Skip to main content

Theory and Modern Applications

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]