MATHEMATICAL ANALYSIS OF A TB TRANSMISSION MODEL WITH DOTS

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17 Number 1 Spring 2009 MATHEMATICAL ANALYSIS OF A TB TRANSMISSION MODEL WITH DOTS S. O. ADEWALE C. N. PODDER AND A. B. GUMEL ABSTRACT. The paper presents a deterministic model for the transmission dynamics of Mycobacterium tuberculosis (TB) in a population in the presence of Directly Observed Therapy Short-course (DOTS). The model which allows for the detection and treatment of individuals with symptoms and uses standard incidence function for the infection rate is rigorously analysed to gain insight into its dynamical features. The analysis reveals that the model undergoes a backward bifurcation where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the associated reproduction threshold (R d ) is less than unity. This phenomenon resulted due to the exogenous re-infection property of TB disease. It is shown that in the absence of such re-infection the model has a globally-asymptotically stable DFE when R d is less than unity. Further the model has a unique endemic equilibrium for a special case whenever the associated threshold quantity exceeds unity. This endemic equilibrium is shown to be globally-asymptotically stable for a special case using a nonlinear Lyapunov function of Goh-Volterra type. The model provides a reasonable fit to a data set for the TB transmission data in Nigeria. Additional numerical simulations show that exogenous re-infection increases the cumulative number of new TB cases regardless of whether or not the DOTS program is implemented (but the corresponding cumulative number of new cases decreases for the case with DOTS program in comparison to the case without DOTS). 1 Introduction Tuberculosis (TB) an airborne-transmitted bacterial disease caused by Mycobacterium tuberculosis remains one of the most important global public health challenges for decades. In addition to affecting at least one-third of the human population (2 billion people) TB is the second greatest contributor of adult mortality amongst Keywords: tuberculosis DOTS equilibria reproduction number stability. Copyright c Applied Mathematics Institute University of Alberta. 1

2 2 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL infectious diseases (causing at least 2 million deaths a year globally) [ ]. Over 80% of all TB patients live in 22 countries mostly in sub-saharan Africa and Asia [20]. Owing to the rising TB mortality and infection rates (especially in developing countries) the World Health Organization (WHO) declared TB as a global public health emergency in 1993 [29 30]. Over the years a number of global initiatives spear-headed by WHO were embarked upon with the hope of minimizing the burden of TB worldwide (in particular to achieve the Millennium Development Goal of halting and beginning to reverse the incidence of TB by 2015). These include the Stop TB Partnership International Standards of Tuberculosis Care and Patient s Care and the Global Plan to Stop TB [20]. A notable medical contribution in TB control was the introduction of antibiotics which resulted in significant decrease in mortality (for instance a 70% reduction in TB-related mortality was recorded in the USA between 1945 to 1955 [ ]). As a consequence of this development TB-infected people can be effectively treated using multiple drugs via the Directly Observed Therapy Short-course (DOTS) strategy [5]. However if not strictly complied to or administered wrongly such therapy may lead to the evolution and development of multi-drug resistant TB (MDR-TB) [7 9]. Numerous mathematical modelling studies have been carried out to gain insight into the transmission dynamics and control of TB spread in human population (see for instance [ ]). For instance Feng et al. [17] presented a model for TB transmission dynamics with exogenous re-infection. A number of studies notably by Guo and Li [21] and McCluskey [11 12] have provided rigorous global asymptotic stability results of the equilibria of TB models with mass action incidence in the absence of exogenous reinfection (using Lyapunov function theory). Similarly Okuonghae and Korobeinikov [31] also gave global stability results for a mass action model of TB in the presence of exogenous re-infection. The purpose of the current study is to provide a rigorous mathematical analysis of a model for TB spread which uses standard incidence function for the infection rate and allows for the exogenous re-infection of latent and recovered cases in the presence of DOTS. The model to be designed is an extension of some of the models presented in the aforementioned studies. The paper is organized as follows. The model is formulated in Section 2 and is qualitatively analysed in Section 3.

3 TB TRANSMISSION MODEL WITH DOTS 3 2 Model formulation The total homogeneously-mixing population at time t denoted by N(t) is sub-divided into mutually-exclusive compartments of susceptible (S(t)) exposed/latent (E(t)) undetected infectious ( (t)) detected infectious (T d (t)) recovered (R(t)) individuals together with treated individuals who failed treatment (F (t)) so that N(t) = S(t) + E(t) + (t) + T d (t) + R(t) + F (t). The susceptible population is increased by the recruitment of people (either by birth or immigration) into the population (all recruited individuals are assumed to be susceptible) at a rate Π. This population is decreased by infection which can be acquired following effective contact with infectious individuals in the undetected ( ) detected (T d ) or failed treatment (F ) categories at a rate λ given by (1) λ = β( + η d T d + η f F ). N In (1) β represents the effective contact rate (i.e. contact capable of leading to TB infection) η d is a modification parameter that compares the transmissibility of detected infectious individuals in relation to undetected infectious individuals. Since detected individuals are assumed to be taking therapeutic treatment it is intuitive to assume that 0 < η d 1. Similarly 0 < η F 1 is a modification parameter that accounts for the reduced transmissibility of infectious individuals in the failed treatment class (F ) in comparison to those in the undetected infectious class ( ). Finally this population decreases by natural death (at a rate µ). Thus the rate of change of the susceptible population is given by (2) ds = Π λs µs. A fraction ξ of the newly-infected individuals are assumed to show no disease symptoms initially. These individuals (known as slow progressors ) are moved to the exposed class (E). The remaining fraction 1 ξ of the newly infected individuals are assumed to immediately display disease symptoms ( fast progressors ) and are moved to the undetected infectious class ( ). The population of exposed individuals is decreased by the progression of exposed individuals to active TB (at a rate κ) and exogenous re-infection (at a rate ψ e λ where ψ e < 1 accounts for the assumption that exposed individuals have reduced infection rate

4 4 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL in comparison to wholly-susceptible individuals (this is to account for the fact that individuals with latent TB infection have partial immunity against exogenous reinfection [25 37])). It is further reduced by natural death (at the rate µ). Thus (3) de = ξλs ψ eλe (κ + µ)e. The population of undetected infectious individuals is increased by the infection of fast progressors (at the rate (1 ξ)λ) and the development of symptoms by exposed individuals (at the rate (1 ω 1 )κ where ω 1 is the fraction of exposed individuals who develop symptoms and are detected). It is further increased by the exogenous re-infection of exposed individuals (at the rate (1 ω 2 )ψ e λ where ω 2 is the fraction of re-infected exposed individuals who are detected) recovered individuals (at the rate (1 ω 3 )ψ r λ where ω 3 is the fraction of re-infected recovered individuals who are detected). This population is decreased by natural recovery (at a rate σ u ) screening and subsequent detection (at a rate γ u ) natural death (at the rate µ) and disease-induced death (at a rate δ u ). Hence (4) d = (1 ξ)λs + (1 ω 1 )κe + (1 ω 2 )ψ e λe + (1 ω 3 )ψ r λr (σ u + γ u + µ + δ u ). The population of detected infectious individuals increases by the detection of a fraction of exposed individuals who develop disease symptoms (at the rate ω 1 κ) exogenous re-infection of exposed and recovered individuals (at the rates ω 2 ψ e λ and ω 3 ψ r λ respectively) and the detection of undetected individuals (at the rate γ u ). The population is decreased by natural recovery (at a rate σ d ) treatment failure (at a rate τ) natural death (at the rate µ) and disease-induced death (at a rate δ d < δ u ). This gives (5) dt d = ω 1 κe + ω 2 ψ e λe + ω 3 ψ r λr + γ u (σ d + τ + µ + δ d )T d. Individuals in the F compartment are those in whom the DOTS treatment has failed. The treatment failure could be due to a number of reasons such as incomplete compliance to the specified DOTS treatment

5 TB TRANSMISSION MODEL WITH DOTS 5 regimen development of resistance etc. This population is generated by the failure of treatment in detected infectious individuals (at the rate τ). In addition to natural death (at the rate µ) and disease-induced mortality (at the rate δ f ) individuals who recovered naturally can leave this (F ) class and move to the recovered class (at a rate σ f ). Thus (6) df = τt d (σ f + µ + δ f )F. Recovery here means recovery from the illness (not from disease; since TB-infected individuals do not completely clear the bacteria from their system. They instead undergo a long latency period which could last many years [26 27] or even a lifetime). The population of recovered individuals is increased by the natural recovery of undetected and detected infectious individuals together with those who failed treatment (at the rates σ u σ d and σ f respectively). This population is decreased by exogenous re-infection (at the rate ψ r λ) and natural death (at the rate µ). Hence dr (7) = σ u + σ d T d + σ f F ψ r λr µr. Thus in summary the TB transmission model in the presence of DOTS treatment strategy is given by the following system of non-linear differential equations (a flow diagram of the model is depicted in Figure 1; associated variables and parameters are described in Tables 1 and 2). (8) ds = Π λs µs de = ξλs ψ eλe (κ + µ)e d = (1 ξ)λs + (1 ω 1 )κe + (1 ω 2 )ψ e λe dt d + (1 ω 3 )ψ r λr (γ u + σ u + µ + δ u ) = ω 1 κe + ω 2 ψ e λe + ω 3 ψ r λr + γ u (σ d + τ + µ + δ d )T d df = τt d (σ f + µ + δ f )F dr = σ u + σ d T d + σ f F ψ r λr µr.

6 6 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL FIGURE 1: Schematic diagram of the model (8) The essential features of the model (8) are that it: (i) allows for disease transmission by individuals in the undetected ( ) detected (T d ) and failed treatment (F ) classes; (ii) allows for the exogenous re-infection of exposed and recovered individuals (at the rates ψ e λ and ψ r λ respectively) and the endogenous re-activation of exposed individuals (at the rate κ); (iii) allows for slow progression (at the rate ξλ) and fast progression (at the rate (1 ξ)λ) to active disease; (iv) allows for the possibility of treatment failure (at the rate τ); (v) distributes the number of exposed individuals who develop symptoms (either due to re-activation or re-infection) into the undetected and detected infectious classes. The model (8) extends the models in many of the aforementioned studies (such as those presented in [ ]) by including a separate compartment (F ) for treated individuals who failed treatment. Furthermore it extends the studies in [11 21] which were based on using mass

7 TB TRANSMISSION MODEL WITH DOTS 7 Parameter Desciption Π Recruitment rate into the population µ Per capita natural mortality rate β Effective contact rate for TB infection η d η f Modification parameters σ u σ d σ f Recovery rate for individuals in T d and F classes respectively ξ Fraction of newly-infected individuals who are slow progressors ψ e ψ r Exogenous re-infection rate for individuals in E and R classes respectively κ Endogenous re-activation rate for exposed individuals ω 1 Fraction of exposed individuals who are detected ω 2 Fraction of re-infected exposed individuals who develop symptoms and are detected ω 3 Fraction of re-infected recovered individuals who are detected γ u Detection rate for undetected infectious individuals τ Treatment failure rate for detected infectious individuals δ u δ d δ f TB-induced mortality rate for individuals in T d and F classes respectively TABLE 1: Description of parameters of the model (8). action incidence and without exogenous re-infection (it also extends the work of Okuonghae [31] by using standard incidence). Also the model offers additional extensions to many of the earlier models by incorporating the slow and fast progression aspect of TB disease (by splitting the number of new infected individuals into the latent (E) and undetected ( ) classes) and also allowing for the screening and detection of infectious individuals (at the rate γ u ). The study further contributes to the literature by carrying out a detailed rigorous analysis of the model (8). Before carrying out any mathematical analysis of the model (8) it is first of all compared with real data (for TB transmission dynamics in Nigeria). The results obtained shows a reasonably good fit (Figure 2) suggesting that the model (8) is suitable for use to realistically study TB transmission dynamics in human populations. 3 Analysis of the model { } Lemma 1. The closed set D = (S E T d F R) R 6 + : N Π µ is positively-invariant and attracting with respect to the model (8).

8 8 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL Parameter Nominal Value (per year) References π 2000 (per population) [31] µ 0.02 [ ] β 6 [10] ξ 0.7 [33] ψ e ψ r Assumed κ [31] ω [31] ω [33] ω Assumed σ u σ d σ f Assumed γ u 0.2 [38 40] τ 0.9 [10] δ u δ d [31] δ f 0.3 Assumed η d η f Assumed TABLE 2: Parameter values 6 x 104 Total number of infected individuals Time (years) FIGURE 2: Comparison of observed TB data for Nigeria [38 40] (solid line) and model prediction (dotted line). Parameter values used are as given in Table 2 with β = 1.8 ψ e = 0.35 and ψ r = 0.35.

9 TB TRANSMISSION MODEL WITH DOTS 9 Proof. Consider the biologically-feasible region D defined above. The rate of change of the total population obtained by adding all the equations of the model (8) is given by (9) It follows that dn it is clear that N(t) Π µ dn = Π µn δ u δ d T d δ f F. Π µn. Therefore all solutions of the model with initial conditions in D remain in D for all t > 0 (i.e. the ω-limits sets of the system (8) are contained in D). Thus D is positivelyinvariant and attracting. < 0 whenever N > Π dn µ. Further since if N(0) Π µ In the region D the model can be considered as being epidemiologically and mathematically well-posed [22]. 3.1 Disease-free equilibrium (DFE) The model (8) has a DFE obtained by setting the right-hand sides of the equations of the model to zero given by (10) E 0 = (S E T u T d F R ) = ( Π µ The stability of the DFE E 0 will be analysed using the next generation method (see [39]). The non-negative matrix P (of the new infection terms) and the nonsingular M-matrix V (of the remaining transfer terms) are given respectively by 0 ξβ ξβη d ξβη f 0 0 (1 ξ)β (1 ξ)βη d (1 ξ)βη f 0 P = K (1 ω 1 )κ K V = ω 1 κ γ u K τ K σ u σ d σ f µ where K 1 = κ + µ K 2 = γ u + σ u + µ + δ u K 3 = σ d + τ + µ + δ d and K 4 = σ f + µ + δ f. The associated reproduction number denoted by R d ).

10 10 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL is given by R d = ρ(p V 1 ) where ρ denotes the spectral radius (dominant eigenvalue in magnititude) of the next generation matrix P V 1. It follows that with R d = β K 1 K 2 K 3 K 4 (A 1 + A 2 + A 3 ) A 1 = (1 ξ)[k 1 K 4 (K 3 + η d γ u ) + K 1 η f τγ u ] A 2 = ξκ(1 ω 1 )[K 4 (K 3 + η d γ u ) + η f τγ u ] A 3 = ξκω 1 K 2 (η d K 4 + η f τ). Hence the result below follows from Theorem 2 of [39]. Lemma 2. The DFE of the model (8) given by (10) is locally asymptotically stable (LAS) if R d < 1 and unstable if R d > 1. The threshold quantity R d is the reproduction number for the model. It measures the average number of new TB infections generated by a single infectious individual in a population where some of the infected individuals are offered DOTS treatment. The epidemiological implication of Lemma 2 is that TB spread can be effectively controlled in the community (when R d < 1) if the initial sizes of the sub-populations of the model are in the basin of attraction of the disease-free equilibrium E 0. Since TB models with exogenous re-infection and standard incidence function are often shown to exhibit the phenomenon of backward bifurcation (see for instance [ ] and some of the references therein) where the stable DFE co-exists with a stable endemic equilibrium when the associated reproduction threshold (R d ) is less than unity it is instructive to determine whether or not the model (8) also exhibits this dynamical feature. This is investigated below. Theorem 1. The model (8) undergoes a backward bifurcation at R d = 1 if the inequality (34) holds. The proof of Theorem 1 which is based on the use of centre manifold theory [ ] is given in Appendix A. The backward bifurcation phenomenon of the model (8) is numerically illustrated in

11 TB TRANSMISSION MODEL WITH DOTS 11 (A1) (A2) (A3) (A4) (A5) (A6) FIGURE 3: Simulations of the model (31) showing backward bifurcation diagrams for populations of (A1) susceptible individuals (S(t)) (A2) exposed individuals (E(t)) (A3) undetected active TB individuals ((t)) (A4) detected active TB individuals (T d (t)) (A5) individuals who failed treatment (F (t)) and (A6) recovered individuals (R(t)). Parameter values used are as given in Table 2 with β = 0.5 τ = γ u = 0.4 ψ e = 10 and ψ r = 10 (so that R d = a = and b = ).

12 12 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL Figure 3 by simulating the model (8) with parameter values such that the inequality (34) is satisfied. It should be noted that in the absence of re-infection (i.e. when ψ e = ψ r = 0) the backward bifurcation coefficient a given in (33) reduces to a = 2βµ π (w 3 + η d w 4 + η f w 5 )[v 3 (1 ξ) + ξv 2 ](w 2 + w 3 + w 4 + w 5 + w 6 ) < 0 since all the model parameters and the eigenvectors w i (i = ) and v i (i = ) are non-negative and 0 < ξ < 1. Thus since the inequality (34) does not hold in this case the model (8) will not undergo backward bifurcation in the absence of exogenous re-infection (this result is consistent with earlier studies on TB transmission dynamics; see for instance [33]). This result is summarized below. Lemma 3. The model (8) does not undergo backward bifurcation at R d = 1 in the absence of exogenous re-infection (i.e. when ψ e = ψ r = 0). 3.2 Global stability of the DFE the of model The result given in Lemma 3 (discounting the possibility of backward bifurcation when ψ e = ψ r = 0) suggests that the DFE E 0 of the model (8) may be globally-asymptotically stable (GAS) when R d < 1 and ψ e = ψ r = 0. This is explored below. Theorem 2. The DFE E 0 of the model (8) with ψ e = ψ r = 0 is GAS in D if R d 1. Proof. Consider the Lyapunov function where F = f 1 E + f 2 + f 3 T d + f 4 F f 1 = κ {(1 ω 1 )[K 4 (K 3 + η d γ u ) + η f τγ u ] + ω 1 K 2 (η d K 4 + η f τ)} f 2 = K 1 [K 4 (K 3 + η d γ u ) + η f τγ u ] f 3 = K 1 K 2 (η d K 4 + η f τ) f 4 = K 1 K 2 K 3 K 4 η f

13 TB TRANSMISSION MODEL WITH DOTS 13 with Lyapunov derivative given by (where a dot represents differentiation with respect to t) F = f 1 Ė + f 2 T u + f 3 T d + f 4 F = f 1 [ ξβ(tu + η d T d + η f F )S N + f 2 [ (1 ξ)β(tu + η d T d + η f F )S N + f 4 (τt d K 4 F ) ( ) S = K 1 K 2 K 3 K 4 N R d 1 ( ) S + F η f K 1 K 2 K 3 K 4 N R d 1 ] K 1 E + f 3 (ω 1 κe + γ u K 3 T d ) + (1 ω 1 )κe K 2 ] ( ) S + T d η d K 1 K 2 K 3 K 4 N R d 1 ) ( ) K 1 K 2 K 3 K 4 (R d 1 + T d η d K 1 K 2 K 3 K 4 R d 1 ) + F η f K 1 K 2 K 3 K 4 (R d 1 since S N in D. Thus F 0 if R d 1 with F = 0 if and only if = T d = F = 0. Also E 0 as t if = T d = F = 0 (since λ = β(tu+η dt d +η f F ) N = 0 in this case). It follows from LaSalle s Invariance Principle [24] that 0 T d 0 and F 0 as t. Further substituting = T d = F = 0 in the first and last equations in (8) shows that S Π µ and R 0 as t. Thus (S E T d F R) ( Π µ ) as t. Hence since D is positively-invariant it follows that the DFE E 0 of the model (8) is GAS in D if R d 1. The epidemiological implication of Theorem 2 is that in the absence of exogenous re-infection (i.e. ψ e = ψ r = 0) the disease will be eliminated from the community if the DOTS strategy can bring (and maintain) the reproduction threshold (R d ) to a value less than or equal to unity. In other words in the absence of exogenous re-infection the classic epidemiological requirement of having the reproduction threshold less than or equal to unity (R d 1) is necessary and sufficient for TB elimination in a community. In the presence of re-infection however disease elimination when R d 1 is dependent on the initial sizes of the

14 14 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL sub-populations of the model (this is a consequence of the phenomenon of backward bifurcation in this case). Figure 4A depicts the solution profiles of the model (8) with ψ e = ψ r = 0 and R d < 1 showing convergence to the DFE (in line with Theorem 2). 3.3 Existence of endemic equilibrium point (EEP): special case In this section the possible existence and stability of endemic (positive) equilibria of the model (8) (i.e. equilibria where at least one of the infected components of the model is non-zero) will be considered for the special case where exogenous re-infection does not occur (i.e. ψ e = ψ r = 0). Let E 1 = (S E Tu Td F R ) represents any arbitrary endemic equilibrium of the model (8) with ψ e = ψ r = 0. Solving the equations of the system at steady-state gives (11) S = Π λ + µ E = ξλ S K 1 T u T d F = = (1 ξ)λ S + (1 ω 1 )κe K 2 = ω 1κE + γ u Tu K 3 τt d K 4 R = σ utu + σ d Td µ + σ f F. The expression for λ defined in (1) at the endemic steady-state denoted by λ is given by (12) λ = β(t u + η d T d + η f F ) N. For mathematical convenience the expressions in (11) are re-written

15 TB TRANSMISSION MODEL WITH DOTS 15 (A) Total number of infected individuals Time (years) (B) 3.5 x 104 Total number of infected individuals Time (years) FIGURE 4: Simulations of the model (8) showing the total number of infected individuals (E + + T d + F ) as a function of time using the parameters in Table 2 with Π = 4000 µ = 0.2 σ u = 0.5 σ d = 0.5 σ f = 0.5 and γ u = 2. (A) β = 1.5 and ψ e = ψ r = 0 (so that R d = 0.62). (B) β = 3 ψ e = ψ r = 0 δ u = δ d = δ f = 0 ω 1 = 0 and ξ = 1 (so that R d2 = ).

16 16 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL in terms of λ S as follows: (13) E = ξλ S K 1 T u T d = (1 ξ)λ S K 2 = ω 1κξλ S K 1 K 3 + (1 ω 1)ξκλ S K 1 K 2 = P 1 λ S + γ up 1 λ S = P 2 λ S K 3 ] [ F = τ ω 1 κξλ S + γ up 1 λ S K 4 K 1 K 3 K 3 = P 3 λ S R = σ up 1 λ S + σ d P 2 λ S + σ f P 3 λ S µ = P 4 λ S where (14) P 1 = (1 ξ) K 2 P 2 = ω 1κξ + γ u K 1 K 3 K 3 { P 3 = τ ω 1 κξ K 4 + (1 ω 1)ξκ K 1 K 2 [ (1 ξ) K 2 K 1 K 3 + γ u K 3 ] + (1 ω 1)ξκ K 1 K 2 [ (1 ξ) + (1 ω 1)ξκ K 2 K 1 K 2 ]} P 4 = σ up 1 λ S + σ d P 2 λ S + σ f P 3. µ Substituting the expressions in (13) with (14) into (12) gives (15) λ ( S + ξλ S + P 1 λ S + P 2 λ S + P 3 λ S K 1 + P 4 λ S ) = βλ S (P 1 + η d P 2 + η f P 3 ). Dividing each term in (15) by λ S (and noting that at the endemic steady-state λ S 0) gives 1 + P 5 λ = β(p 1 + η d P 2 + η f P 3 )

17 TB TRANSMISSION MODEL WITH DOTS 17 where P 5 = ξ K 1 + P 1 + P 2 + P 3 + P 4 > 0. Thus 1 + P 5 λ = β K 1 K 2 K 3 K 4 (A 1 + A 2 + A 3 ) = R d. Hence 1 + P 5 λ = R d so that (16) λ = R d 1 P 5 > 0 whenever R d > 1. The components of the unique endemic equilibrium (E 1 ) can then be obtained by substituting the unique value of λ given in (16) into the expressions in (11). Thus the following result has been established. Lemma 4. The model (8) with ψ e = ψ r = 0 has a unique endemic equilibrium given by E 1 whenever R d > Local stability of EEP: Special case The local stability of the unique EEP E 1 will now be explored for the special case where the disease-induced mortality is negligible (i.e. δ u = δ d = δ f = 0) no fast progression to active disease (i.e. ξ = 1) and exogenous re-infection does not occur (so that ψ e = ψ r = 0). Setting δ u = δ d = δ f = ψ e = ψ r = 0 in the model (8) shows that (17) dn(t) = Π µn(t). Hence it follows from (17) that N(t) Π/µ = N as t. Further using the substitution S = N E T d F R (and noting that δ u = δ d = δ f = ψ e = ψ r = 0) and ξ = 1 in the model (8) gives the

18 18 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL following reduced model de = β( + η d T d + η f F )(N E T d F R) N K 1 E (18) d = (1 ω 1 )κe K 12 dt d = ω 1 κe + γ u K 13 T d df = τt d K 14 F dr = σ u + σ d T d + σ f F µr where K 12 = γ u + σ u + µ K 13 = σ d + τ + µ and K 14 = σ f + µ. For the reduced model (18) the associated reproduction number denoted by R d1 is given by with R d1 = β K 1 K 12 K 13 K 14 (A 11 + A 12 ) A 11 = κ(1 ω 1 )[K 14 (K 13 + η d γ u ) + η f τγ u ] A 12 = κω 1 K 12 (η d K 14 + η f τ). It can be shown using the approach in Section 3.3 that the system (18) has a unique endemic equilibrium given by E 2 = E 1 δu=δ d =δ f =ψ e=ψ r=0ξ=1 whenever R d1 > 1. Theorem 3. The unique endemic equilibrium E 2 of the reduced model (18) is LAS whenever R d1 > 1. Proof. The proof is based on using the technique in [22] (see also [15 16]) which employs a Krasnoselskii sub-linearity trick. The approach essentially entails showing that the linearization of the system (18) around the equilibrium E 2 has no solutions of the form (19) Z(t) = Z 0 e τt

19 TB TRANSMISSION MODEL WITH DOTS 19 with Z 0 C n \ {0} τ C Z i C and Re (τ) 0. The consequence of this is that the eigenvalues of the characteristic polynomial associated with the linearized method will have negative real part (in which case the unique endemic equilibrium E 2 is LAS). Linearizing the model (18) around the endemic equilibrium E 2 gives de = ( P 11 K 1 )E + (P 12 P 11 ) + (η d P 12 P 11 )T d + (η f P 12 P 11 )F P 11 R (20) d = (1 ω 1 )κe K 12 dt d = ω 1 κe + γ u K 13 T d df = τt d K 14 F dr = σ u + σ d T d + σ f F µr where (21) P 11 = P 12 = β N S. β (T N u + η d Td + η f F ) Substituting a solution of the form (19) into the linearized system of (18) around the equilibrium E 2 gives the following linear system (22) τz 1 = ( P 11 K 1 )Z 1 + (P 12 P 11 )Z 2 + (η d P 12 P 11 )Z 3 + (η f P 12 P 11 )Z 4 P 11 Z 5 τz 2 = (1 ω 1 )κz 1 K 12 Z 2 τz 3 = ω 1 κz 1 + γ u Z 2 K 13 Z 3 τz 4 = τz 3 K 14 Z 4 τz 5 = σ u Z 2 + σ d Z 3 + σ f Z 4 µz 5. Firstly all the negative terms in the last three equations of system (22) are moved to their respective left-hand sides. Each of the last three

20 20 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL equations is then re-written in terms of Z 1 and Z 2 and the results substituted into the remaining equations of the system (22). Finally all the negative terms of the remaining (first two) equations are moved to the right-hand sides. These algebraic manipulations result in the following system: (23) where (24) with and Z 1 [1 + F 1 (τ)] + Z 2 [1 + F 2 (τ)] = (M Z) 1 + (M Z) 2 Z 3 [1 + F 3 (τ)] = (M Z) 3 Z 4 [1 + F 4 (τ)] = (M Z) 4 Z 5 [1 + F 5 (τ)] = (M Z) 5 F 1 (τ) = τ + P 11 K 1 + P 11T 1 K 1 F 2 (τ) = 1 + τ + (1 + T 2)P 11 K 12 K 1 F 3 (τ) = τ F 4 (τ) = τ F 5 (τ) = τ K 13 K 14 µ M = 0 βs N K 1 η d βs N K 1 η f βs N K 1 0 (1 ω 1)κ K ω 1κ K 13 γ u K τ K σ u µ σ d µ σ f µ 0 T 1 = ω 1κ τω 1 κ + τ + K 13 (τ + K 13 )(τ + K 14 ) + σ d ω 1 κ τ + µ τ + K 13 + σ f τω 1 κ τ + µ (τ + K 13 )(τ + K 14 ) γ u τγ u T 2 = + τ + K 13 (τ + K 13 )(τ + K 14 ) + σ u τ + µ + σ d τ + µ + σ f τγ u τ + µ (τ + K 13 )(τ + K 14 ). γ u τ + K 13

21 TB TRANSMISSION MODEL WITH DOTS 21 In the above calculations the notation M(Z) i (with i = ) denotes the ith coordinate of the vector M(Z). It should further be noted that the matrix M has non-negative entries and the equilibrium E 2 satisfies E 2 = ME 2. Furthermore since the coordinates of E 2 are all positive it follows then that if Z is a solution of (23) then it is possible to find a minimal positive real number s such that (25) Z se 2 where Z = ( Z 1 Z 2 Z 3 Z 4 Z 5 ) with the lexicographic order and is a norm in C. The main goal is to show that Re (τ) < 0. Assume the contrary (i.e. Re (τ) 0). We then need to consider two cases: τ = 0 and τ 0. Assume the first case (i.e. τ = 0). Then (22) is a homogeneous linear system in the variables Z i (i = ). The determinant of the system (22) corresponds to that of the Jacobian of the system (18) evaluated at E 2 which is given by (26) = + η d Td + η f F N B 1 ) K 1 K 12 K 13 K 14 µ (1 S N R d1 where B 1 = γ u κ(1 ω 1 )[σ u K 14 + τµ + τσ f + K 14 µ + K 13 K 14 (σ u + µ)] + ψk 12 [ω 1 κ(τσ f + σ d K 14 + K 14 µ + τµ) + K 13 K 14 µ]. By solving (18) at the endemic steady-state E 2 and using the first equation of (18) it can be shown that S N = 1 R d1. Thus < 0. Consequently the system (22) can only have the trivial solution Z = 0 (which corresponds to the DFE E 0 ). Consider next the case τ 0. In this case Re (F i (τ)) 0 (i = ) since by assumption Re (τ) 0. It is easy to see that this implies 1 + F i (τ) > 1 for all i. Define F (τ) = min 1 + F i (τ) (for i = s ) then F (τ) > 1 and hence < s. The minimality of s implies that Z > F (τ) s F (τ) E 2. On the other hand taking norms of both

22 22 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL sides of the second equation of (22) and using the fact that the matrix M is non-negative gives (27) F (τ) Z 3 M( Z ) 3 s(m E 2 ) 3 st u. s F (τ) T Then it follows from the above inequality that Z 3 u which contradicts Re (F i (τ)) 0. Hence Re (τ) < 0 so that the endemic equilibrium E 2 is LAS if R d1 > 1. The epidemiological implication of Theorem 3 is that the disease will persist in the community if the reproduction threshold (R d1 ) exceeds unity. 3.5 Global stability of EEP: Special case The global asymptotic stability of EEP E 2 of the reduced model (18) is considered for the case when there is no detection of exposed individuals (i.e. ω 1 = 0). Setting ω 1 = 0 into the reduced model (18) and using the substitution β 1 = β N it can be shown that the associated reproduction number of the reduced model (18) with ξ = 1 and ω 1 = 0 denoted by R d2 is given by (28) R d2 = β 1κ(K 13 K 14 + η d γ u K 14 + η f τγ u ) K 1 K 12 K 13 K 14. Furthermore using the approach in Section 3.3 it can be shown that the reduced system (18) with ω 1 = 0 has a unique EEP of the form E 3 = E 2 ω1=0 = (S E Tu T d F R ) > 0 and R > 0 when- where S > 0 E > 0 Tu > 0 Td ever R d2 > 1. Let D 0 = {(S E T d F R) D : E = = T d = F = R = 0}. Theorem 4. The unique EEP E 3 of the reduced model (18) with ω 1 = 0 is GAS in D \ D 0 whenever R d2 > 1.

23 TB TRANSMISSION MODEL WITH DOTS 23 Proof. Let ω 1 = 0 and R d2 > 1 so that the EEP E 3 exists. Consider the following non-linear Lyapunov function: (29) L = ( S S S ln S ) ( S + E E E ln E ) E + β 1S [(σ f + µ)(τ + σ d + µ) + η d (µ + σ f )γ u + η f τγ u ] (σ f + µ)(τ + σ d + µ)(γ u + σ u + µ) ( Tu ln T ) u Tu + β 1S (η d (µ + σ f ) + η f τ) (σ f + µ)(τ + σ d + µ) ( T d Td Td ln T ) d Td + β 1S ( η f F F F ln F (σ f + µ) F with Lyapunov derivative given by L = (1 S S ) Ṡ + (1 E E ) Ė ) + β 1S [(σ f + µ)(τ + µ) + η d (µ + σ f )γ u + η f τγ u ] (σ f + µ)(τ + σ d + µ)(γ u + σ u + µ) (1 ) T u + β 1S [η d (µ + σ f ) + η f τ] (1 T ) d T d + β 1S η f (σ f + µ)(τ + σ d + µ) T d µ + σ f (1 F ) F = (1 S S F ) [β 1 (Tu + η d Td + η f F )S + µs β 1 ( + η d T d + η f F )S µs] ) + (1 E [β 1 ( + η d T d + η f F )S (κ + µ)e] E

24 24 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL + β 1S [(σ f + µ)(τ + σ d + µ) + η d (µ + σ f )γ u + η f τγ u ] (σ f + µ)(τ + σ d + µ)(γ u + σ u + µ) (1 ) [κe (γ u + σ u + µ) ] + β 1S [η d (σ f + µ) + η f τ] (σ f + µ)(τ + σ d + µ) (1 T d ) [γ u (τ + σ d + µ)t d ] T d + β 1S η f µ + σ f (1 F F ) [τt d (σ f + µ)f ] = µs (2 S S S S ) + 2β 1(Tu + η dtd + η f F )S β 1 (T u + η d T d + η f F ) S2 S + β 1 ( + η d T d + η f F )S β 1 (T u + η d T d + η f F ) S E E β 1 ( + η d T d + η f F ) SE E β 1 S β 1 S E T E + β 1 S η d T d u 2 E E β 1S η d Td + β 1S T u + β 1 S T u Tu E E β 1 S η d T d + β 1 S η d T d + β 1 S η f F E E β 1S η f F T u β 1 S η f F E Tu E β 1 S η d T d β 1 S η d T u β 1 S η f T d Td + β 1 S Tu η f F Td 2 T d F β 1 S η f T u + β 1 S η d T d + β 1 S T d η f F β 1 S η f F β 1 S η f Td E Tu E + β 1 S η d T d Tu β 1 S η f T d T d F + β 1 S η f F Tu F T d F 2 Td F + β 1S η f F

25 TB TRANSMISSION MODEL WITH DOTS 25 = µs ( S E S E S E S E 2 S S S S ) T u T d T d T d F Td F S E S E ) + β 1 S T u ( 3 S S E Tu E ( + β 1 S η d Td 4 S S E Tu E ) ( + β 1 S η f F 5 S S E Tu E ) F F. T d T d T d T d T u T u Since the arithmetic mean exceeds the geometric mean it follows then that 5 S S E Tu E 3 S S E 4 S S E Tu E T d T d T u Tu E T d T d 2 S S S S 0 S E S E Tu 0 Tu S E S E T d F Td F S E S E T d Td 0 F 0 F so that L 0 for R d2 > 1. Hence L is a Lyapunov function of the system (18) with ω 1 = 0 on D \ D 0. In other words lim ( T d F ) = t (Tu Td F ). Setting = Tu T d = Td and F = F in the equation for dr/ in (18) shows that R R as t. Thus by the Lyapunov function L and LaSalle s Invariance Principle [24] every solution to the equations in the reduced model (18) with ω 1 = 0 approaches E 3 as t for R d2 > 1. Figure 4B depicts the solution profiles of the reduced model (18) converging to the EEP when R d2 > 1 (in line with Theorem 4). The effect of exogenous re-infection on TB transmission dynamics is monitored by simulating the model (8) with the parameters in Table 2 in the presence and absence of re-infection. The results obtained depicted in Figure 5 show that re-infection increases the cumulative number of new cases regardless of whether or not DOTS is implemented. It should

26 26 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL be mentioned that the cumulative number of new TB cases is decreased if DOTS program is implemented (see Figure 5A in comparison to the results depicted in Figure 5B). (A) 14 x 104 Cumulative number of new infections with reinfection and DOTS without reinfection and DOTS (B) Time (years) 18 x 104 Cumulative number of new infections with reinfection and no DOTS without reinfection and no DOTS Time (years) FIGURE 5: Simulations of the model (8) showing the cumulative number of new TB infections as a function of time using the parameters in Table 2. (A) in the presence and absence of re-infection with DOTS and (B) in the presence and absence of re-infection without DOTS.

27 TB TRANSMISSION MODEL WITH DOTS 27 Figure 6 shows a decrease in the cumulative number of new TB cases if the undetected TB-infected individuals (in the class) are detected within 6 months (i.e. γ u = 2) the cumulative number of new cases will be about within 20 years. However if the detection rate is further decreased to a month (i.e. γ u = 12) the cumulative number of new cases decreases to about (Figure 6). Cumulative number of new infections 18 x γ u = and 12 respectively (from top to bottom) Time (years) FIGURE 6: Simulations of the model (8) showing the cumulative number of new TB infections as a function of time for various values of γ u. Other parameter values used are as in Table 2. Conclusions A deterministic model for TB transmission dynamics in the presence of DOTS is presented and rigorously analysed. Some of the main findings of the study include: (i) (ii) The model reasonably fits data for TB transmission in Nigeria; The model undergoes the phenomenon of backward bifurcation which is shown to arise due to the exogenous re-infection property of TB disease; (iii) In the absence of re-infection the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold (R d ) is less than unity;

28 28 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL (iv) In the absence of re-infection and disease-induced mortality the model has a unique endemic equilibrium whenever the associated reproduction threshold (R d1 ) exceeds unity. This equilibrium is shown to be locally-asymptotically stable whenever there is no fast progression to active disease (i.e. ξ = 1) in the community. Furthermore if additionally there is no detection of exposed individuals (ω 1 = 0) this endemic equilibrium is globally-asymptotically stable whenever the associated reproduction threshold (R d2 ) is greater than unity; (v) Exogenous re-infection increases the cumulative number of new TB cases regardless of whether or not the DOTS program is implemented (but the number of new cases decreases for the case when DOTS program is implemented); (vi) The cumulative number of new TB cases decreases with increasing detection rate. Appendix A: Proof of Theorem 1 To prove Theorem 1 it is convenient to let S = x 1 E = x 2 = x 3 T d = x 4 F = x 5 and R = x 6 so that N = x 1 +x 2 +x 3 +x 4 +x 5 +x 6. Further by introducing the vector notation x = (x 1 x 2 x 3 x 4 x 5 x 6 ) T the model (8) can be written in the form dx/ = F (x) where F = (f 1 f 2 f 3 f 4 f 5 f 6 ) T as follows: (30) dx 1 dx 2 dx 3 = f 1 = Π = f 2 = β(x 3 + x 4 η d + x 5 η f ) (x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) x 1 µx 1 ξβ(x 3 + x 4 η d + x 5 η f ) (x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) x 1 ψ e β(x 3 + x 4 η d + x 5 η f ) (x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) x 2 (κ + µ)x 2 = f 3 = (1 ξ)β(x 3 + x 4 η d + x 5 η f ) (x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) x 1 + (1 ω 1 )κx 2 + (1 ω 2)ψ e β(x 3 + x 4 η d + x 5 η f ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 2 + (1 ω 3)ψ r β(x 3 + x 4 η d + x 5 η f ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 6 (σ u + γ u + µ + δ u )x 3

29 TB TRANSMISSION MODEL WITH DOTS 29 (30) dx 4 dx 5 dx 6 = f 4 = ω 1 κx 2 + ω 2ψ e β(x 3 + x 4 η d + x 5 η f ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 2 + ω 3ψ r β(x 3 + x 4 η d + x 5 η f ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 6 + γ u x 3 (σ d + τ + µ + δ d )x 4 = f 5 = τx 4 (σ f + µ + δ f )x 5 = f 6 = σ u x 3 + σ d x 4 + σ f x 5 + ψ rβ(x 3 + x 4 η d + x 5 η f ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 6 µx 6. The Jacobian of the system (30) or equivalently (8) at the DFE (E 0 ) is given by µ 0 β βη d βη f 0 0 K 1 ξβ ξβη d ξβη f 0 J(E 0 ) = 0 (1 ω 1 )κ (1 ξ)β K 2 (1 ξ)βη d (1 ξ)βη f 0 0 ω 1 κ γ u K τ K σu σd σf µ from which it can be shown (as before) that (31) R d = ρ(p V 1 ) = β K 1 K 2 K 3 K 4 (A 1 + A 2 + A 3 ) where A i (i = 1 2 3) and K i (i = ) are as defined in Section 3.1. The above linearized system of the transformed system (30) with β = β has a zero eigenvalue which is simple. Hence the Center Manifold theory [6] can be used to analyze the dynamics of (30) near β = β. In particular Theorem 4.1 in [8] reproduced below for convenience will be used. Theorem 5 (Castillo-Chavez and Song [8]). Consider the following general system of ordinary differential equations with a parameter φ dx/ = f(x φ) f : R n R R n andf C 2 (R n R) where 0 is an equilibrium point of the system (that is f(0 φ) 0 for all φ) and

30 30 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL 1. A = D x f(0 0) = ( fi x j (0 0)) is the linearization matrix of the system around the equilibrium 0 with φ evaluated at 0; 2. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts; 3. Matrix A has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let f k be the k th component of f and (32) a = b = n kij=1 n ki=1 2 f k v k w i w j (0 0) x i x j 2 f k v k w i (0 0). x i φ Then the local dynamics of the system around the equilibrium point 0 is totally determined by the signs of a and b. Particularly if a > 0 and b > 0 then a backward bifurcation occurs at φ = 0. Theorem 5 is applied as follows. First of all suppose β is chosen as a bifurcation parameter. Setting R d = 1 from (31) and solving for β gives: β = β = K 1K 2 K 3 K 4 A 1 + A 2 + A 3. Secondly the following computations are carried out. Eigenvectors of J(E 0 ) β=β It can be shown that the Jacobian of the system (30) at β = β (denoted by J(E 0 ) β=β = J β ) has a right eigenvector (corresponding to the zero eigenvalue) given by where w = (w 1 w 2 w 3 w 4 w 5 w 6 ) T w 1 = (βw 3 + βη d w 4 + βη f w 5 ) µ w 2 = free w 3 = free w 4 = ω 1κw 2 + γ u w 3 K 3

31 TB TRANSMISSION MODEL WITH DOTS 31 w 5 = τw 4 K 4 w 6 = σ uw 3 + σ d w 4 + σ f w 5. µ Further the Jacobian J β zero eigenvalue) given by has a left eigenvector (associated with the v = (v 1 v 2 v 3 v 4 v 5 v 6 ) with v 1 = 0 v 2 = (1 ω 1)κv 3 + ω 1 κv 4 K 1 v 3 = free v 4 = free v 5 = ξβη f v 2 + (1 ξ)βη f v 3 K 4 v 6 = 0. Computations of a and b For the system (30) the associated non-zero second partial derivatives of F (at the DFE (E 0 )) are given by 2 f 2 x 2 x 3 = βµ(ξ + ψe) Π 2 f 2 = βη f µ(ξ + ψ e) x 2 x 5 Π 2 f 2 = ξβµ(1 + η d) x 3 x 4 Π 2 f 2 = βη dµ(ξ + ψ e) x 2 x 4 Π 2 f 2 x 3 x 3 = 2ξβµ Π 2 f 2 = ξβµ(1 + η f ) x 3 x 5 Π 2 f 2 = ξβµ x 3 x 6 Π 2 f 2 = βη dµ(ξ + ψ e) x 4 x 2 Π 2 f 2 = ξβµ(1 + η d) x 4 x 3 Π 2 f 2 = 2ξβµη d x 4 x 4 Π

32 32 S. O. ADEWALE C. N. PODDER AND A. B. GUMEL 2 f 2 = ξβµ(η d + η f ) x 4 x 5 Π 2 f 2 = βη f µ(ξ + ψ e) x 5 x 2 Π 2 f 2 = ξβµ(η d + η f ) x 5 x 4 Π 2 f 2 x 4 x 6 = ξβµη d Π 2 f 2 = ξβµ(η f + 1) x 5 x 3 Π 2 f 2 = 2ξβµη f x 5 x 5 Π 2 f 2 x 5 x 6 = ξβµη f Π 2 f 3 x 2 x 3 = βµ(1 ξ ψe + ψeω2) Π 2 f 3 = βη dµ(1 ξ ψ e + ψ eω 2) 2 f 3 = βη f µ(1 ξ ψ e + ψ eω 2) x 2 x 4 Π x 2 x 5 Π 2 f 3 x 3 x 3 = 2βµ(1 ξ) Π 2 f 3 = βµ(1 ξ)(1 + η f ) x 3 x 5 Π 2 f 3 = 2βη dµ(1 ξ) x 4 x 4 Π 2 f 3 = βµ(1 ξ)(1 + η d) x 3 x 4 Π 2 f 3 x 3 x 6 = βµ(1 ξ ψr + ψrω3) Π 2 f 3 = βµ(1 ξ)(η d + η f ) x 4 x 5 Π 2 f 3 = βη dµ(1 ξ ψ r + ψ rω 3) 2 f 3 = 2βη f µ(1 ξ) x 4 x 6 Π x 5 x 5 Π 2 f 3 = βη f µ(1 ξ ψ r + ψ rω 3) 2 f 4 = ω2ψeβµ x 5 x 6 Π x 2 x 3 π 2 f 4 = ω2ψeβη dµ x 2 x 4 π 2 f 4 = ω3ψrβµ x 3 x 6 π 2 f 4 = ω3ψrβη f µ x 5 x 6 π 2 f 6 = ψrβη dµ x 4 x 6 Π 2 f 4 = ω2ψeβη f µ x 2 x 5 π 2 f 4 = ω3ψrβη dµ x 4 x 6 π 2 f 6 x 3 x 6 = ψrβµ Π 2 f 6 = ψrβη f µ x 5 x 6 Π Using the above expressions for the partial derivatives in the equation for a in (32) gives: (33) a = 6 kij=1 2 f k v k w i w j x i x j = 2βµ π (w 3 + η d w 4 + η f w 5 )[(ξv 3 + v 1 ξv 2 v 3 )

33 TB TRANSMISSION MODEL WITH DOTS 33 (w 2 + w 3 + w 4 + w 5 + w 6 ) + (w 2 ψ e ω 2 + w 6 ψ r ω 3 )(v 4 v 3 ) + w 2 ψ e (v 3 v 2 ) + w 6 ψ r (v 3 v 6 )] from which it can be shown that a > 0 if (34) B 1 > B 2 where B 1 = (w 3 + η d w 4 + η f w 5 )[(w 2 ψ e ω 2 + w 6 ψ r ω 3 )v 4 + w 2 ψ e v 3 + w 6 ψ r v 3 ] B 2 = (w 3 + η d w 4 + η f w 5 ){[v 3 (1 ξ) + ξv 2 ](w 2 + w 3 + w 4 + w 5 + w 6 ) + v 3 (w 2 ψ e ω 2 + w 6 ψ r ω 3 ) + w 2 ψ e v 2 }. For the sign of b it can be shown that the associated nonvanishing derivatives of F are 2 f 2 x 3 β = ξ 2 f 2 x 4 β = ξη d 2 f 3 x 3 β = (1 ξ) 2 f 3 x 4 β = η d(1 ξ) 2 f 2 x 5 β = ξη f 2 f 3 x 5 β = η f (1 ξ) so that b = 6 2 f k v k w i x i β ki=1 = [v 3 (1 ξ) + v 2 ξ](w 1 + w 2 + w 3 + w 4 + w 5 + w 6 ) (w 3 + η d w 4 + η f w 5 ) > 0. Thus the transformed system (30) or equivalently system (8) undergoes backward bifurcation at R d = 1 as required. Acknowledgements One of the authors (ABG) acknowledges with thanks the support in part of the Natural Sciences and Engineering Research Council (NSERC) and Mathematics of Information Technology and Complex Systems (MITACS) of Canada. SOA acknowledges the support in part of the Department of Mathematics University of Manitoba during his research visit in CNP acknowledges the support of the Manitoba Health Research Council. The authors are grateful to the referees and the handling editors for their constructive comments.

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