Supplemental Material for: The effect of the incidence function on the existence of backward bifurcation

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1 Supplemental Material for: The effect of the incidence function on the existence of backward bifurcation David J Gerberry Department of Mathematics Xavier University Cincinnati, Ohio, 4527, USA Andrew M Philip Department of Mathematics Department of Biology Xavier University Cincinnati, Ohio, 4527, USA Contents 1 General theorem for the existence of backward bifurcation 1 2 Backward bifurcation caused by imperfect vaccination 2 21 Backward bifurcation threshold; stard incidence 2 22 Backward bifurcation threshold; mass action 3 3 Combining exogenous reinfection imperfect vaccination 5 31 Backward bifurcation threshold; stard incidence 5 32 Backward bifurcation threshold; mass action 6 1 General theorem for the existence of backward bifurcation In order to prove the existence of backward bifurcation, we use the result of Castillo-Chavez Song 1] restated here in slightly modified form for direct applicability to our system) which is closely related to that of van den Driessche Watmough 3] Theorem 1 Theorem 41 from Castillo-Chavez Song 1]) Consider a system of ordinary differential equations with parameter φ: dx dt = fx, φ), f: Rn R R n, f C 2 R n R) 1) where x is the disease-free equilibrium for system 1) for all values of the parameter φ, that is fx, φ) for all φ Assume ) A1: J = D x fx, φ fi x j x, φ ) is the linearization of system 1) around the equilibrium x with φ evaluated at φ Zero is a simple eigenvalue of J all other eigenvalues of J have negative real parts A2: Matrix J has a right eigenvector w a left eigenvector v corresponding to the zero eigenvalue 1 1 In 1], the authors assume that w is nonnegative but later remark that this condition can be relaxed to w j for all j such that x j = 1

2 Let f k be the k th component of f a = w j x, φ ), 2) x i x j x i φ x, φ ) 3) The local dynamics of system 1) around x are totally determined by a b i a >, b > When φ < φ with φ φ 1, x is locally asymptotically stable, there exists an unstable endemic equilibrium When φ > φ with φ φ 1, x is unstable there exists a negative locally asymptotically stable equilibrium ii a <, b < When φ < φ with φ φ 1, x is unstable When φ > φ with φ φ 1, x is locally asymptotically stable, there exists an unstable endemic equilibrium iii a >, b < When φ < φ with φ φ 1, x is unstable, there exists a locally asymptotically stable negative equilibrium When φ > φ with φ φ 1, x is stable, an unstable endemic equilibrium appears iv a <, b > When φ φ changes from negative to positive, x changes its stability from stable to unstable Correspondingly, a negative unstable equilibrium becomes a locally asymptotically stable endemic equilibrium In our work, we examine the local dynamics of infectious disease models around the disease-free equilibrium ie x = DFE) use the transmission coefficient as the bifurcation parameter ie φ = β) Noting that b < in Theorem 1 implies that x is unstable for φ < φ locally asymptotically stable for φ > φ, we see that b < indicates that R is a decreasing function of the bifurcation parameter As this is not the case for our bifurcation parameter β, our analyses will show that b > that the direction of the bifurcation is completely determined by the sign of a Specifically, we see that the case a <, b > produces the typical behavior for mathematical models of infectious disease: as R becomes less than 1 a stable endemic equilibrium gives way to a stable DFE ie forward bifurcation) When a >, b >, backward bifurcation is exhibited Here, as R becomes less than 1 an unstable endemic equilibrium is formed 2 Backward bifurcation caused by imperfect vaccination We begin by establishing the condition for the existence of backward bifurcation for the Kribs-Zaleta et al model as proposed in 2] ie the stard incidence formulation) which is given by System 13) of the main text with λx, I, N β XI N, for X {S, E, T } We use the general theorem for the existence of backward bifurcation of Castillo-Chavez Song 1] which is restated in Theorem 1 in a modified form for more direct applicability to our systems 21 Backward bifurcation threshold; stard incidence The basic reproductive number, R, of the model is given by R = β ψγ + µ + θ) µ + r + d) µ + θ + γ) 4) As backward bifurcation describes the local behavior of the DFE at the reproductive number threshold, we begin our analysis by finding the disease-free equilibrium DFE) β, the transmission rate for which R = 1 to get S, I, V ) µ + θ) Λ µ µ + θ + γ),, γλ, β = µ µ + θ + γ) 2 µ + r + d) µ + θ + γ) ψγ + µ + θ

3 Linearizing around the DFE, we calculate the Jacobian of the system at the DFE to get β+r)θ+ β+r)µ+rγ µ γ µ+θ+γ θ µ J = 2 +β γ θ d r)µ+ψβ d r)γ θd β+r) µ+θ+γ γ µ θ ψβγ µ+θ+γ As we are concerned with the system s behavior near R = 1, we evaluate the Jacobian at β = β calculate its eigenvalues to get λ 1,2,3,4 =, µ, µ θ γ The left right eigenvectors, v w, respectively, corresponding to the eigenvalue λ =, are then found to be ψγµ + dψγ + ψrµ + ψdµ + ψµ 2 + µ 2 + µθ + dµ + dθ ) ] γ v =, µ µ 2 + 2µθ + θ 2 + ψγµ + θψγ + µγ + θγ + ψγ 2, ) w = γψdθ+µθψγ µrψγ+µθ 2 +µ 3 +dθ 2 +2θdµ+2µ 2 θ+dµ 2 ψγµ+dψγ+ψrµ+ψdµ+ψµ 2 +µ 2 +µθ+dµ+dθ)γ µµ2 +2µθ+θ 2 +ψγµ+θψγ+µγ+θγ+ψγ 2 ) ψγµ+dψγ+ψrµ+ψdµ+ψµ 2 +µ 2 +µθ+dµ+dθ)γ In accordance with the conditions of Theorem 1, we have that v w = 1 With this condition met, we rewrite our system in the form x = f x) for a clearer application of Theorem 1 by renaming the state variables as S = x 1, I = x 2, V = x 3 Our system becomes x βx 1 x 2 1 = Λ µ + γ) x 1 + rx 2 + θx 3, x 1 + x 2 + x 3 x βx 1 x 2 2 = + ψβx 3x 2 µ + r + d) x 2, x 1 + x 2 + x 3 x 1 + x 2 + x 3 x 3 = γx 1 ψβx 3x 2 x 1 + x 2 + x 3 µ + θ) x 3 From this reformulated system, we may directly calculate a b as described in the Theorem 1 as a = w j x, β ) x i x j µ + θ + γ) ψ 2 γ 2 + ψ µ + r + d) ψ d + 2θ + µ r) γ + µ + θ) 2) µ + r + d) µ 2 = 2, Λ ψγ + µ + θ) 1 x i β x, β ψγ + µ + θ µ + θ + γ 6) We immediately see that b > for all biologically reasonable parameters, thus the existence of backwards bifurcation hinges solely upon a > Using this condition, we establish the following threshold recovery rate necessary for the stard incidence version of the Kribs-Zaleta et al model to exhibit backward bifurcation: r > rsi = ψ2 γd + ψ 2 γµ + ψ 2 γ 2 dψγ + ψγµ + 2θψγ + µ 2 + 2µθ + θ 2 7) ψγ 1 ψ) 22 Backward bifurcation threshold; mass action The basic reproductive number, R, of the model is given by R = β ψγ + µ + θ) Λ µ + r + d) µ + θ + γ) µ 8) 3 T 5)

4 As backward bifurcation describes the local behavior of the DFE at the reproductive number threshold, we begin our analysis by finding the disease-free equilibrium DFE) β, the transmission rate for which R = 1 to get S, I, V ) µ + θ) Λ µ µ + θ + γ),, γλ µ µ + θ + γ) β = µ + r + d) µ + θ + γ) µ Λ ψγ + µ + θ) Linearizing around the DFE, we calculate the Jacobian of the system at the DFE to get µ γ βµ+θ)λ µµ+θ+γ) + r θ βµ+θ)λ J = µµ+θ+γ) + ψβγλ µµ+θ+γ) µ r d γ ψβγλ µµ+θ+γ) µ θ As we are concerned with the system s behavior near R = 1, we evaluate the Jacobian at β = β calculate its eigenvalues to get λ 1,2,3,4 =, µ, µ θ γ The left right eigenvectors, v w, respectively, corresponding to the eigenvalue λ =, are then found to be v =, 1 µ µ 2 + 2µθ + θ 2 + ψγµ + θψγ + µγ + θγ + γ 2 ψ), ], w = ψθγµ ψγµr + ψθγd + µ 3 + 2µdθ + 2µ 2 θ + µθ 2 + dµ 2 + dθ 2 µ µ 2 + 2µθ + θ 2 + ψγµ + θψγ + µγ + θγ + γ 2 ψ ) ψγd + ψγµ + ψrµ + ψµd + ψµ 2 + dµ + dθ + µθ + µ 2) γ T In accordance with the conditions of Theorem 1, we have that v w = 1 With this condition met, we rewrite our system in the form x = f x) for a clearer application of Theorem 1 by renaming the state variables as S = x 1, I = x 2, V = x 3 Our system becomes x 1 = Λ βx 1 x 2 µ + γ) x 1 + rx 2 + θx 3, x 2 = βx 1 x 2 + ψβx 3 x 2 µ + r + d) x 2, x 3 = γx 1 ψβx 3 x 2 µ + θ) x 3 From this reformulated system, we may directly calculate a b as described in the Theorem 1 as a = w j x, β x i x j 2 µ + r + d) µ µ + θ + γ) A, 9) Λ µ + θ + ψγ) x i β x, β Λ ψγ + µ + θ) µ µ + θ + γ), 1) where A = 2ψθγµ ψγµr + 2ψθγd + dµ 2 + 2µdθ + 2µ 2 θ + µθ 2 + µ 3 + dθ 2 + γ 2 ψ 2 d + γ 2 ψ 2 µ + ψ 2 γµr + ψ 2 γµd + ψ 2 γµ 2 + ψγµd + ψγµ 2 As was the case in the model formulated for stard incidence, b > for all biologically reasonable parameters, thus the existence of backwards bifurcation depends only on a > Using this condition once again, we establish the following threshold recovery rate necessary for the mass action version of the Kribs-Zaleta et al model to exhibit backward bifurcation: r >rma = γ2 ψ 2 d+2ψθγd+dθ 2 + ψ 2 γd+2θψγ+θ 2 +2dθ+ψγd+γ 2 ψ 2) µ+ d+ψ 2 γ+2θ+ψγ ) µ 2 +µ 3 11) ψγµ 1 ψ) 4

5 3 Combining exogenous reinfection imperfect vaccination 31 Backward bifurcation threshold; stard incidence To established the threshold for the existence of backward bifurcation for the stard incidence version of the combination model ie System 18) of the main text with λx, I, N β XI N, for X {S, E, V, T } in the main text), we begin by calculating the basic reproductive number: ) β θ+µ) θ+µ+γ + βψ γ θ+µ+γ k R = µ + r + d) µ + k = kβ θ + µ + ψγ) µ + r + d) µ + k) θ + µ + γ) As backward bifurcation describes the local behavior of the DFE at the reproductive number threshold, we must calculate the disease-free equilibrium DFE) β, the transmission rate for which R = 1 to get ) Λ θ + µ) S, V, E, I, T µ θ + µ + γ), Λγ,,, µ θ + µ + γ) β = µ + r + d) µ + k) θ + µ + γ) k θ + µ + ψγ) Linearizing around the DFE, we calculate the Jacobian of the system at the DFE to get µ γ θ βθ+µ) θ+µ+γ γ θ µ βψγ θ+µ+γ J = βθ+µ+ψγ) µ k θ+µ+γ k µ r d r µ As we are concerned with the system s behavior near R = 1, we evaluate the Jacobian at β = β calculated its eigenvalues to get λ 1,2,3,4,5 =, d 2µ k r, θ µ γ, µ, µ The left right eigenvectors, v w, respectively, corresponding to the eigenvalue λ 1 =, are then found to be ] kr v =,, µ k + r + d + 2µ), r µ + k) µ k + r + d + 2µ),, w = Γ µ 2 + 2µθ + θ θ + ψγ) ) µ + r + d) µ, Γγ ψ + 1) µ + θ + ψγ),, µ ] kr r, 1, µ+k)µ+r+d) where Γ = rkθ+µ+ψγ)θ+µ+γ) Confirming that v w = 1, we proceed to write our system in the form x = f x) for a clearer application of Theorem 1 by renaming the state variables as S = x 1, V = x 2, E = x 3, 5

6 I = x 4, T = x 5 The system becomes x βx 1 x 4 1 = Λ µx 1 γx 1 +θx 2, x 1 +x 2 +x 3 +x 4 +x 5 x βψx 4 x 2 2 = γx 1 θx 2 µx 2, x 1 +x 2 +x 3 +x 4 +x 5 x βx 1 x 4 βψx 4 x 2 3 = + x 1 +x 2 +x 3 +x 4 +x 5 x 1 +x 2 +x 3 +x 4 +x 5 x pβx 3 x 4 4 = +x 3 k µ+r+d) x 4, x 1 +x 2 +x 3 +x 4 +x 5 x σβx 5 x 4 5 = rx 4 µx 5 x 1 +x 2 +x 3 +x 4 +x 5 We may then calculate a b as described in Theorem 1 to get a = pβx 3 x 4 σβx 5 x 4 µ+k) x 3 +, x 1 +x 2 +x 3 +x 4 +x 5 x 1 +x 2 +x 3 +x 4 +x 5 w j x, β 2µ µ + k) µ + r + d) A x i x j Λk 2 r θ + µ + ψγ) 2 2µ + k + r + d), 12) x i β x, β kc ψγ + µ + θ) θ + µ + γ) d + k + 2µ + r), 13) where A = c + c 1 θ + c 2 r c 3 σ c 4 p is such that c = kdµθ 2 + µ 3 kψγ + k 2 µθ 2 + kµ 2 θ 2 + kdµ 3 + k 2 µ 3 + µ 4 k + kdµ 2 ψγ + γk 2 µ 2 ψ + γ 2 kµ 2 ψ 2 + γk 2 µ 2 ψ 2 + γkµ 3 ψ 2 + γ 2 k 2 ψ 2 µ + dγ 2 kµψ 2 + dγk 2 µψψ 1) + dγkµ 2 ψ 2, c 1 = 2kµ ψγ + µ) d + k + µ), c 2 = k µ + k) µ 2 + ψ 2 + ψ ) γ + 2θ ) µ + θ + ψγ) 2), c 3 = k 2 r θ + µ + γ) θ + µ + ψγ), c 4 = µ 2 θ + µ + γ) θ + µ + ψγ) µ + r + d) 14) As the existence of backward bifurcation requires a <, we see from 12) that A < is an equivalent condition Notably, as c 2 > in 14), we note that a high recovery rate, r, alone is not capable of producing backward bifurcation in the combination model despite being able to do so in the Kribs-Zaleta et al model From the form of A, it is apparent that the rate of reinfection, both after recovery σ) exogenous p), are the drivers of backward bifurcation in the combination model To make comparisons with the findings of Section 2 for the Feng et al model, we focus on the threshold level of exogenous reinfection necessary to produce backward bifurcation in the stard incidence version of the combination model: 32 Backward bifurcation threshold; mass action p > p SI = c + c 1 θ + c 2 r c 3 σ c 4 15) To established the threshold for the existence of backward bifurcation for the mass action version of the combination model ie System 18) with λx, I, N βxi, for X {S, E, V, T } in the main text), we begin by calculating the basic reproductive number: R = βλ µ θ+µ) θ+µ+γ + βψλ γ µ θ+µ+γ µ + r + d) ) k µ + k = Λkβ θ + µ + ψγ) µ µ + r + d) µ + k) θ + µ + γ) 6

7 As backward bifurcation describes the local behavior of the DFE at the reproductive number threshold, we must calculate the disease-free equilibrium DFE) β, the transmission rate for which R = 1 to get ) Λ θ + µ) S, V, E, I, T µ θ + µ + γ), Λγ,,, µ θ + µ + γ) β = µ + r + d) µ + k) θ + µ + γ) µ k θ + µ + ψγ) Λ Linearizing around the DFE, we calculate the Jacobian of the system at the DFE to get J = µ γ θ βθ+µ)λ µθ+µ+γ) γ θ µ βψγλ µθ+µ+γ) µ k βλθ+µ+ψγ) µθ+µ+γ) k µ r d r µ As we are concerned with the system s behavior near R = 1, we evaluate the Jacobian at β = β calculated its eigenvalues to get λ 1,2,3,4,5 =, d 2µ k r, θ µ γ, µ, µ 16) The left right eigenvectors, v w, respectively, corresponding to the eigenvalue λ 1 =, are then found to be ] rk v =,, µ d + 2µ + k + r), µ + k) r µ d + 2µ + k + r), w = Γ µ 2 + 2µθ + θ θ + ψγ) ) µ + r + d) µ, Γγ µψ + µ + θ + ψγ),, µ ] rk r, 1, µ+r+d)µ+k) where Γ = θ+µ+ψγ)rθ+µ+γ)k Noting that v w = 1, we rewrite our system in the form x = f x) for a clearer application of Theorem 1 by renaming the state variables as S = x 1, V = x 2, E = x 3, I = x 4, T = x 5 The system becomes x 1 = βcx 1 x 4 µx 1 γx 1 + θx 2 + Λ, x 2 = βcψx 4 x 2 µx 2 + γx 1 θx 2, x 3 = pβcx 3 x 4 + βcψx 4 x 2 + βcx 1 x 4 µ + k) x 3 + σβcx 5 x 4, x 4 = pβcx 3 x 4 + kx 3 µ + r + d) x 4, x 5 = σβcx 5 x 4 µx 5 + rx 4 Calculating a b as described in Theorem 1, we get a = w j x, β 2µ µ + k) µ + r + d) A x i x j r + d + 2µ + k) θ + µ + ψγ) 2 rk 2 Λ, 17) x i β x, β kc ψγ + µ + θ) Λ d + k + 2µ + r) µ θ + µ + γ), 18) 7

8 where A = c + c 1 θ + c 2 r c 3 σ c 4 p is such that c = kµ 4 + k 2 µ 3 + dk 2 θ 2 + dk 2 µ 2 + k 2 µ θ 2 + kµ 2 θ 2 + kµ 3 d + γ ψ 2 kµ 2 d + γ ψ 2 k 2 µ d + γ ψ kµ 2 d + γ ψ k 2 dµ + γ 2 ψ 2 kµ d + γ ψ 2 kµ 3 + γ ψ 2 k 2 µ 2 + γ ψ kµ 3 + γ ψ k 2 µ 2 + γ 2 ψ 2 kµ 2 + γ 2 ψ 2 k 2 µ + γ 2 ψ 2 k 2 d + kµ dθ 2, c 1 = 2 kµ ψ γ + µ) d + k + µ), c 2 = k µ + k) µ 2 + ψ 2 + ψ ) γ + 2 θ ) µ + θ + ψ γ) 2), c 3 = k 2 r θ + µ + γ) θ + µ + ψ γ), c 4 = µ 2 θ + µ + γ) θ + µ + ψ γ) µ + r + d) 19) As the existence of backward bifurcation requires a <, we see from 17) that to establish the threshold condition we must solve A = The resulting threshold for the existence of backward bifurcation for the mass action version of the combination model is given by: References p > p MA = c + c 1 θ + c 2 r c 3 σ c 4 2) 1] C Castillo-Chavez B Song Dynamical models of tuberculosis their applications Math Biosci Eng, 12):361 44, Sept 24 2] C M Kribs-Zaleta J X Velasco-Hernández A simple vaccination model with multiple endemic states Math Biosci, 1642):183 21, Apr 2 3] P van den Driessche J Watmough Reproduction numbers sub-threshold endemic equilibria for compartmental models of disease transmission Math Biosci, 18:29 48, 22 8