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Epidemic Modellingmodelling: Expectationexpectation of time of infection given the distribution of transmission and recovery

TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution. If not, is it at least possible to express: \begin{equation} I=\int_0^\infty \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moments of $ \psi_\text{rec}$ and $\psi_\text{inf}$?

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

Epidemic Modelling: Expectation of time of infection given the distribution of transmission and recovery

TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution. If not, is it at least possible to express: \begin{equation} I=\int_0^\infty \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moments of $ \psi_\text{rec}$ and $\psi_\text{inf}$?

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

Epidemic modelling: expectation of time of infection given the distribution of transmission and recovery

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution. If not, is it at least possible to express: \begin{equation} I=\int_0^\infty \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moments of $ \psi_\text{rec}$ and $\psi_\text{inf}$?

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

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papad
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TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution. If not, is it at least possible to express: \begin{equation} I=\int_0^\infty \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moments of $ \psi_\text{rec}$ and $\psi_\text{inf}$?

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution.

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution. If not, is it at least possible to express: \begin{equation} I=\int_0^\infty \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moments of $ \psi_\text{rec}$ and $\psi_\text{inf}$?

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?

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papad
papad
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Epidemic Modelling: Expectation of time of infection given the distribution of transmission and recovery

TLDR;

Can I express the expected value of \begin{equation} \langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation} in terms of the moment(s) of the two other distributions $\psi_\text{inf},\psi_\text{rec}$ where $\Psi_\text{rec}(\tau)$ is the survival function of the recover distribution.

Consider the following model: An infected individual will attempt to transmit the disease to a healthy neighbour at a random time whose distribution is $\psi_\text{inf}(\tau)$ such that $\langle \tau\rangle_\text{inf}$ is the expected time of transmission after initially contracting the disease. The probability that an individual stays infected for a duration $\tau$ without recovering and transmits the disease to the healthy neighbour in the next infinitesimal time interval $\mathrm{d}\tau$ is given by: \begin{equation} \psi_\text{total}(\tau)\mathrm{d}\tau= \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau \end{equation}

Can I express the expected time of a successful transmission $\langle \tau\rangle_\text{total}$ in terms of the moments of the times of (attempted) infection and recovery: $\langle \tau\rangle_\text{inf},\langle \tau\rangle_\text{rec}$?