Background
Methods: model description
Vector dispersal model
Name | Description | Unit | Range |
---|---|---|---|
b
| Integer number of female eggs laid per oviposition | − | 50–300 |
\(\psi ^\mathsf {W}_{\zeta }\)
| 50% of the eggs are assumed to hatch into female mosquitoes parameter represent water availability in cell \(\zeta\) | − | 0.0–1.0 |
\(\psi ^\mathsf {H}_{\zeta }\)
| Binary parameter represent human presence in cell \(\zeta\) | − | 0–1 |
\(\rho _E\)
| Egg hatching rate into larvae | day\(^{-1}\) | 0.33–1.0 |
\(\rho _L\)
| Rate at which larvae develop into pupae | day\(^{-1}\) | 0.08–0.17 |
\(\rho _P\)
| Rate at which pupae develop into adult/emergence rate | day\(^{-1}\) | 0.33–1.0 |
\(\mu _E\)
| Egg mortality rate | day\(^{-1}\) | 0.32–0.80 |
\(\mu _{L_1}\)
| Natural mortality rate of larvae | day\(^{-1}\) | 0.30–0.58 |
\(\mu _{L_2}\)
| Density-dependent mortality rate of larvae | day\(^{-1}\)mosq\(^{-1}\) | 0.0–1.0 |
\(\mu _P\)
| Pupae mortality rate | day\(^{-1}\) | 0.22–0.52 |
\(\rho _{A_h}\)
| Rate at which host-seeking mosquitoes enter the resting state | day\(^{-1}\) | 0.322–0.598 |
\(\rho _{A_r}\)
| Rate at which resting mosquitoes enter oviposition searching state | day\(^{-1}\) | 0.30–0.56 |
\(\rho _{A_o}\)
| Oviposition rate | day\(^{-1}\) | 3.0–4.0 |
\(\mu _{A_h}\)
| Mortality rate of mosquitoes of searching for hosts | day\(^{-1}\) | 0.125–0.233 |
\(\mu _{A_r}\)
| Mortality rate of resting mosquitoes | day\(^{-1}\) | 0.0034–0.01 |
\(\mu _{A_o}\)
| Mortality rate of mosquitoes searching for oviposition sites | day\(^{-1}\) | 0.41–0.56 |
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The rate of change in egg population (\(E_{\zeta }\)) as a function of oviposition, egg mortality and hatching:where b is the average number of eggs laid during an oviposition with 1:1 sex ratio; \(\rho _{A_o}\) is the rate at which eggs are oviposited by gravid mosquitoes; \(\mu _{E}\) is egg mortality rate; and \(\rho _{E}\) is the hatching rate into larvae. The term \(\psi ^\mathsf {W}_{\zeta }\) on the right hand side represents water availability in a particular cell \(\zeta\) and is discussed later.$$\begin{aligned} \frac{dE_{\zeta }}{dt} = b \psi ^\mathsf {W}_{\zeta } \rho _{A_o} A_{o,\zeta } - \left( \mu _{E} + \rho _{E} \right) E_{\zeta } \end{aligned}$$(1a)
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The rate of change in larvae population (\(L_{\zeta }\)) as a function of egg population, larval mortality and maturation into pupae:where \(\rho _{L}\) is the progression rate from larvae to pupae; \(\mu _{L_1}\) and \(\mu _{L_2}\) represent natural and density-dependent death rates of larvae, respectively.$$\begin{aligned} \frac{dL_{\zeta }}{dt} = \rho _{E} E_{\zeta } - \left( \mu _{L_1} + \mu _{L_2} L_{\zeta } + \rho _{L} \right) L_{\zeta } \end{aligned}$$(1b)
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The rate of change in pupae population (\(P_{\zeta }\)) as a function of larval maturation, pupal mortality, and emergence into adults:where \(\mu _{P}\) is the mortality rate of pupae, and \(\rho _{P}\) represent the rate of emergence from pupae into adults.$$\begin{aligned} \frac{dP_{\zeta }}{dt} = \rho _{L} L_{\zeta } - \left( \mu _{P} + \rho _{P} \right) P_{\zeta } \end{aligned}$$(1c)
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The rate of change in population of host-seeking adults (\(A_{h,\zeta }\)) as a function of pupal emergence, oviposition, mortality, and blood feeding rates:where \(\mu _{A_h}\) is the death rate of host-seeking adults; and \(\rho _{A_h}\) is the rate at which they enter a resting state after blood feeding. Host-seeking adults can spread to adjacent cells for searching human host in which \(\omega ^\mathsf {H} _{\zeta _1:\zeta _2}\) represents their movement rate from cell \(\zeta _1\) to \(\zeta _2\) modelled as a decreasing exponential function of human population and average area of cell \(\zeta _1\) and \(\zeta _2\) (see [28]); \(\mathcal {N}\) denotes the neighbors of the cell under consideration. The last two terms in (1d) represent the movements of vectors from all neighbors \(\zeta '\) into cell \(\zeta\) and vice versa, respectively. After laying eggs, gravid mosquitoes return to the host-seeking state for subsequent blood feeding.$$\begin{aligned} \begin{array}{l} \frac{{d{A_{h,\zeta }}}}{{dt}} = {\rho _P}{P_\zeta } + \psi _\zeta ^W{\rho _{{A_o}}}{A_{o\zeta }} - \left( {{\mu _{{A_h}}} + \psi _\zeta ^H{\rho _{{A_h}}}} \right) {A_{h,\zeta }}\\ \qquad \quad+ \left( {\sum \limits _{\zeta ' \in {\mathcal {N}}} {\omega _{\zeta ':\zeta }^H} } \right) {A_{h,\zeta '}} - \left( {\sum \limits _{\zeta ' \in \mathcal{N}} {\omega _{\zeta :\zeta '}^H} \times {A_{h,\zeta }}} \right) \end{array} \end{aligned}$$(1d)
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The rate of change in population of resting adults (\(A_{r,\zeta }\)) as a function of blood feeding, mortality, and protein digestion rates:where \(\mu _{A_r}\) is the death rate of resting adults; and \(\rho _{A_r}\) is the progression rate at which the survivors enter the oviposition site searching phase. In the resting state, female mosquitoes are usually dormant to digest protein.$$\begin{aligned} \frac{dA_{r,\zeta }}{dt} = \psi ^\mathsf {H}_{\zeta } \rho _{A_h} A_{h,\zeta } - \left( \mu _{A_r} + \rho _{A_r} \right) A_{r,\zeta } \end{aligned}$$(1e)
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The rate of change in population of oviposition site searching adults (\(A_{o,\zeta }\)) as a function of emergence, oviposition, mortality, and digestion rates:where \(\mu _{A_o}\) is the death rate of gravid female mosquitoes. Vectors in ovipositing state also spread in space to find water for oviposition in which \(\omega ^\mathsf {W} _{\zeta _1:\zeta _2}\) represents their movement rate from cell \(\zeta _1\) to \(\zeta _2\) modelled as a decreasing exponential function of surface soil moisture and average area of cell \(\zeta _1\) and \(\zeta _2\) [28].$$\begin{aligned} \frac{dA_{o,\zeta }}{dt}&= \rho _{r} A_{r,\zeta } - (\mu _{A_o} + \psi ^\mathsf {W}_{\zeta } \rho _{A_o}) A_{o,\zeta } \nonumber \\&\quad + \left( \sum _{\zeta ' \in \mathcal {N}} \omega ^{\mathsf {W}}_{\zeta ':\zeta } \right) A_{o,\zeta '} - \left( \sum _{\zeta ' \in \mathcal {N}} \omega _{\zeta :\zeta '}^{\mathsf {W}} \times A_{o,\zeta } \right) \end{aligned}$$(1f)
l
| Change, \(\Delta X^{l}_{k,\zeta }(t)\) | Probability, \(p^{l}_{k,\zeta }(t)\) | Description |
---|---|---|---|
1 |
\([1, 0, 0, 0, 0, 0]^T\)
|
\(b \psi ^{\mathsf {W}}_{\zeta } \rho _{A_o} \mathcal {A}_{o,\zeta } \Delta t\)
| A new egg E is deposited by \(A_o\) |
2 |
\([-1, 0, 0, 0, 0, 0]^T\)
|
\(\mu _E \mathcal {E}_{\zeta } \Delta t\)
| An egg E dies |
3 |
\([-1, 1, 0, 0, 0, 0]^T\)
|
\(\rho _E \mathcal {E}_{\zeta } \Delta t\)
| An egg E hatches into a larva L |
4 |
\([0, -1, 0, 0, 0, 0]^T\)
|
\((\mu _{L_1} + \mu _{L_2} \mathcal {L}_{\zeta } ) \mathcal {L}_{\zeta } \Delta t\)
| A larva L dies |
5 |
\([0, -1, 1, 0, 0, 0]^T\)
|
\(\rho _L \mathcal {L}_{\zeta } \Delta t\)
| A larva L develops into a pupa P |
6 |
\([0, 0, -1, 0, 0, 0]^T\)
|
\(\mu _P \mathcal {P}_{\zeta } \Delta t\)
| A pupa P dies |
7 |
\([0, 0, -1, 1, 0, 0]^T\)
|
\(\rho _P \mathcal {P}_{\zeta } \Delta t\)
| A pupa P develops into a host-seeking adult \(A_h\) |
8 |
\([0, 0, 0, 1, 0, -1]^T\)
|
\(\psi ^\mathsf {W}_{\zeta } \rho _{A_o} \mathcal {A}_{o,\zeta } \Delta t\)
| An oviposition adult \(A_o\) enters host-seeking state |
9 |
\([0, 0, 0, -1, 0, 0]^T\)
|
\(\mu _{A_h} \mathcal {A}_{h,\zeta } \Delta t\)
| A host-seeking adult \(A_h\) dies |
10 |
\([0, 0, 0, -1, 1, 0]^T\)
|
\(\psi ^\mathsf {H}_{\zeta } \rho _{A_h} \mathcal {A}_{h,\zeta } \Delta t\)
| A host-seeking adult \(A_h\) enters resting state |
11 |
\([0, 0, 0, 0, -1, 0]^T\)
|
\(\mu _{A_r} \mathcal {A}_{r,\zeta } \Delta t\)
| A resting adult \(A_r\) dies |
12 |
\([0, 0, 0, 0, -1, 1]^T\)
|
\(\rho _{A_r} \mathcal {A}_{r,\zeta } \Delta t\)
| A resting adult \(A_r\) enters oviposition searching state |
13 |
\([0, 0, 0, 0, 0, -1]^T\)
|
\(\mu _{A_o} \mathcal {A}_{o,\zeta } \Delta t\)
| An oviposition searching adult \(A_o\) dies |
Malaria transmission model
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The rate of change in susceptible host (\(S_{h,\zeta }\)) as a function of immigration, birth, human infection, recovery from infection, and human mortality:where \(\Lambda _h\) is immigration rate; and \(\psi _h\) is per capita birth rate of humans; \(\rho _h\) is per capita rate of losing acquired temporary immunity. Acquired temporary immunity represents the enhancement of the defense mechanism of the human host as a result of a previous encounter with the pathogen [32]. \(N_{h,\zeta } = S_{h,\zeta } + E_{h,\zeta } + I_{h,\zeta } + R_{h,\zeta }\) is total population size for humans in each cell \(\zeta\); \(f_h(N_{h,\zeta }) = \mu _{1h} + \mu _{2h} N_{h,\zeta }\) is the human per capita death rate; and \(\lambda _{h,\zeta }\) is the infection rate from mosquitoes to humans defined as:$$\begin{aligned} \frac{dS_{h,\zeta }}{dt} =\,& \Lambda _h + \psi _h N_{h,\zeta } + \rho _h R_{h,\zeta }\\& - \lambda _{h,\zeta }(t) S_{h,\zeta } - f_h(N_{h,\zeta }) S_{h,\zeta } \end{aligned}$$(10a)in which \(N_{v,\zeta } = S_{v,\zeta } + E_{v,\zeta } + I_{v,\zeta }\) is total population of mosquitoes in cell \(\zeta\); \(\sigma _v\) represents the number of times one mosquito attempt to bite humans per unit time; \(\sigma _h\) is the maximum number of mosquito bites a human can have per unit time; and \(\beta _{hv}\) is the probability of infection transmission from an infectious mosquito to a susceptible human, given that a contact between the two occurs.$$\begin{aligned} \lambda _{h,\zeta } = \frac{\sigma _v \sigma _h}{\sigma _v N_{v,\zeta } + \sigma _h N_{h,\zeta }} \times \beta _{hv} I_{v,\zeta } \end{aligned}$$
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The rate of change in exposed host (\(E_{h,\zeta }\)) as a function of new host infections, latent period, and human mortality:where \(\nu _h\) is per capita rate of progression of humans from exposed to infectious state.$$\begin{aligned} \frac{dE_{h,\zeta }}{dt} = \lambda _{h,\zeta }(t) S_{h,\zeta } - \nu _h E_{h,\zeta } - f_h(N_{h,\zeta }) E_{h,\zeta } \end{aligned}$$(10b)
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The rate of change in infected host (\(I_{h,\zeta }\)) as a function of latent period, recovery and human mortality rates:where \(\gamma _h\) represents per capita recovery rate for humans from infectious to recovered states; and \(\delta _h\) represents per capita disease-induced death rate for humans.$$\begin{aligned} \frac{dI_{h,\zeta }}{dt} = \nu _h E_{h,\zeta } - \gamma _h I_{h,\zeta } - f_h(N_{h,\zeta }) I_{h,\zeta } - \delta _h I_{h,\zeta } \end{aligned}$$(10c)
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The rate of change in recovered host (\(R_{h,\zeta }\)) as a function of recovery, immunity loss, and human mortality:$$\begin{aligned} \frac{dR_{h,\zeta }}{dt} = \gamma _h I_{h,\zeta } - \rho _h R_{h,\zeta } - f_h(N_{h,\zeta }) R_{h,\zeta } \end{aligned}$$(10d)
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The rate of change in susceptible vector (\(S_{v,\zeta }\)) as a function of reproduction, vector infection, and vector mortality:where \(\psi _v\) represents per capita birth rate of the vectors; \(f_v(N_{v,\zeta }) = \mu _{1v} + \mu _{2v} N_{v,\zeta }\) is the per capita death rate for vectors in each cell \(\zeta\); and \(\lambda _{v,\zeta }\) is the infection rate from humans to mosquitoes defined as:$$\begin{aligned} \frac{dS_{v,\zeta }}{dt} = \psi _v N_{v,\zeta } - \lambda _{v,\zeta }(t) S_{v,\zeta } - f_v(N_{v,\zeta }) S_{v,\zeta } \end{aligned}$$(10e)where \(\beta _{vh}\) and \(\tilde{\beta }_{hv}\) represent the transmission probability of infection from an infectious and a recovered human, respectively, to a susceptible mosquito, given that a contact between them occurs.$$\begin{aligned} \lambda _{v,\zeta } = \frac{\sigma _v \sigma _h}{\sigma _v N_{v,\zeta } + \sigma _h N_{h,\zeta }} \times \left( \beta _{vh} I_{h,\zeta } + \tilde{\beta _{vh}} R_{h,\zeta } \right) \end{aligned}$$
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The rate of change in exposed vector (\(E_{v,\zeta }\)) as a function of new vector infection, vector latent period, and vector mortality:where \(\nu _v\) is per capita rate of progression of mosquitoes from the exposed state to the infectious state.$$\begin{aligned} \frac{dE_{v,\zeta }}{dt} = \lambda _{v,\zeta }(t) S_{v,\zeta } - \nu _v E_{v,\zeta } - f_v(N_{v,\zeta }) E_{v,\zeta } \end{aligned}$$(10f)
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The rate of change in infected vector (\(I_{v,\zeta }\)) as a function of latent period and mortality:$$\begin{aligned} \frac{dI_{v,\zeta }}{dt} = \nu _v E_{v,\zeta } - f_v(N_{v,\zeta }) I_{v,\zeta } \end{aligned}$$(10g)
Name | Description | Unita |
---|---|---|
\(\Lambda _h\)
| Immigration rate of humans | H \(\times\) T\(^{-1}\) |
\(\psi _h\)
| Per capita birth rate of humans | T\(^{-1}\) |
\(\psi _v\)
| Per capita birth rate of mosquitoes | T\(^{-1}\) |
\(\sigma _v\)
| Number of times one mosquito would want to bite humans per unit time, if humans were freely available. This is a function of the mosquito’s gonotrophic cycle (the amount of time a mosquito requires to produce eggs) and its anthropophilic rate (its preference for human blood) | T\(^{-1}\) |
\(\sigma _h\)
| The maximum number of mosquito bites a human can have per unit time. This is a function of the human’s exposed surface area | T\(^{-1}\) |
\(\beta _{hv}\)
| Probability of transmission of infection from an infectious mosquito to a susceptible human, given that a contact between the two occurs | − |
\(\beta _{vh}\)
| Probability of transmission of infection from an infectious human to a susceptible mosquito, given that a contact between the two occurs | − |
\(\tilde{\beta }_{hv}\)
| Probability of transmission of infection from a recovered (asymptomatic carrier) human to a susceptible mosquito, given that a contact between the two occurs | − |
\(\nu _h\)
| Per capita rate of progression of humans from the exposed state to the infectious state. \(1/\nu _h\) is the average duration of the latent period | T\(^{-1}\) |
\(\nu _v\)
| Per capita rate of progression of mosquitoes from the exposed state to the infectious state. \(1/\nu _v\) is the average duration of the latent period | T\(^{-1}\) |
\(\gamma _h\)
| Per capita recovery rate for humans from the infectious state to the recovered state. \(1/\gamma _h\) is the average duration of the infectious period | T\(^{-1}\) |
\(\delta _h\)
| Per capita disease-induced death rate for humans | T\(^{-1}\) |
\(\rho _h\)
| Per capita rate of loss of acquired temporary immunity for humans. \(1/\rho _h\) is the average duration of the immune period | T\(^{-1}\) |
\(\mu _{1h}\)
| Density-independent part of the death (and emigration) rate for humans | T\(^{-1}\) |
\(\mu _{2h}\)
| Density-dependent part of the death (and emigration) rate for humans | H \(\times\) T\(^{-1}\) |
\(\mu _{1h}\)
| Density-independent part of the death (and emigration) rate for mosquitoes | T\(^{-1}\) |
\(\mu _{2h}\)
| Density-dependent part of the death (and emigration) rate for mosquitoes | M \(\times\) T\(^{-1}\) |
l
| Change, \(\Delta Y^{l}_{k,\zeta }(t)\) | Probability, \(p^{l}_{k,\zeta }(t)\) | Description |
---|---|---|---|
1 |
\([1, 0, 0, 0, 0, 0, 0]^T\)
|
\((\Lambda _h + \psi _h N_{h,\zeta }) \Delta t\)
| A new host enters the human susceptible class |
2 |
\([1, 0, 0, -1, 0, 0, 0]^T\)
|
\(\rho _h R_{h,\zeta } \Delta t\)
| A recovered host becomes susceptible again |
3 |
\([-1, 1, 0, 0, 0, 0, 0]^T\)
|
\(\frac{\sigma _v \sigma _h \beta _{hv}I_{v,\zeta }S_{h,\zeta }}{\sigma _v N_{v,\zeta } + \sigma _h N_{h,\zeta }}\Delta t\)
| A susceptible host enters exposed state |
4 |
\([-1, 0, 0, 0, 0, 0, 0]^T\)
|
\((\mu _{1h} + \mu _{2h} N_{h,\zeta })S_{h,\zeta } \Delta t\)
| A susceptible host dies |
5 |
\([0, -1, 1, 0, 0, 0, 0]^T\)
|
\(\nu _h E_{h,\zeta } \Delta t\)
| An exposed host enters infectious state |
6 |
\([0, -1, 0, 0, 0, 0, 0]^T\)
|
\((\mu _{1h} + \mu _{2h} N_{h,\zeta })E_{h,\zeta } \Delta t\)
| An exposed host dies |
7 |
\([0, 0, -1, 1, 0, 0, 0]^T\)
|
\(\gamma _h I_{h,\zeta } \Delta t\)
| An infectious host enters recovered state |
8 |
\([0, 0, -1, 0, 0, 0, 0]^T\)
|
\((\mu _{1h} + \mu _{2h} N_{h,\zeta } + \delta _h) I_{h,\zeta } \Delta t\)
| An infectious host dies |
9 |
\([0, 0, 0, -1, 0, 0, 0]^T\)
|
\((\mu _{1h} + \mu _{2h} N_{h,\zeta }) R_{h,\zeta } \Delta t\)
| A recovered host dies |
10 |
\([0, 0, 0, 0, 1, 0, 0]^T\)
|
\(\psi _v N_{v,\zeta } \Delta t\)
| A new mosquito enters the vector susceptible class |
11 |
\([0, 0, 0, 0, -1, 1, 0]^T\)
|
\(\frac{\sigma _v \sigma _h \beta _{hv}I_{v,\zeta } S_{h,\zeta }}{\beta _{vh} I_{h,\zeta } + \tilde{\beta }_{vh} R_{h,\zeta }} \Delta t\)
| A susceptible vector enters exposed state |
12 |
\([0, 0, 0, 0, -1, 0, 0]^T\)
|
\((\mu _{1v} + \mu _{2v} N_{v,\zeta }) S_{v,\zeta } \Delta t\)
| A susceptible vector dies |
13 |
\([0, 0, 0, 0, 0, -1, 1]^T\)
|
\(\nu _v E_{v,\zeta } \Delta t\)
| An exposed vector enters infectious state |
14 |
\([0, 0, 0, 0, 0, -1, 0]^T\)
|
\((\mu _{1v} + \mu _{2v} N_{v,\zeta }) E_{v,\zeta } \Delta t\)
| An exposed vector dies |
15 |
\([0, 0, 0, 0, 0, 0, -1]^T\)
|
\((\mu _{1v} + \mu _{2v} N_{v,\zeta }) I_{v,\zeta } \Delta t\)
| An infectious vector dies |