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Article Contents

# A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity

• For an intervention against the spread of communicable diseases, the idealized situation is when individuals fully comply with the intervention and the exposure to the infectious agent is comparable across all individuals. Some level of non-compliance is likely where the intervention is widely implemented. The focus is on a more accurate view of its effects population-wide. A frailty model is applied. Qualitative analysis, in mathematical terms, reveals how large variability in compliance renders the intervention less effective. This finding sharpens our vague, intuitive and empirical notions. An effective reproduction number in the presence of frailty is defined and is shown to be invariant with respect to the time-scale of disease progression. This makes the results in this paper valid for a wide spectrum of acute and chronic infectious diseases. Quantitative analysis by comparing numerical results shows that they are also robust with respect to assumptions on disease progression structure and distributions, such as with or without the latent period and the assumed distributions of latent and infectious periods.

Mathematics Subject Classification: Primary: 92B05, 60E10, 60E15.

 Citation:

• Figure 1.  Length-biasness in a prevalence cohort: the red sections represent the infectious period. Only 2 individuals with longer infectious periods are included in the prevalence cohort.

Figure 2.  Verbal and graphic presentation for the convex order showing that $X_{2}$ is more "spread out" than $X_{1}$.

Figure 3.  Schematic presentation of $R_{v}(\phi)/R_{0}$ and $R_{c}(\phi)/R_{0}$ as two survival functions standardized by the scale parameter $\lambda.$

Figure 4.  A schmatic presentation of $G(z;R_{0},f_{G}).$

Figure 5.  Shapes of $\xi(z)$ and $\overline{F}^{(frailty)}(x)=\left( 1+\phi xv\right) ^{-1/v}$.

Figure 6.  Comparing $G_{Gamma}(z)$ given by (16) and $G_{inv-Gaussian}(z)$ given by (18) against $G^{\ast}(z)=\frac{R_{0}}{1+\left( R_{0}-1\right) z}.$

Figure 7.  Integrand in (17) at $R_{0}=3;$ $\theta$ from $0.01$ to $\infty$ $.$

Figure 8.  $f_{G}(x)=\frac{1}{\mu}\overline{F}_{I}(x)$ when the infectious period is Gamma distributed.

Figure 9.  Shapes of $f_{G}(x)$ in the special cases in the presence of a latent period.

Figure 10.  The left panel is for the value $R_{v}(\rho)$ and the right panel is for the correponding final sizes. At each level of $R_{0}$, there are $16$ points plotted at each $v$.

Figure 11.  At each $R_{0}$ level, there are $16$ points plotted at $v=0.00,...,4.00$ by $0.5$ increment with trend lines as the averages.

Table 1.  Tabulation of $R_{v}^{\ast}(R_{0})$ in (14) along with $R_{v}(R_{0},\theta)$ with respect to Gamma and inverse-Gaussian distributed infectious period at $R_{0}=2$ and $R_{0}=3,$ noticing that $R_{v}^{\ast}(R_{0})$ is identical to Gamma distributed infectious period with $\theta=1.$

 $R_{0}=2$ $v$ $R_{v}^{\ast}$ $\theta$ $\theta>0:$ Gamma $\theta>0:$ inv-Gaussian $\rightarrow0$ $0.2$ $1$ $2$ $10$ $0.2$ $1$ $2$ $10$ $0.25$ $1.059$ $1.061$ $1.061$ $1.059$ $1.057$ $1.055$ $1.060$ $1.056$ $1.054$ $1.049$ $0.50$ $1.109$ $1.113$ $1.112$ $1.109$ $1.108$ $1.105$ $1.112$ $1.106$ $1.103$ $1.095$ $1.00$ $1.193$ $1.196$ $1.195$ $1.193$ $1.191$ $1.188$ $1.195$ $1.189$ $1.185$ $1.176$ $2.00$ $1.311$ $1.313$ $1.313$ $1.311$ $1.310$ $1.308$ $1.312$ $1.308$ $1.304$ $1.297$ $R_{0}=3$ $R_{v}^{\ast}$ $\theta$ $\theta>0:$ Gamma $\theta>0:$ inv-Gaussian $\rightarrow0$ $0.2$ $1$ $2$ $10$ $0.2$ $1$ $2$ $10$ $0.25$ $1.109$ $1.132$ $1.123$ $1.109$ $1.103$ $1.095$ $1.123$ $1.105$ $1.095$ $1.077$ $0.50$ $1.211$ $1.244$ $1.232$ $1.211$ $1.201$ $1.188$ $1.231$ $1.204$ $1.187$ $1.156$ $1.00$ $1.384$ $1.425$ $1.411$ $1.384$ $1.372$ $1.355$ $1.410$ $1.374$ $1.352$ $1.308$ $2.00$ $1.637$ $1.677$ $1.663$ $1.637$ $1.624$ $1.606$ $1.662$ $1.626$ $1.603$ $1.554$

Table 2.  Constant infectious period without latent period.

 $R_{0}=2,$ $\rho=1.594/\mu$ $R_{0}=3,$ $\rho=2.821/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.061$ $0.113$ $1.120$ $0.25$ $1.132$ $0.224$ $1.190$ $1.00$ $1.196$ $0.309$ $1.577$ $1.00$ $1.426$ $0.531$ $2.024$ $2.00$ $1.313$ $0.436$ $2.510$ $2.00$ $1.678$ $0.681$ $4.253$ $3.00$ $1.394$ $0.506$ $4.056$ $3.00$ $1.847$ $0.750$ $9.352$ $4.00$ $1.454$ $0.551$ $6.668$ $4.00$ $1.971$ $0.789$ $21.512$

Table 3.  Exponential infectious period without latent period.

 $R_{0}=2,$ $\rho=1/\mu$ $R_{0}=3,$ $\rho=2/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.059$ $0.108$ $1.133$ $0.25$ $1.109$ $0.190$ $1.186$ $1.00$ $1.193$ $0.305$ $1.639$ $1.00$ $1.384$ $0.498$ $2.022$ $2.00$ $1.311$ $0.434$ $2.670$ $2.00$ $1.637$ $0.661$ $4.257$ $3.00$ $1.393$ $0.506$ $4.370$ $3.00$ $1.811$ $0.737$ $9.325$ $4.00$ $1.454$ $0.552$ $7.229$ $4.00$ $1.940$ $0.780$ $21.295$

Table 4.  Gamma distributed infectious period in four scenarios.

 Gamma distributed infectious period, variance/mean ratio =2 $R_{0}=2,$ $\theta=0.5,$ $\rho=0.719/\mu$ $R_{0}=3,$ $\theta=0.5,\rho=1.5/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.057$ $0.106$ $1.136$ $0.25$ $1.103$ $0.181$ $1.185$ $1.00$ $1.191$ $0.303$ $1.660$ $1.00$ $1.372$ $0.488$ $2.030$ $2.00$ $1.310$ $0.433$ $2.731$ $2.00$ $1.624$ $0.655$ $4.295$ $3.00$ $1.393$ $0.505$ $4.501$ $3.00$ $1.800$ $0.732$ $9.425$ $4.00$ $1.454$ $0.551$ $7.481$ $4.00$ $1.930$ $0.777$ $21.507$ Gamma distributed infectious period, variance/mean ratio =0.5 $R_{0}=2,$ $\theta=2,$ $\rho=1.236/\mu$ $R_{0}=3,$ $\theta=2,\rho=2.372/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.060$ $0.110$ $1.129$ $0.25$ $1.116$ $0.201$ $1.186$ $1.00$ $1.194$ $0.306$ $1.618$ $1.00$ $1.397$ $0.509$ $2.017$ $2.00$ $1.312$ $0.435$ $2.611$ $2.00$ $1.650$ $0.668$ $4.234$ $3.00$ $1.393$ $0.506$ $4.249$ $3.00$ $1.823$ $0.741$ $9.272$ $4.00$ $1.454$ $0.552$ $7.006$ $4.00$ $1.950$ $0.783$ $21.215$

Table 5.  Constant latent period and constant infectious period in six scenarios.

 The latent period is half the infectious period: $l=1/2$ $R_{0}=2,$ $l=0.5,$ $\rho=0.714/\mu$ $R_{0}=3,$ $l=0.5,\rho=1.153/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.057$ $0.105$ $1.097$ $0.25$ $1.136$ $0.229$ $1.156$ $1.00$ $1.184$ $0.294$ $1.463$ $1.00$ $1.428$ $0.533$ $1.842$ $2.00$ $1.298$ $0.421$ $2.209$ $2.00$ $1.676$ $0.680$ $3.685$ $3.00$ $1.378$ $0.493$ $3.444$ $3.00$ $1.844$ $0.748$ $7.968$ $4.00$ $1.438$ $0.540$ $5.537$ $4.00$ $1.967$ $0.788$ $18.345$ The latent period equals to the infectious period: $l=1$ $R_{0}=2,$ $l=1,$ $\rho=0.468/\mu$ $R_{0}=3,$ $l=1,\rho=0.748/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.056$ $0.104$ $1.094$ $0.25$ $1.136$ $0.230$ $1.153$ $1.00$ $1.182$ $0.292$ $1.451$ $1.00$ $1.429$ $0.533$ $1.829$ $2.00$ $1.296$ $0.419$ $2.182$ $2.00$ $1.677$ $0.681$ $3.658$ $3.00$ $1.376$ $0.491$ $3.398$ $3.00$ $1.845$ $0.749$ $7.919$ $4.00$ $1.436$ $0.539$ $5.461$ $4.00$ $1.968$ $0.788$ $18.257$ The latent period is twice the infectious period: $l=2$ $R_{0}=2,$ $l=2,$ $\rho=0.279/\mu$ $R_{0}=3,$ $l=2,\rho=0.443/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.056$ $0.103$ $1.093$ $0.25$ $1.136$ $0.230$ $1.152$ $1.00$ $1.182$ $0.291$ $1.446$ $1.00$ $1.430$ $0.534$ $1.823$ $2.00$ $1.295$ $0.418$ $2.170$ $2.00$ $1.678$ $0.681$ $3.647$ $3.00$ $1.375$ $0.491$ $3.377$ $3.00$ $1.845$ $0.749$ $7.899$ $4.00$ $1.435$ $0.538$ $5.428$ $4.00$ $1.968$ $0.788$ $18.222$

Table 6.  Exponentially distributed latent period and exponentially distributed infectious period.

 Mean latent period equals to the mean infectious period: $l=1$ $R_{0}=2,$ $\rho=0.414/\mu$ $R_{0}=3,$ $\rho=0.732/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.059$ $0.110$ $1.116$ $0.25$ $1.126$ $0.215$ $1.174$ $1.00$ $1.191$ $0.303$ $1.548$ $1.00$ $1.413$ $0.521$ $1.930$ $2.00$ $1.307$ $0.430$ $2.414$ $2.00$ $1.663$ $0.674$ $3.923$ $3.00$ $1.387$ $0.501$ $3.831$ $3.00$ $1.833$ $0.745$ $8.463$ $4.00$ $1.447$ $0.547$ $6.209$ $4.00$ $1.958$ $0.785$ $19.310$ Either $\mu_{E}=\mu,$ $\mu_{I}=2\mu$ or $\mu_{E}=2\mu,$ $\mu_{I}=\mu$ $R_{0}=2,$ $\rho=0.281/\mu$ $R_{0}=3,$ $\rho=0.5/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.059$ $0.109$ $1.117$ $0.25$ $1.124$ $0.213$ $1.174$ $1.00$ $1.191$ $0.302$ $1.552$ $1.00$ $1.410$ $0.519$ $1.931$ $2.00$ $1.307$ $0.430$ $2.425$ $2.00$ $1.660$ $0.673$ $3.926$ $3.00$ $1.387$ $0.501$ $3.851$ $3.00$ $1.831$ $0.744$ $8.463$ $4.00$ $1.448$ $0.547$ $6.242$ $4.00$ $1.956$ $0.784$ $19.298$

Table 7.  Constant latent period and exponentially distributed infectious period.

 The latent period is half the infectious period: $l=1/2$ $R_{0}=2,$ $\rho=0.53249/\mu$ $R_{0}=3,$ $\rho=0.90659/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.057$ $0.106$ $1.106$ $0.25$ $1.127$ $0.216$ $1.160$ $1.00$ $1.186$ $0.296$ $1.501$ $1.00$ $1.413$ $0.521$ $1.855$ $2.00$ $1.301$ $0.424$ $2.289$ $2.00$ $1.661$ $0.673$ $3.701$ $3.00$ $1.381$ $0.495$ $3.581$ $3.00$ $1.831$ $0.744$ $7.953$ $4.00$ $1.441$ $0.542$ $5.757$ $4.00$ $1.956$ $0.784$ $18.210$ The latent period equals to the infectious period: $l=1$ $R_{0}=2,$ $\rho=0.3748/\mu$ $R_{0}=3,$ $,\rho=0.61764/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.057$ $0.105$ $1.100$ $0.25$ $1.131$ $0.223$ $1.156$ $1.00$ $1.184$ $0.294$ $1.476$ $1.00$ $1.421$ $0.527$ $1.840$ $2.00$ $1.298$ $0.421$ $2.232$ $2.00$ $1.669$ $0.6769$ $3.673$ $3.00$ $1.378$ $0.493$ $3.482$ $3.00$ $1.838$ $0.746$ $7.921$ $4.00$ $1.438$ $0.540$ $5.596$ $4.00$ $1.962$ $0.786$ $18.201$ The latent period is twice the infectious period: $l=2$ $R_{0}=2,$ $\rho=0.2393/\mu$ $R_{0}=3,$ $\rho=0.386/\mu.$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.056$ $0.104$ $1.096$ $0.25$ $1.134$ $0.227.$ $1.154$ $1.00$ $1.183$ $0.292$ $1.458$ $1.00$ $1.426$ $0.531$ $1.831$ $2.00$ $1.297$ $0.420$ $2.196$ $2.00$ $1.674$ $0.679$ $3.658$ $3.00$ $1.377$ $0.492$ $3.421$ $3.00$ $1.842$ $0.748$ $7.909$ $4.00$ $1.437$ $0.539$ $5.498$ $4.00$ $1.966$ $0.787$ $18.215$

Table 8.  Exponentially distributed latent period and constant infectious period.

 The latent period is half the infectious period: $l=1/2$ $R_{0}=2,$ $\rho=0.7788/\mu$ $R_{0}=3,$ $\rho=0.1325/\mu$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.059$ $0.109$ $1.109$ $0.25$ $1.131$ $0.222$ $1.169$ $1.00$ $1.189$ $0.300$ $1.517$ $1.00$ $1.422$ $0.528$ $1.907$ $2.00$ $1.304$ $0.427$ $2.340$ $2.00$ $1.671$ $0.678$ $3.865$ $3.00$ $1.385$ $0.499$ $3.694$ $3.00$ $1.840$ $0.747$ $8.351$ $4.00$ $1.445$ $0.545$ $5.972$ $4.00$ $1.964$ $0.787$ $19.123$ The latent period equals to the infectious period: $l=1$ $R_{0}=2,$ $\rho=0.5432/\mu$ $R_{0}=3,$ $,\rho=0.9426/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.058$ $0.108$ $1.112$ $0.25$ $1.127$ $0.216$ $1.168$ $1.00$ $1.189$ $0.300$ $1.528$ $1.00$ $1.414$ $0.522$ $1.903$ $2.00$ $1.304$ $0.427$ $2.362$ $2.00$ $1.664$ $0.674$ $3.849$ $3.00$ $1.385$ $0.499$ $3.731$ $3.00$ $1.834$ $0.745$ $8.298$ $4.00$ $1.445$ $0.545$ $6.030$ $4.00$ $1.958$ $0.785$ $18.965$ The latent period is twice the infectious period: $l=2$ $R_{0}=2,$ $\rho=0.3455/\mu$ $R_{0}=3,$ $\rho=0.6186/\mu$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $v$ $R_{c}(\rho,\rho)$ $\eta$ $\rho^{\prime}/\rho$ $0.00$ $1.000$ $0.000$ $1.000$ $0.00$ $1.000$ $0.000$ $1.000$ $0.25$ $1.058$ $0.108$ $1.116$ $0.25$ $1.122$ $0.209$ $1.170$ $1.00$ $1.189$ $0.300$ $1.548$ $1.00$ $1.405$ $0.515$ $1.910$ $2.00$ $1.305$ $0.428$ $2.410$ $2.00$ $1.655$ $0.670$ $3.860$ $3.00$ $1.386$ $0.500$ $3.816$ $3.00$ $1.826$ $0.742$ $8.301$ $4.00$ $1.446$ $0.546$ $6.173$ $4.00$ $1.951$ $0.783$ $18.915$
•  R. Anderson and  R. May,  Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, 1991. O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2013. K. Dietz, Some problems in the theory of infectious diseases transmission and control, in Epidemic Models: their Structure and Relation to Data (ed. Denis Mollison), Cambridge University Press, (1995), 3-16. M. Greenwood  and  G. Yule , An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to occurrence of multiple attacks of diseases or of repeated accidents, Journal of the Royal Statistical Society, 83 (1920) , 255-279.  doi: 10.2307/2341080. S. Goldstein , Operational representation of Whittaker's confluent hypergeometric function and Weber's parabolic cylinder function, Proc. London Math. Soc., 2 (1932) , 103-125.  doi: 10.1112/plms/s2-34.1.103. P. Hougaard , Life table methods for heterogenous populations: Distributions describing the heterogeneity, Biometrika, 71 (1984) , 75-83.  doi: 10.1093/biomet/71.1.75. P. Hougaard , Frailty models for survival data, Lifetime Data Analysis, 1 (1995) , 255-273.  doi: 10.1007/BF00985760. E. K. Lenzi , E. P. Borges  and  R. S. Mendes , A q-generalization of Laplace transforms, Journal of Physics A: Mathematical and General, 32 (1999) , 8551-8561.  doi: 10.1088/0305-4470/32/48/314. K. S. Lomax , Business failures, another example of the analysis of failure data, Journal of the American Statistics Association, 49 (1954) , 847-852.  doi: 10.1080/01621459.1954.10501239. J. Ma  and  D. Earn , Generality of the final size formula for an epidemic of a newly invading infectious disease, Bulletin of Mathematical Biology, 68 (2006) , 679-702.  doi: 10.1007/s11538-005-9047-7. A. W. Marshall and I. Olkin, Life Distributions, Structure of Nonparametric, Semiparametric and Parametric Families, Springer, 2007. S. R. Naik, The q-Laplace transforms and applications, Chapter 7 of Pathway Distributions, Autoregressive Processes and Their Applications, PhD Thesis, Mahatima Gandhi University, India, (2008). A. Olivieri , Heterogeneity in survival models, applications to pensions and life annuities, Belgian Actuarial Bulletin, 6 (2006) , 23-39.  doi: 10.2139/ssrn.913770. S. Picoli , R. S. Mendes , L. C. Malacarne  and  R. P. B. Santos , q-distributions in complex systems: A brief review, Brazilian Journal of Physics, 39 (2009) , 468-474.  doi: 10.1590/S0103-97332009000400023. S. Ross, Stochastic Processes, Second Edition, Wiley and Sons Inc, 1996. R. K. Saxena , A study of the generalized Stieltjes transform, Lecturer in Mathematics, M.B. College, Udaipur, 25 (1959) , 340-355. J. F. Steffensen , Deux problèms du calcul des probabilités, Ann. Inst. H. Poincaré, 3 (1933) , 319-344. J. W. Vaupel , K. G. Manton  and  E. Stallard , The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography, 16 (1979) , 439-354.  doi: 10.2307/2061224. H. W. Watson  and  F. Galton , On the probability of extinction of families, J. Anthropol. Inst. Great Britain and Ireland, 4 (1874) , 138-144. P. Yan  and  Z. Feng , Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness, Mathematical Biosciences, 224 (2010) , 43-52.  doi: 10.1016/j.mbs.2009.12.007. O. Yürekli , A theorem on the generalized Stieltjes transform and its applications, Journal of Mathematical Analysis and Applications, 168 (1992) , 63-71.  doi: 10.1016/0022-247X(92)90189-K.
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