Estimating the threshold number for exponential survival rate

da a da a e a Q B e a Q B I da dI e d a Q B I da I e a Q B I I f I V a I V V H I u H H H H a da e a Q B H H H H a da e a Q B H H H H H H H V H a H I V V V H a H I V V V H ∫ ∫ ∫ ∫ ∞ ∞ − − + ∞ ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ − − + ∞ ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ − − + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ → = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − → = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − → = → ∞ − ∞ − 1 1 lim 1 1 lim 1 lim lim β β β β β β β 21 with 1 lim lim 1 = → = → ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ − − + − + ∞ − a da e a Q B H I u H a H I V V V H H e I e I β β ; . 1 lim = → − + a I H H V e I β 22 Hence we have lim ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = → ∫ ∫ ∞ ∞ + da da a aQ a aQ B B I f I V H V H V H H H H β β . 23 Similarly, we obtain lim = ∞ → H H H I f I . 24 This will ensure that a host endemic equilibrium exists if and only if H I 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∫ ∫ ∞ ∞ da da a aQ a aQ B B V H V H V H β β . 25 The LHS of 25 will be refereed as a threshold number R of the model.

4. Estimating the threshold number for exponential survival rate

It is well known in the literatures that in the single species model the threshold number R can be estimated by the ratio of life expectancy and mean age at infection. In this section we will show that, to some extent, it still true in the host-vector model. Let and . In Diekman and Heesterbeek 2000, Mean age at infection is given by a H H e a Q μ − = a V V e a Q μ − = H V H H I a μ β + = 1 or H H H H H V H a a a I 1 1 μ μ β − = − = , and similarly by V H V V I a μ β + = 1 or V V V V V H V a a a I 1 1 μ μ β − = − = . Hence, by combining the equations 19 and 25 we will have 35 ∫ ∫ ∫ ∞ ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ − − ∞ ∞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∞ − 1 1 da I e a Q B da da a aQ a aQ B B R H a da e a Q B H H V H V H V H a H I V V V H β β β β 26 which can be written as H V H V V V V V H H V H V H V a a B H H V H V H V a da e a Q B H V H H V H V a da e a Q B H V H H V H V a a B I B da e a Q I B da e a Q da a aQ I B da e a Q da da a aQ a aQ I B R V V V V H a V a V a V V V H a H I V V V H μ μ μ μ μ β μ μ β β μ μ β β μ β β β β μ μ β β β μ β 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 1 2 1 1 − − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∫ ∫ ∫ ∫ ∫ ∫ ∞ − − ∞ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∫ − − ∞ ∞ ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ − − ∞ ∞ ∞ − − ∞ − . 27 Finally, we end up to H V H V V V V H V V V V H V H V V V V H V H V a a a a a a I B R μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ 1 1 1 1 1 1 1 1 1 2 2 2 2 − − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = − − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 28 Since it is symmetrical, we also have V V H H H H H H V H H H a a a a R μ μ μ μ μ μ μ μ 1 1 1 1 1 2 2 − − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = 29 Which can be written as 2 2 2 2 1 1 1 H H V H H H V H V H H H V H H V H V V H H H a a a a a a R μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ − + = − + = − − − = 30 36 If we assume that vector life expectancy H V L L , that is very short compared to host life expectancy, then we can consider the limiting case as ∞ → V μ as follows H H H H H H V H H H V V a L a a a R = = − + ∞ → ≈ μ μ μ μ μ μ μ 1 1 lim 2 31 The last equation shows that R indicating infection from host to host for the host-vector model can be estimated by the ratio of host life expectancy and host mean age at infection. References Brauer, F. 2002. A Model for an SI Disease in an Age-Structured Population. Discrete and Continuous Dynamical Systems – Series

B. 2, 257-264.