Potential for international spread of wild poliovirus via travelers

Abstract

Background

The endgame of polio eradication is hampered by the international spread of poliovirus via travelers. In response to ongoing importations of poliovirus into polio-free countries, on 5 May 2014, WHO’s Director-General declared the international spread of wild poliovirus a public health emergency of international concern. Our objective was to develop a mathematical model to estimate the international spread of polio infections.

Methods

Our model took into account polio endemicity in polio-infected countries, population size, polio immunization coverage rates, infectious period, the asymptomatic-to-symptomatic ratio, and also the probability of a traveler being infectious at the time of travel. We applied our model to three scenarios: (1) number of exportations of both symptomatic and asymptomatic polio infections out of currently polio-infected countries, (2) the risk of spread of poliovirus to Saudi Arabia via Hajj pilgrims, and (3) the importation risk of poliovirus into India.

Results

Our model estimated 665 polio exportations (>99 % of which were asymptomatic) from nine polio-infected countries in 2014, of which 78.3 % originated from Pakistan. Our model also estimated 21 importations of poliovirus into Saudi Arabia via Hajj pilgrims and 20 poliovirus infections imported to India in the same year.

Conclusion

The extent of importations of asymptomatic and symptomatic polio infections is substantial. For countries that are vulnerable to polio outbreaks due to poor national polio immunization coverage rates, our newly developed model may help guide policy-makers to decide whether imposing an entry requirement in terms of proof of vaccination against polio would be justified.

Box 1. Model equations as explained in the main text

The general equation for the variation in polio cases in country i, in year t, I _i (t), is given by:

$$ \frac{d{I}_i(t)}{dt}={\lambda}_i(t){S}_i(t)-\left({\mu}_i+{\gamma}_i\right){I}_i(t) $$

(1)

where λ _i (t) is the force of infection (per capita number of new infections per time), S _i (t) is the number of susceptible individuals at time _t , $\frac{1}{\mu_i}$ is the life expectancy of the population in country _i , and $\frac{1}{\gamma_i}$ is the average duration of infection in country i. The number of susceptible individuals is given by the size of the population multiplied by one minus the proportion of polio vaccination coverage. Equation (1) describes the dynamics of infectious individuals. It expresses the time variation in the number of infective people and it contains two terms, one income (positive) term provided by the disease incidence (with rate λ _i (t)S _i (t)) and one outcome (negative) term provided by natural deaths (with rate μ_i), and the recovery of infected individuals (with rate γ _i ).

Equation (1) can be integrated by standard methods, resulting in:

$$ {I}_i(t)={I}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right){e}^{-\left({\mu}_i+{\gamma}_i\right)\left(t-{t}^{\prime}\right)}dt $$

(2)

We consider ${\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)dt$, the reported number of cases along the year t in country i, and assume this annual incidence as constant in each year. Equation (2), therefore, simplifies to

$$ {I}_i(t)={I}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\varLambda}_i(t)\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{\left({\mu}_i+{\gamma}_i\right)}\right\} $$

(3)

Where ${\varLambda}_i(t)={\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)dt$ for each individual year t.

Dividing Equation (3) by the number of susceptible individuals S _i (t) in year t in country i, we obtain the prevalence p _i (t) in year t in country i:

$$ {p}_i(t)={p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\varLambda}_i(t)\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)\left({\mu}_i+{\gamma}_i\right)}\right\} $$

(4)

Multiplying p _i (t) by the number of non-immune travelers from country i, we obtain the expected number of polio cases among travelers.

Box 2. Sensitivity of the model to the parameters

For a small variation of parameter Par _i , ΔPar _i , the variation in the parameter-dependent variable π, Δπ, is given by the well-known error propagation formula²⁶:

$$ \varDelta \pi ={\displaystyle \sum_i\frac{\partial \pi }{\partial Pa{r}_i}\times \varDelta Pa{r}_i} $$

(5)

The relative variation in the risk π, Δπ/π, as a function of the relative variation in the parameters ΔPar _i /Par _i , is therefore:

$$ \frac{\varDelta \pi }{\pi }={\displaystyle \sum_i Pa{r}_i\frac{\partial \pi }{\partial Pa{r}_i}\times \frac{\varDelta Pa{r}_i}{Pa{r}_i}\times \frac{1}{\pi }} $$

(6)

We calculated the sensitivity of Equation (3) of Box 1 to the natural mortality rate of hosts, μ:

$$ \frac{\varDelta p(t)}{p(t)}=\frac{\partial p(t)}{\partial \mu}\times \frac{\varDelta \mu }{\mu}\times \frac{\mu }{p(t)} $$

(7)

Where

$$ \frac{\partial p(t)}{\partial \mu }=-t{p}_i(0){e}^{-\left({\mu}_2+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\frac{\partial \lambda }{\partial \gamma } $$

(8)

is the partial derivative of time-dependent prevalence with respect to the natural mortality rate of hosts μ.

An identical result is obtained for $\frac{\partial p(t)}{\partial \gamma }$.

The result of $\frac{\partial p(t)}{\partial \gamma }$ is much simpler:

$$ \frac{\partial p(t)}{\partial \lambda }=\frac{\left[1- \exp \left(-\left(\mu +\gamma \right)t\right)\right]}{S(t)\left(\mu +\gamma \right)} $$

(9)

Therefore, the equations for the sensitivity analysis are:

$$ \begin{array}{c}\hfill \frac{\varDelta p(t)}{\mu }=-t{p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\hfill \\ {}\hfill \frac{\partial \lambda }{\partial \mu}\times \left\{\frac{\varDelta \mu }{\mu}\times \frac{p(t)}{\pi}\right\}\hfill \end{array} $$

(10)

$$ \begin{array}{l}\frac{\varDelta p(t)}{\gamma }=-t{p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\\ {}\frac{\partial \lambda }{\partial \mu}\times \left\{\frac{\varDelta \mu }{\gamma}\times \frac{p(t)}{\gamma}\right\}\end{array} $$

(11)

$$ \frac{\varDelta p(t)}{\lambda }=\frac{\left[1- \exp \left(-\left(\mu +\gamma \right)t\right)\right]}{S(t)\left(\mu +\gamma \right)}\times \frac{\varDelta \mu }{\lambda}\times \frac{p(t)}{\lambda } $$

(12)

As the sensitivity of the prevalence to the parameters is an explicit function of time, as time passes (t → ∞), the values settle to:

$$ \frac{\varDelta p}{\lambda}\cong \frac{\varDelta \mu }{\lambda } $$

(13)

that is, for every 1 % variation in the force of infection there will be approximately 1 % variation in the time-dependent prevalence.

For the other two parameters μ and γ the results are:

$$ \frac{\varDelta p}{\mu}\cong \frac{1}{12}\frac{\varDelta \mu }{\mu } $$

(14)

And

$$ \frac{\varDelta p}{\gamma}\cong -\frac{1}{12}\frac{\varDelta \gamma }{\gamma } $$

(15)

that is, the model is about 12 times less sensitive to μ and γ than to λ.