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Table 1 Parameter values

From: Optimal control of visceral, cutaneous and post kala-azar leishmaniasis

Parameter Definition Value Source
\(\Gamma _{h}\) Human birth rate 0.0015875 day−1 [17, 18]
\(\Gamma _{r}\) Dog birth rate 0.073 day−1 [19]
\(\Gamma _{v}\) Sandfly birth rate 0.299 day−1 [20]
\(\mu _{h}\) Human natural mortality rate 0.00004 day−1 [20]
\(\mu _{r}\) Dog natural death rate 0.000181 day−1 [21]
\(\mu _{v}\) Natural mortality rate of sandflies 0.0714 day−1 [22]
\(k_{1}\) Inverse of \(C_{l}\)’s latent period 0.0157871 day−1 [6]
\(k_{2}\) Inverse of \(V_{l}\)’s latent period 0.007789 day−1 [2]
\(k_{3}\) Inverse of \(C_{l}\)’s latent period after \(V_{l}\) recovery 0.011111 day−1 [6]
\(k_{4}\) Inverse of \(V_{l}\)’s latent period after \(C_{l}\) recovery 0.005111 day−1 [2]
\(k_{6}\) Inverse of dormant period before the development of PKDL 0.0049259 [2]
a Biting rate of sandflies 0.125 day−1 [23]
\(\gamma _{4}\) Recovery rate from \(C_{l}\), after being recovered from \(V_{l}\) 0.008 day−1 [14]
\(\gamma _{5}\) \(V_{l}\)’s treatment rate for \(C_{l}\)’s recovered 0.03 day−1 Free to choose
\(\delta _{1}\) \(V_{l}\)-induced expiry in humans 0.011 day−1 [24]
\(\delta _{2}\) \(V_{l}\)-induced expiry in humans for \(C_{l}\) recovered humans 0.006 day−1 [14, 24]
\(c_{1}\) Trans. prob. of \(C_{l}\) in sandflies 0.22 day−1 assumed
\(b_{1}\) Trans. prob. of \(C_{l}\) in humans 0.0714 day−1 assumed
\(c_{2}\) Trans. prob. of \(V_{l}\) in sandflies 0.22 day−1 [25]
\(b_{2}\) Trans. prob. of \(V_{l}\) in humans 0.0714 day−1 [26]
c Trans. prob. of either strain in a fly from reservoir 0.22 day−1 [25]
b Trans. prob. of either strain in dogs from a fly 0.0714 day−1 [26]
\(\gamma _{1}\) Per capita rate of recovery from \(C_{l}\) 0.0056 day−1 [14]
\(\gamma _{2} \) \(V_{l}\)’s treatment rate 0.03 day−1 Free to choose
\(\delta _{3}\) PKDL-induced expiry rate in humans 0.0006 day−1 [27]
\(\alpha _{d}\) Rate of \(V_{l}\)-induced expiry in dogs 0.00181 day−1 [23]
\(\tau _{d}\) Natural recovery of dogs 0.000274 day−1 [23]
\(\beta _{2}\) PKDL’s natural healing for \(C_{l}\)’s recovered human 0.01 day−1 [14, 28]
\(\varrho _{v}\) Inverse vectors’s incubation period 0.1428 day−1 [22]
\(\omega _{d}\) Rate of immunity loss in dogs 0.00274 day−1 [23]
k Rate of immunity loss in humans 0.000548 day−1 [23]
\(\beta _{1}\) PKDL’s natural healing in humans 0.00556 day−1 [28]
ρ Cross-immunity between \(C_{l}\) and \(V_{l}\) (0,1) [14]