (ii) a unique endemic equilibrium if b 0 < 0, and c 0 = 0 or b 2 0 4a 0 c 0 = 0; (iii) two endemic equilibria if c 0 > 0, b 0 < 0 and b 2 0 4a 0 c 0 > 0; (iv) no endemic equilibrium otherwise. It is clear from Theorem 1 (Case i) that the model has a unique endemic equilibrium whenever R 0 > 1. Further, Case (iii) indicates the possibility of backward bifurcation (where the locally-asymptotically stable DFE co-exists with a locally-asymptotically stable endemic equilibrium when R 0 < 1; see, for instance, [3, 10, 12, 23, 43]) in the model (6) when R 0 < 1. To check for this, the discriminant b 2 0 4a 0 c 0 is set to zero and solved for the critical value of R 0, denoted by R c, given by R c = 1 b 2 0 4a 0 Q 2 1Q 2 2Q 3 Q 4 µ 2 H µ V Π H. (17) Thus, backward bifurcation would occur for values of R 0 such that R c < R 0 < 1. This 257 is illustrated by simulating the model with the following set of parameter values (it 258 should be stated that these parameters are chosen for illustrative purpose only, and may 259 not necessarily be realistic epidemiologically): η H = 0.99, η V = 0.98, Π H = 10, Π V = , C HV = 0.068, δ H = 0.99, δ V = , µ H = , σ H = 0.53, σ V = 0.2, µ V = and τ H = 0.2 (see Table 1 for the units of the aforemention parameters). With 262 this set of parameters, R c = < 1 and R 0 = < 1 (so 263 that, R c < R 0 < 1). The associated bifurcation diagram is depicted in Figure 2A. A time series of λ H is also plotted in Figure 2B, showing the DFE (corresponding to 265 λ H = 0) and two endemic equilibria (corresponding to λ H = and λ H = ). Further, Figure 2B shows that one of the endemic equilibria (λ ) is LAS, the other (λ H = ) is unstable (saddle), and the 268 DFE is LAS. This clearly shows the co-existence of two locally-asymptotically stable 269 equilibria when R 0 < 1, confirming that the model (6) undergoes the phenomenon of 270 backward bifurcation. Thus, the following result is established. H = 10

12 Lemma 3 The basic model (6) undergoes backward bifurcation when Case (iii) of Theorem 1 holds and R c < R 0 < 1. The epidemiological significance of the phenomenon of backward bifurcation is that the classical requirement of R 0 < 1 is, although necessary, no longer sufficient for disease elimination. In such a scenario, disease elimination would depend on the initial sizes of the sub-populations (state variables) of the model. That is, the presence of backward bifurcation in the dengue transmission model (6) suggests that the feasibility of controlling dengue when R 0 < 1 could be dependent on the initial sizes of the subpopulation of the model (6). Although the phenomenon of backward bifurcation has been established in many epidemiological settings (see [3, 10, 12, 23, 43] and the references therein), to the authors knowledge, this is the first time such a phenomenon has been theoretically shown in the transmission dynamics of dengue disease. Further, as a consequence, it is instructive to try to determine the cause of the backward bifurcation phenomenon in the model (6). This is explored below by considering the mass action equivalent of the model (6). 2.5 Analysis of the Mass Action Model Consider the model (6) with constant total human population, N H (t) = N H = constant. Thus, the associated forces of infection, λ H and λ V in (4) and (5), respectively, reduce to λ m H = C HV (η V E V + I V ) and λ m V = C HV (η H E H + I H ), (18) where, now, C HV is scaled down by the constant N H. The resulting (mass action) model, obtained by using (18) in (6), has the same DFE given by (9). The associated next generation matrices, F m and V m, are, respectively, given by 0 0 C HV SH η V C HV SH Q F m = C HV SV η H C HV SV 0 0, V σ m = H Q Q σ V Q 4 It follows that the associated reproduction number for the mass action model, denoted by R m 0 = ρ(f m Vm 1 ), is given by R m 0 = C 2 HV Π HΠ V (η H Q 2 + σ H )(η V Q 4 + σ V ) µ H µ V Q 1 Q 2 Q 3 Q 4. 11

13 Here, too, the following result is established (using Theorem 2 of [47]): Theorem 2 For basic dengue model with mass action incidence, given by (6) with (18), the DFE, (9), is LAS if R m 0 < 1, and unstable if R m 0 > Non-existence of endemic equilibria for R m In this section, the non-existence of endemic equilibria of the model when R m 0 be explored. We claim the following: 1 will Theorem 3 The basic dengue model with mass action incidence, given by (6) with (18), has no endemic equilibrium when R m 0 1, and has a unique endemic equilibrium otherwise. Proof. Solving the equations in the model (6) in terms of (λ m H ) and (λ m V ), gives: S H = Π H (λ m H ) + µ H, E H = (λ m H ) Π H Q 1 [(λ m H ) + µ H ], I H = σ H (λ m H ) Π H Q 1 Q 2 [(λ m H ) + µ H ], R H = τ H σ H (λ m H ) Π H Q 1 Q 2 µ H [(λ m H ) + µ H ], S V = Π V (λ m V ) + µ V, E V = (λ m V ) Π V Q 3 [(λ m V ) + µ V ], (19) I V = σ V (λ m V ) Π V Q 3 Q 4 [(λ m V ) + µ V ] Substituting (19) into (18), and simplifying, shows that the non-zero equilibria of the mass action model satisfy the linear equation a 1 (λ m H) + b 1 = 0, (20) 312 where, a 1 = Q 3 Q 4 [Π H C HV (η H Q 2 + σ H ) + µ V Q 1 Q 2 ], b 1 = Q 1 Q 2 Q 3 Q 4 µ H µ V [1 (R m 0 ) 2 ] Clearly, a 1 > 0 and b 1 0 whenever R m 0 1, so that (λ m H ) = b 1 a 1 0. Therefore, the mass action model, (6) with (18), has no endemic equilibrium whenever R m 0 1. The above result suggests the impossibility of backward bifurcation in the mass action model, since no endemic equilibrium exists when R m 0 1 (and backward bifurcation requires the presence of at least two endemic equilibria when R m 0 1). A global stability result is established for the DFE of the mass action model below (to completely rule out backward bifurcation). 12

14 Global Stability of the DFE Define the region D = { } (S H, E H, I H, R H, S V, E V, I V ) R 7 + : S H + E H + I H + R H Π H µh, S V + E V + I V Π V µ V, 323 and, D = {(S H, E H, I H, R H, S V, E V, I V ) D : S H S H, S V S V } We claim the following. Lemma 4 The region D is positively-invariant and attracting. 331 ds H dt = Π H λ H S H µ H S H Π H µ H S H, = µ H S H µ H S H, = µ H (S H S H ). Hence, S H (t) SH [S H S H(0)]e µ H(t), and, it follows that either S H (t) approaches SH asymptotically, or there is some finite time after which S H(t) SH (note that < 0 if S H (t) > SH ). Further, ds H 333 dt ds V dt = Π V λ V S V µ V S V Π V µ V S V, = µ V S V µ V S V, = µ V (S V S V ). Proof. It was shown (Lemma 1) that the region D is positively-invariant and attracting. Using the approach in [43], we show that the set D is also positively-invariant and attracting. Assuming N H (0) Π H µh and N V (0) Π V µ V, it follows that SH = Π H µh and SV = Π V µ V at the (DFE). Applying these bounds to the equations for S H and S V in (6) gives 334 Hence, S V (t) SV [S V S V (0)]e µ V (t). Thus, either S V (t) approaches SV asymptoti- cally, or S V (t) becomes, and remains, less than SV V 335 dt < 0 if 336 S V (t) > SV ). Thus, the region D is positively-invariant and attracting. 337 We claim the following result Theorem 4 The DFE, E 0, of the mass action model, given by (6) with (18), is globally asymptotically stable (GAS) in D if R m

15 The proof is based on using the following Lyapunov function (see Appendix A for the detailed calculations): F = g 1 E H + g 2 I H + g 3 E V + g 4 I V, where, g 1 = Π V µ H Q 4 C HV (η V Q 4 + σ V )(η H Q 2 + σ H ), g 2 = Π V µ H Q 1 Q 4 C HV (η V Q 4 + σ V ), g 3 = µ V µ H Q 1 Q 2 Q 4 R m 0 (η V Q 4 + σ V ), g 4 = µ V µ H Q 1 Q 2 Q 3 Q 4 R m 0. The above result shows that, for the mass action model (given by basic model (6) with (18)), dengue disease can be eliminated from the community if the associated threshold quantity, R m 0, can be brought to a value less than unity (that is, unlike in the basic model (6), where standard incidence formulation is used, the classical epidemiological requirement of R m 0 < 1 is both necessary and sufficient for dengue elimination from the community if mass action incidence is used in the model formulation). The result in Theorem 4 is presented in the context of backward bifurcation as below. Theorem 5 The mass action model, given by (6) with (18), does not undergo backward bifurcation. Proof. It follows from Theorem 3, where no endemic equilibrium exists whenever R m 0 1; and Theorem 4, where E 0 is GAS whenever R m 0 1. Thus, the substitution of standard incidence with mass action incidence in the basic dengue model (6) removes the backward bifurcation phenomenon of the model. Since data strongly suggests that standard incidence formulation is more suited for modelling human diseases [1, 2, 34], the above result shows that the phenomenon of backward bifurcation is an important, and perhaps inherent, property of dengue transmission dynamics. It should be mentioned that a similar situation was reported by Sharomi et al. [43] for some HIV transmission models that undergo backward bifurcation in the presence of an imperfect vaccine A Vaccination Model for Dengue As stated earlier, although there is no effective vaccine for dengue at the moment, efforts are underway to develop one [6, 7, 13, 14, 22, 35, 37, 42, 49, 50]. Further, it is expected that any future dengue vaccine would be imperfect (that is, it would not offer 100% protection against infection in all people [14]). Thus, it is instructive to assess, 14

16 via mathematical modelling, the potential impact of an imperfect dengue vaccine. This is done below by extending the model (6) to incorporate such a vaccine. 3.1 Model Formulation The basic model (6) is extended to include a population of vaccinated individuals, denoted by P H (t) (so that the total human population, N H, is now given by N H = S H +P H +E H +I H +R H ). This population is generated by the vaccination of susceptible individuals (at a rate ξ). Since the vaccine is assumed to be imperfect, vaccinated individuals acquire infection at a rate λ H (1 ɛ), where 0 < ɛ < 1 is the vaccine efficacy. It is assumed that the vaccine wanes (at a rate ω). Using these assumptions and definitions, together with the equations in (6), gives the following vaccination model for dengue transmission dynamics 379 ds H = Π H + ωp H ξs H λ H S H µ H S H, dt dp H = ξs H λ H (1 ɛ)p H ωp H µ H P H, dt de H = λ H [S H + P H (1 ɛ)] σ H E H µ H E H, dt di H = σ H E H τ H I H µ H I H δ H I H, dt dr H = τ H I H µ H R H, dt ds V = Π V λ V S V µ V S V, dt de V = λ V S V σ V E V µ V E V, dt di V = σ V E V µ V I V δ V I V. dt The dynamics of the vaccination model (21) will be considered in the following regions (21) Γ = { (S H, P H, E H, I H, R H, S V, E V, I V ) R 8 + : N H Π H, N V Π } V, µ H µ V 380 and, Γ = {(S H, P H, E H, I H, R H, S V, E V, I V ) D : S H S H, P H P H, S V S V } As in Section 2.4.2, the region Γ can be shown to be positively-invariant and attracting. The DFE of the vaccination model (21) is given by 15

17 E v 0 = (S H, P H, E H, I H, R H, S V, E V, I V ) = ( SH, PH, 0, 0, 0, Π ) V, 0, 0, (22) µ V where, S H = (µ H + ω)π H µ H (µ H + ω + ξ), P H = ξπ H µ H (µ H + ω + ξ). Further, the associated next generation matrices are given by F = C HV η V [S H +P H (1 ɛ)] N H C HV [S H +P H (1 ɛ)] N H C HV S V η H N H C HV S V N H µ H + σ H σ V = H µ H + δ H + τ H µ V + σ V σ V µ V + δ V For computational convenience, let K 1 = µ H +ξ, K 2 = µ H +ω, K 3 = µ H +σ H, K 4 = µ H + δ H + τ H, K 5 = µ V + σ V and K 6 = µ V + δ V. It follows that the vaccinated reproduction number, denoted by R vac, is given by,, R vac = ρ(f V 1 ) = CHV 2 Π V µ 2 H (η HK 4 + σ H )(η V K 6 + σ V )[SH + P H (1 ɛ)] Π 2 H µ. V K 3 K 4 K 5 K The following result is established (using Theorem 2 of [47]). Lemma 5 The DFE of the model with vaccination (21), E0, v is LAS if R vac < 1 and unstable if R vac > 1. The threshold quantity, R vac, represents the average number of secondary cases that one case can produce if introduced into a human community where a fraction of the susceptible population has been vaccinated. 3.2 Threshold Analysis and Vaccine Impact Since a future dengue vaccine is expected to be imperfect, it is instructive to determine whether or not its widespread use in a community will always be beneficial (or not). 16

18 The effect of such a vaccine on the disease dynamics can, first of all, be qualitatively assessed by differentiating the expression for R vac with respect to fraction of individuals vaccinated at steady-state (p = PH /N H ). Considering R vac as a function of p (that is, R vac = R vac (p)), it can be shown that R vac p = R 0ɛ 2(1 pɛ). Since 0 < ɛ < 1, it follows that R vac / p < 0 for 0 p < 1. Thus, R vac is a decreasing function of p. Since a reduction in reproduction number is well-known to signify reduction in disease burden (measured in terms of new cases, hospitalization and mortality), the above analysis shows that an imperfect dengue vaccine would have a positive impact (in reducing disease burden) for any p > 0 and ɛ > 0 (that is, for a given vaccine efficacy, ɛ > 0, vaccinating any fraction of susceptible individuals at steady-state would result in a decrease in disease burden). Furthermore, there is a unique p c such that R vac (p c ) = 1 given by p c = 1 ɛ [1 1R0 ] Lemma 6 The DFE of the model with vaccination (21), E0, v is LAS if p > p c, and unstable if p < p c. Proof. It is clear that R vac < 1 when p > p c, so that the DFE E0 v is LAS (Lemma 5). Similarly, R vac > 1 for p < p c and the DFE is unstable. This lemma clearly implies that if the fraction of individuals vaccinated at steadystate exceeds the threshold level (p c ), then the DFE is locally asymptotically stable; and, consequently, the disease could be eliminated in this case (Lemma 5; although, as shown in Section 3.3, the presence of backward bifurcation in the vaccination model (21) requires smaller values of R vac < 1 (so that the condition for backward bifurcation, given by a > 0; with a defined in (30), is not satisfied) to guarantee such elimination. Figure 3 shows the simulation of the vaccination model (21) for values of p > p c and p < p c to illustrate the result given in Lemma 6. The figure shows a decrease in the number of dengue cases for p > p c and a corresponding increase for the case p < p c. An alternative approach for assessing the impact of the imperfect vaccine entails re-writing the effective reproduction number ( R vac ) as [ R 2 vac = R P ( )] H 1 R2 ov, (23) NH R 2 0 where R ov is the reproduction number for a dengue model with a wholly-vaccinated population (model in which every member of the population is vaccinated), given by 17

19 R ov = C 2 HV Π V µ H (η H K 4 + σ H )(η V K 6 + σ V )(1 ɛ) Π H µ V K 3 K 4 K 5 K 6. Using the notations in [8, 23], a measure of the vaccine impact for the model (21) is defined in terms of an vaccine impact factor, denoted by Ω, where We claim the following result. Ω = P ( ) H 1 R2 ov. (24) NH R 2 0 Theorem 6 The use of an imperfect dengue vaccine will have (i) positive impact in the community if Ω > 0 (R vac < R 0 ); (ii) no impact if Ω = 0 (R vac = R 0 ); (iii) negative impact if Ω < 0 (R vac > R 0 ). Proof. Using (24) in (23) gives R 2 vac = R 2 0(1 Ω), so that Ω = R2 vac. R 2 0 Thus, whenever R vac < R 0, then 1 Ω < 1; so that Ω > 0 (and the vaccine has positive impact). Similarly, whenever R vac > R 0, then 1 Ω > 1; so that Ω < 0 (and the vaccine would have negative impact in this case). Finally, if R vac = R 0, then 1 Ω = 1; so that Ω = 0 (and the vaccine has no impact). It is easy to show that (noting that Q 1 = K 3, Q 2 = K 4, Q 3 = K 5, and Q 4 = K 6 ), R 2 0 R 2 ov = C2 HV Π V µ H (η H K 4 + σ H )(η V K 6 + σ V )ɛ Π H µ V K 3 K 4 K 5 K 6 > 0, so that R 0 > R ov. Thus, it follows from (24) that Ω is always positive. In other words, the vaccine will always have positive impact. This result is summarized below. Lemma 7 The dengue vaccine will always have positive impact. Although the above result is intuitively expected, it should be mentioned that the use of vaccination program can have detrimental impact especially if their widespread use induce increase in risky behavior (see, for instance, [23]). This result is illustrated by simulating the vaccine model (21) using a reasonable set of parameter values (chosen within the ranges in Table 1). A plot of the vaccine impact (Ω) as a function of dengue 18

20 prevalence is depicted in Figure 6, from which it is evident that prevalence decreases with increasing Ω (and greater reduction in prevalence is recorded for increasing values of Ω). It is worth noting also that R vac Ω = R Ω < 0, for 0 < Ω < 1. Thus, the effective reproduction number decreases with increasing vaccine impact (Ω). Consequently, increasing the value of Ω will lead to a corresponding decrease in disease burden. 3.3 Backward Bifurcation Analysis It is instructive to determine whether the vaccination model (21), like the basic model (6), also undergoes the phenomenon of backward bifurcation under certain conditions. However, the method used to establish backward bifurcation in the basic model (6) may not be easily applied here (its application results in higher order degree polynomials, making the associated Routh-Hurwitz computations less tractable). Consequently, a different method, based on the use of Centre Manifold theory [11, 12, 47], will be used to investigate the possibility of backward bifurcation in the vaccination model (21). We claim the following result (the proof is given in Appendix B): Theorem 7 The vaccination model (21) exhibits backward bifurcation whenever the coefficient a, given by equation (30), is positive. The backward bifurcation phenomenon of (21) is illustrated, using a suitable set of parameter values (satisfying the condition a > 0), in Figure 4. Furthermore, a contour plot of the reproduction threshold R vac, as a function of vaccine efficacy (ɛ) and fraction of individuals vaccinated at steady state (p = P H ), is depicted in Figure 5. This figure NH is generated using a set of biologically-plausible parameter values, mostly obtained from the literature (as tabulated in Table 1). The contours show a marked decrease in vaccination reproduction number (R vac ) with increasing vaccine efficacy (ɛ) and fraction of individuals vaccinated (p). Significantly high efficacy and vaccine coverage are needed to effectively control the disease in the community (i.e., achieve R vac < 1). In particular, even if 90% of susceptible individuals are vaccinated at steady state, an efficacy level of at least 90% would be needed to effectively control the spread of dengue in a community. It should be emphasized here that, owing to the phenomenon of backward bifurcation in the vaccination model (21), dengue elimination when R vac < 1 would depend on the initial sizes of the sub-populations of the vaccination model. 19

21 Vaccination Model with Mass Action Incidence As in the basic model (6), the objective here is to check whether the dengue vaccination model (21) will exhibit backward bifurcation if mass action incidence is used instead of the standard incidence function. Consider, now, the vaccination model with mass action incidence, given by (21) with (18). This (mass action vaccination) model has the same DFE (E0 V ) as the model (21). The associated next generation matrices, Fm p and Vm, p are, respectively, given by 0 0 C HV η V [SH + P H (1 ɛ)] C HV [SH + P H (1 ɛ)] Fm p = C HV SV η H C HV SV 0 0, K Vm p σ = H K K σ V K 6 For this model, the vaccination reproduction number, denoted by R m vac, is given by R m vac = CHV 2 Π V (η H K 4 + σ H )(η V K 6 + σ V )[SH + P H (1 ɛ)], µ V K 3 K 4 K 5 K 6 so that the following local stability result is established (using Theorem 2 of [47]). Lemma 8 The DFE, E0 V, of the mass action vaccination model, (21) with (18), is LAS if R m vac < 1, and unstable if R m vac > Non-existence of endemic equilibria for R m vac 1 We claim the following (the proof is given in Appendix C): Theorem 8 The vaccination model with mass action, given by (21) with (18), has no endemic equilibrium when R m vac 1, and has a unique endemic equilibrium otherwise. As in Section 2.4.1, the result above indicates the impossibility of backward bifurcation in the mass action model, since it has no endemic equilibria when R m vac 1. A global stability result for the DFE (E0 V ) is given below Global stability of the DFE As in Section 2.4.2, the dynamics of the mass action model, (21) with (18) will be considered in the positively-invariant and attracting region, Γ. 20

22 Theorem 9 The DFE of the mass action vaccination model, (21) with (18), is GAS in Γ whenever R m vac 1. The proof is based on using the following Lyapunov function (see Appendix D for the detailed calculations): F = C HV S V (η HK 4 + σ H )(η V K 6 + σ V ) R m vack 3 K 4 K 5 E H + C HV S V (η V K 6 + σ V ) R m vack 4 K 5 I H + (η V K 6 + σ V ) K 5 E V + I V Hence, the following result is established. Theorem 10 The mass action vaccination model, given by (21) with (18), does not undergo backward bifurcation. Proof. It follows from Theorem 7, where no endemic equilibrium exists whenever R m vac 1; and Theorem 8, where E0 v is GAS in Γ whenever R m vac Conclusions This paper presents a deterministic model for the transmission dynamics of a single strain of dengue disease. The model, which realistically adopts a standard incidence formulation, allows dengue transmission by exposed humans and vectors. The model was extended to include an imperfect vaccine for dengue. The two models, together with their mass action equivalents, were rigorously analysed to gain insights into their qualitative dynamics. The following results were obtained: (i) The basic (vaccination-free) model has a locally-stable disease-free equilibrium whenever the associated reproduction number is less than unity. (ii) The basic model, with standard incidence formulation, undergoes the phenomenon of backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. The backward bifurcation property can be removed by replacing the standard incidence function in the model with a mass action formulation. Since data suggests that standard incidence is more suited for modelling human diseases than mass action incidence, this study suggests that the phenomenon of backward bifurcation is, perhaps, an inherent feature of dengue transmission dynamics. This feature has direct consequence on whether or not dengue disease can be controlled even when the associated reproduction number (R 0 ) is less than unity; (iii) The vaccination model, like the vaccination-free model, undergoes backward bifurcation (which can be removed as in the case of the basic model); 21

23 (iv) An imperfect dengue vaccine would always have positive epidemiological impact in the community (by reducing disease burden), and the degree of its impact is dependent on the size of (a) the fraction of individuals vaccinated at steady-state or (b) a certain quantity defined as vaccine impact factor (Ω); (v) Simulations of the vaccination model show that the use of an imperfect vaccine can result in the effective control of dengue in a community provided the vaccine efficacy and vaccine coverage are high enough (at least 90% each). Acknowledgments One of the authors (ABG) acknowledges, with thanks, the support in part of the Natural Science and Engineering Research Council (NSERC) and Mathematics of Information Technology and Complex Systems (MITACS) of Canada. SMG acknowledges the support, in part, of the Department of Mathematics and the Institute of Industrial Mathematical Sciences, University of Manitoba. The authors are grateful to O. Sharomi and C.N. Podder (Manitoba) for their useful comments in the preparation of the manuscript. The authors are grateful to the two anonymous Reviewers and the Handling Editor for their constructive comments, which have enhanced the paper. 22

24 Appendix A: Proof of Theorem 4 Proof. Consider the Lyapunov function F = g 1 E H + g 2 I H + g 3 E V + g 4 I V, 565 where, g 1 = Π V µ H Q 4 C HV (η V Q 4 + σ V )(η H Q 2 + σ H ), g 2 = Π V µ H Q 1 Q 4 C HV (η V Q 4 + σ V ), g 3 = µ V µ H Q 1 Q 2 Q 4 R m 0 (η V Q 4 + σ V ), g 4 = µ V µ H Q 1 Q 2 Q 3 Q 4 R m The Lyapunov derivative is given by (where a dot represents differentiation with respect to t) F = g 1 Ė H + g 2 I H + g 3 E V + g 4 IV, = Π V µ H Q 4 C HV (η H Q 2 + σ H )(η V Q 4 + σ V )[C HV (η V E V + I V )S H Q 1 E H ] + Π V µ H Q 1 Q 4 C HV (η V Q 4 + σ V )(σ H E H Q 2 I H ) + µ V µ H Q 1 Q 2 Q 4 R m 0 (η V Q 4 + σ V )[C HV (η H E H + I H )S V Q 3 E V ] + µ V µ H Q 1 Q 2 Q 3 Q 4 R m 0 (σ V E V Q 4 I V ), = [ Π V µ H Q 1 Q 4 C HV (η H Q 2 + σ H )(η V Q 4 + σ V ) + Π V µ H Q 1 Q 4 C HV σ H (η V Q 4 + σ V ) + µ V µ H Q 1 Q 2 Q 4 C HV η H R m 0 (η V Q 4 + σ V )S V ]E H + [ Π V µ H Q 1 Q 2 Q 4 C HV (η V Q 4 + σ V ) + µ V µ H Q 1 Q 2 Q 4 C HV R m 0 (η V Q 4 + σ V )S V ] I H + [Π V µ H Q 4 CHV 2 η V (η H Q 2 + σ H )(η V Q 4 + σ V )S H µ V µ H Q 1 Q 2 Q 3 Q 4 R m 0 (η V Q 4 + σ V ) + µ V µ H σ V Q 1 Q 2 Q 3 Q 4 R m 0 ]E V + [ Π V µ H Q 4 CHV 2 (η H Q 2 + σ H )(η V Q 4 + σ V )S H µ V µ H Q 1 Q 2 Q 3 Q 2 4R m ] 0 IV, = µ H Q 1 Q 4 C HV (η V Q 4 + σ V )[ Π V (η H Q 2 + σ H ) + Π V σ H + µ V Q 2 η H R m 0 S V ]E H µ H Q 1 Q 2 Q 4 C HV (η V Q 4 + σ V )[Π V µ V R m 0 S V ]I H { + µ H ΠV CHV 2 η V Q 4 (η H Q 2 + σ H )(η V Q 4 + σ V )S H + µ V Q 1 Q 2 Q 3 Q 4 R m 0 [σ V (η V Q 4 + σ V )] } E V + µ H Q 4 [Π V CHV 2 (η H Q 2 + σ H )(η V Q 4 + σ V )S H µ V Q 1 Q 2 Q 3 Q 4 R m 0 ]I V, Π V µ H η H Q 1 Q 2 Q 4 C HV (η V Q 4 + σ V )(R m 0 1)E H + µ H Π V Q 1 Q 2 Q 4 C HV (η V Q 4 + σ V )(R m 0 1)I H [ + µ H µv Q 1 Q 2 Q 3 η V Q 2 4(R m 0 ) 2 µ V Q 1 Q 2 Q 3 η V Q 2 4R m ] 0 EV + µ H Q 4 [µ V Q 1 Q 2 Q 3 Q 4 (R m 0 ) 2 µ V Q 1 Q 2 Q 3 Q 4 R m 0 ]I V, since S H S H and S V S V, = η H Π V µ H C HV Q 1 Q 2 Q 4 (η V Q 4 + σ V )(R m 0 1)E H + Π V µ H C HV Q 1 Q 2 Q 4 (η V Q 4 + σ V )(R m 0 1)I H + η V µ V µ H Q 1 Q 2 Q 3 Q 2 4R m 0 (R m 0 1)E V + µ V µ H Q 1 Q 2 Q 3 Q 2 4R m 0 (R m 0 1)I V, = µ H Q 1 Q 2 Q 4 [Π V C HV (η V Q 4 + σ V )(η H E H + I H ) + µ V Q 3 Q 4 R m 0 (η V E V + I V )] (R m 0 1). 23

25 Thus, F 0 if R m 0 1 with F = 0 if and only if E H = I H = E V = I V = 0. Further, 569 the largest compact invariant set in 570 {(S V, E H, I H, R H, S V, E V, I V ) D : F = 0} is the singleton {E 0 }. It follows 571 from the LaSalle Invariance Principle (Chapter 2, Theorem 6.4 of [39]) that every 572 solution to the equations in (6) with initial conditions in D converge to DFE E 0 as 573 t. That is, (E H (t), I H (t), E V (t), I V (t)) (0, 0, 0, 0) as t. Substituting 574 E H = I H = E V = I V = 0 in to the first and the fifth equations of the basic model (6) gives S H (t) SH and S V (t) SV as t. Thus, 576 [S H (t), E H (t), I H (t), R H (t), S V (t), E V (t), I V (t)] (SH V, 0, 0) as t 577 for R m 0 1, so that E 0 is GAS in D if R m Appendix B: Backward Bifurcation in Vaccination Model (21) In this Appendix, we shall give the proof of Theorem 6, using Centre Manifold theory. To apply this method, the following simplification and change of variables are made on the vaccination model (21) first of all. Let S H = x 1, P H = x 2, E H = x 3, I H = x 4, R H = x 5, S V = x 6, E V = x 7, and I V = x 8, so that N H = x 1 + x 2 + x 3 + x 4 + x 5 and N V = x 6 + x 7 + x 8. Further, by using the vector notation X = (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8 ) T, the vaccination model (21) can be written in the form dx = (f dt 1, f 2, f 3, f 4, f 5, f 6, f 7, f 8 ) T as follows 586 dx 1 dt = f C HV (η V x 7 + x 8 ) 1 = Π H x 1 K 1 x 1 + wx 2, x 1 + x 2 + x 3 + x 4 + x 5 dx 2 dt = f C HV (η V x 7 + x 8 ) 2 = ξx 1 (1 ɛ)x 2 K 2 x 2, x 1 + x 2 + x 3 + x 4 + x 5 dx 3 dt = f C HV (η V x 7 + x 8 ) 3 = [x 1 + x 2 (1 ɛ)] K 3 x 3, x 1 + x 2 + x 3 + x 4 + x 5 dx 4 dt = f 4 = σ H x 3 K 4 x 4, dx 5 dt = f 5 = τ H x 4 µ H x 5, dx 6 dt = f C HV (η H x 3 + x 4 ) 6 = Π V x 6 µ V x 6, x 1 + x 2 + x 3 + x 4 + x 5 dx 7 dt = f C HV (η H x 3 + x 4 ) 7 = x 6 K 5 x 7, x 1 + x 2 + x 3 + x 4 + x 5 dx 8 dt = f 8 = σ V x 7 K 6 x 8, with the forces of infection given by (25) 24