Evaluating vaccination effectiveness of group-specific fractional-dose strategies

Zhimin Chen; Kaihui Liu; Xiuxiang Liu

doi:10.3934/dcdsb.2021062

Article Contents

2022, Volume 27, Issue 2: 717-748. Doi: 10.3934/dcdsb.2021062

This issue Previous Article Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response Next Article Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

Evaluating vaccination effectiveness of group-specific fractional-dose strategies

1.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China
2.
Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, China
3.
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

^* Corresponding author: Zhimin Chen
^* Corresponding author: Zhimin Chen

Received: September 2020

Revised: January 2021

Early access: February 2021

Published: February 2022

Supported in part by the International Training Project for Outstanding Young Scientific Research Talents in Guangdong Universities in 2018 at South China Normal University, the NNSF of China, the Research Grants of Jiangsu University, the NSF of Guangdong Province and the General Program of the Natural Science Foundation of Guangdong Province of China

Abstract Full Text(HTML) Figure(8) / Table(5) Related Papers Cited by

Abstract

In this paper, we formulate a multi-group SIR epidemic model with the consideration of proportionate mixing patterns between groups and group-specific fractional-dose vaccination to evaluate the effects of fractionated dosing strategies on disease control and prevention in a heterogeneously mixing population. The basic reproduction number $ \mathscr{R}_0 $, the final size of the epidemic, and the infection attack rate are used as three measures of population-level implications of fractionated dosing programs. Theoretically, we identify the basic reproduction number, $ \mathscr{R}_0 $, establish the existence and uniqueness of the final size and the final size relation with $ \mathscr{R}_0 $, and obtain explicit calculation expressions of the infection attack rate for each group and the whole population. Furthermore, the simulation results suggest that dose fractionation policies take positive effects in lowering the $ \mathscr{R}_0 $, decreasing the final size and reducing the infection attack rate only when the fractional-dose influenza vaccine efficacy is high enough rather than just similar to standard-dose. We find evidences that fractional-dose vaccination in response to influenza vaccine shortages take negative community-level effects. Our results indicate that the role of fractional dose vaccines should not be overestimated even though fractional dosing strategies could extend the vaccine coverage.

Keywords:

Mathematics Subject Classification: Primary: 37N25, 34C60; Secondary: 92D40.

Citation:

Full Text(HTML)

Figure 1. The transition diagram of the group-specific fractional-dose vaccination model with $ K $ mixed groups. In $ l $-th group, the flux of new infected individuals in subgroups $ I^{(l)}_u $ and $ I^{(l)}_v $ are denoted as $ F^{(l)}_{UI} = \sum^{K}_{m = 1}\frac{c_l\beta_{lm}}{N}\left(I^{(m)}_u+p^{(m)}_i \varepsilon^{(m)}_i\left(n_m\right)I^{(m)}_v\right)S^{(l)}_u $ and $ F^{(l)}_{VI} = \sum^{K}_{m = 1}\frac{c_l\beta_{lm}}{N}\left(I^{(m)}_u+p^{(m)}_i \varepsilon^{(m)}_i\left(n_m\right)I^{(m)}_v\right)p^{(l)}_s \varepsilon^{(l)}_s\left(n_l\right)S^{(l)}_v $ respectively, while the flux of new recovered individuals in compartment $ R $ from subgroup $ I^{(l)}_u $ and subgroup $ I^{(l)}_v $ are respectively represented as $ F^{(l)}_{UR} = \gamma_lI^{(l)}_u $ and $ F^{(l)}_{VR} = \gamma_lp^{(l)}_r \varepsilon^{(l)}_r\left(n_l\right)I^{(l)}_v $, where $ l = 1,\dots,K $

Download: Full-size image PowerPoint slide

Figure 2. Illustration of the distributions of the basic reproduction number $ \mathscr{R}_0 $ (2(a)), the final size related quantity $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ (2(b)), the outbreak size (2(c)), the peaking time (2(d)), the $ IAR $ (2(e)) and the number of vaccines (2(f)) along standard-dose vaccine coverage for different target vaccination groups. In each plot, the red line with square markers indicates scenario $ 1 $ (vaccinating only group $ 1 $), the red line with asterisk markers scenario $ 2 $ (vaccinating only group $ 2 $), the green line with diamond markers scenario $ 3 $ (vaccinating only group $ 3 $), the magenta line with circle markers scenario $ 4 $ (vaccinating groups $ 1 $ and $ 2 $), the blue line with downward-pointing triangle markers scenario $ 5 $ (vaccinating groups $ 1 $ and $ 3 $), the cyan line with upward-pointing triangle markers scenario $ 6 $ (vaccinating groups $ 2 $ and $ 3 $), and the black line with star markers scenario $ 7 $ (vaccinating the entire population)

Download: Full-size image PowerPoint slide

Figure 3. The variations of the basic reproduction number $ \mathscr{R}_0 $ ((g1)), the final size related quantity $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ ((h1)), the outbreak size ((i1)), the peaking time ((j1)), and the $ IAR $ ((k1)) over the standard-dose vaccine coverage for the entire population with different vaccine distributions in the first policy. In each plot, the red solid line with plus sign markers indicates the case of first group $ 1 $ then group $ 2 $ and last group $ 3 $, the green dashed line with right-pointing triangle markers the case of first group $ 1 $ then group $ 3 $ and last group $ 2 $, the black dash-dotted line with left-pointing triangle markers the case of first group $ 2 $ then group $ 1 $ and last group $ 3 $, the blue solid line with point markers the case of first group $ 2 $ then group $ 3 $ and last group $ 1 $, the magenta dotted line with six-pointed star markers the case of first group $ 3 $ then group $ 1 $ and last group $ 2 $, the cyan dashed line with cross markers the case of first group $ 3 $ then group $ 2 $ and last group $ 1 $

Download: Full-size image PowerPoint slide

Figure 4. The variations of the basic reproduction number $ \mathscr{R}_0 $ ((g2)), the final size related quantity $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ ((h2)), the outbreak size ((i2)), the peaking time ((j2)), and the $ IAR $ ((k2)) with respect to the standard-dose vaccine coverage for the entire population with different vaccine distributions in the second policy. In each plot, the cyan dash-dotted line with upward-pointing triangle markers indicates the case of first groups $ 1 $ and $ 2 $ and then group $ 3 $, the green dotted line with asterisk markers the case of first group $ 3 $ and then groups $ 1 $ and $ 2 $, the red dashed line with square markers the case of first groups $ 1 $ and $ 3 $ and then group $ 2 $, the blue solid line with circle markers the case of first group $ 2 $ then groups $ 1 $ and $ 3 $, the magenta dashed line with downward-pointing triangle markers the case of first groups $ 2 $ and $ 3 $ and then group $ 1 $, the black solid line with diamond markers the case of first group $ 1 $ and then groups $ 2 $ and $ 3 $

Download: Full-size image PowerPoint slide

Figure 5. Simulations showing the effect of changing $ \varepsilon^{(2)}_{i,s}(5) $ and $ \varepsilon^{(2)}_{e,f,r}(5) $ on $ \mathscr{R}_0 $ ((q1)), $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ ((r1)) and IAR ((s1)) when first group $ 1 $ then group $ 2 $ and last group $ 3 $ distribution is carried out. The white lines are contour lines and the black lines are the baselines which stand for corresponding values under the standard-dose only strategy in each subfigure. The values of $ p_1 $, $ p_2 $ and $ p_3 $ are $ 1 $, $ 0.2 $ and $ 0.5 $, respectively

Download: Full-size image PowerPoint slide

Figure 6. Simulations showing the effect of changing $ \varepsilon^{(2)}_{i,s}(5) $ and $ \varepsilon^{(2)}_{e,f,r}(5) $ on $ \mathscr{R}_0 $ ((q2)), $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ ((r2)) and IAR ((s2)) when first group $ 1 $ then group $ 3 $ and last group $ 2 $ distribution is carried out. The white lines are contour lines and the black lines are the baselines which stand for corresponding values under the standard-dose only strategy in each subfigure. The values of $ p_1 $, $ p_2 $ and $ p_3 $ are $ 1 $, $ 0.1 $ and $ 1 $, respectively

Download: Full-size image PowerPoint slide

Figure 7. Simulations showing the effect of changing $ \varepsilon^{(2)}_{i,s}(5) $ and $ \varepsilon^{(2)}_{e,f,r}(5) $ on $ \mathscr{R}_0 $ ((q3)), $ \sum^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ ((r3)) and IAR ((s3)) when first group $ 1 $ then groups $ 2 $ and $ 3 $ distribution is carried out. The white lines are contour lines and the black lines are the baselines which stand for corresponding values under the standard-dose only strategy in each subfigure. The values of $ p_1 $, $ p_2 $ and $ p_3 $ are $ 1 $, $ 0.01*2000/(1170+250) $ and $ 0.01*2000/(1170+250) $, respectively

Download: Full-size image PowerPoint slide

Figure 8. Sensitivity analyses of $ \sum\limits^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ and IAR under the three optimal vaccine distribution schemes. (t1) and (u1), (t2) and (u2), and (t3) and (u3), present the effect of a change in each of $ \varepsilon^{(2)}_{e}(5) $, $ \varepsilon^{(2)}_{f}(5) $, $ \varepsilon^{(2)}_{i}(5) $, $ \varepsilon^{(2)}_{s}(5) $, and $ \varepsilon^{(2)}_{r}(5) $ (abbreviated as e, f, i, s, and r, respectively) on the $ \sum\limits^3_{l = 1}\left(S^{(l)}_u(\infty)+S^{(l)}_v(\infty)\right) $ and IAR under the first optimal distribution scheme with $ p_1 = 1 $, $ p_2 = 0.2 $ and $ p_3 = 0.5 $, the second optimal distribution strategy with $ p_1 = 1 $, $ p_2 = 0.1 $ and $ p_3 = 1 $, and the third optimal distribution measure with $ p_1 = 1 $, $ p_2 = p_3 = 0.01*2000/(1170+250) $, respectively. The size of the change is chosen by increasing and decreasing each parameter $ 10\% $ off its baseline value

Download: Full-size image PowerPoint slide

Table 1. List of the compartments in the model $ \left(1 \leq l \leq K\right) $

Compartment	Definition	Initial Value
$ S^{(l)}_u(t) $	Total number of susceptible individuals in the $ l $-th group who are unvaccinated or experiencing vaccine failure (i.e., the vaccine takes no effect in protecting the vaccinated individual from infection) at time $ t $	$ S^{(l)}_{u0} $
$ I^{(l)}_u(t) $	Total number of individuals in the $ l $-th group and in the infectious stage who are not vaccinated or experiencing vaccine failure at time $ t $	$ I^{(l)}_{u0} $
$ S^{(l)}_v(t) $	Total number of susceptible individuals in the $ l $-th group who are vaccinated and then gain partial protection at time $ t $	$ S^{(l)}_{v0} $
$ I^{(l)}_v(t) $	Total number of individuals in the $ l $-th group and in the infectious stage who are vaccinated and then gain partial protection at time $ t $	$ I^{(l)}_{v0} $
$ V^{(l)}(t) $	Total number of individuals in the $ l $-th group who are vaccinated and then gain full protection at time $ t $	$ V^{(l)}_0 $
$ R(t) $	Total number of recovered individuals at time $ t $	$ 0 $

| Show Table

DownLoad: CSV

Table 2. Description of the parameters in the model $ \left(1 \leq l, m \leq K\right) $

Parameter	Definition	Range
$ N $	Number of individuals in the total population	$ (0, \infty) $
$ p $	Proportion of population that standard-dose vaccines can coverage for the total population	$ (0, 1] $
$ p_l $	Vaccine coverage achievable with standard-dose vaccines for $ l $-th group subpopulation	$ (0, 1] $
$ n_l $	Fractionation number by each standard-dose vaccine for $ l $-th group subpopulation ($ n_lp_l\leq 1 $)	$ [1, 5] $
$ c_l $	Number of contacts per unit time an individual in the $ l $-th group makes	$ (0, \infty) $
$ \beta_{lm} $	Probability of infection given contact between a susceptible individual in the $ l $-th group and an infected individual in the $ m $-th group	$ (0,1] $
$ \gamma_l $	Recovery rate of infective individuals without vaccine protection in the $ l $-th group	$ (0, \infty) $
$ p^{(l)}_e $	Probability that a standard-dose vaccine takes effects for individuals in the $ l $-th group (providing full or partial protection)	$ (0,1] $
$ \varepsilon^{(l)}_e\left(n_l\right) $	The ratio of the probability that a $ n_l $-fold fractional-dose vaccine taking effects in the $ l $-th group relative to that a standard-dose vaccine taking effects in the same group	$ (0, 1] $
$ p^{(l)}_f $	Probability that a successfully vaccinated individual with a standard-dose vaccine gains full protection in the $ l $-th group	$ (0,1] $
$ \varepsilon^{(l)}_f\left(n_l\right) $	The ratio of the probability that a successfully vaccinated individual in the $ l $-th group gains full protection with a $ n_l $-fold fractional-dose vaccine relative to that with a standard-dose vaccine in the same group	$ (0, 1] $
$ p^{(l)}_i $	The ratio of the transmissibility of a vaccinated individual with a standard-dose vaccine in the $ l $-th group relative to that of an unvaccinated individual in the same group	$ (0,1] $
$ \varepsilon^{(l)}_i\left(n_l\right) $	The ratio of the transmissibility of a vaccinated individual with an $ n_l $-fold fractional-dose vaccine in the $ l $-th group relative to that with a standard-dose vaccine in the same group	$ [1, 1/p^{(l)}_i] $
$ p^{(l)}_s $	The ratio of susceptibility of a vaccinated individual in the $ l $-th group with a standard dose vaccine relative to that of an unvaccinated individual in the same group	$ (0,1] $
$ \varepsilon^{(l)}_s\left(n_l\right) $	The ratio of the susceptibility of a vaccinated individual with an $ n_l $-fold fractional-dose vaccine in the $ l $-th group relative to that with a standard-dose vaccine in the same group	$ [1, 1/p^{(l)}_s] $
$ p^{(l)}_r $	The ratio of the recoverability of a vaccinated individual in the $ l $-th group with a standard dose vaccine relative to that of an unvaccinated individual in the same group	$ [1, \infty) $
$ \varepsilon^{(l)}_r\left(n_l\right) $	The ratio of the recoverability of a vaccinated individual with an $ n_l $-fold fractional-dose vaccine in the $ l $-th group relative to that with a standard-dose vaccine in the same group	$ [1/p^{(l)}_r, 1] $

| Show Table

DownLoad: CSV

Table 3. Data of parameters and variables

Symbol	Description	Value	Unit	Reference
$ N $	Total population	$ 2,000 $	Individuals	Estimated
$ N^{(1)}_0 $	Children aged 0–17 years (group $ 1 $)	$ 580 $	Individuals	Estimated
$ N^{(2)}_0 $	Young adults aged 18–60 years (group $ 2 $)	$ 1170 $	Individuals	Estimated
$ N^{(3)}_0 $	Older adults aged more than $ 60 $ years (group $ 3 $)	$ 250 $	Individuals	Estimated
$ S^{(1)}_0 $	Susceptible persons in group $ 1 $	$ 577 $	Individuals	Estimated
$ S^{(2)}_0 $	Susceptible persons in group $ 2 $	$ 1163 $	Individuals	Estimated
$ S^{(3)}_0 $	Susceptible persons in group $ 3 $	$ 248 $	Individuals	Estimated
$ I^{(1)}_0 $	Infected persons in group $ 1 $	$ 3 $	Individuals	[24]
$ I^{(2)}_0 $	Infected persons in group $ 2 $	$ 7 $	Individuals	[24]
$ I^{(3)}_0 $	Infected persons in group $ 3 $	$ 2 $	Individuals	[24]
$ p $	Vaccine coverage for the total population	$ variable $	Dimensionless	Estimated
$ \gamma $	Average recover rate for overall population ($ \gamma_l=\gamma $ for $ l=1 $, $ 2 $, $ 3 $)	$ 1/4.1 $	days$ ^{-1} $	[18]

| Show Table

DownLoad: CSV

Table 4. List of estimated values and simulation results

Symbol	Value	Unit
$ c_1 $	$ 7 $	Dimensionless
$ c_2 $	$ 5 $	Dimensionless
$ c_3 $	$ 3 $	Dimensionless
$ \beta_1 $	$ 0.098 $	Dimensionless
$ \beta_2 $	$ 0.037 $	Dimensionless
$ \beta_3 $	$ 0.064 $	Dimensionless
$ \mathcal{IAR}^{(1)} $	$ 59\% $	Dimensionless
$ \mathcal{IAR}^{(2)} $	$ 29\% $	Dimensionless
$ \mathcal{IAR}^{(3)} $	$ 45\% $	Dimensionless
$ \mathcal{IAR} $	$ 40\% $	Dimensionless
$ \mathscr{R}_0 $	$ 1.35 $	Dimensionless

| Show Table

DownLoad: CSV

Table 5. Definition of the constant parameters

Symbol	Value	Unit	Reference
$ p^{(1)}_e $	$ 0.75 $	Dimensionless	Estimated
$ p^{(2)}_e $	$ 0.79 $	Dimensionless	[16]
$ p^{(3)}_e $	$ 0.77 $	Dimensionless	Estimated
$ p^{(1)}_f $	$ 0.59 $	Dimensionless	Estimated
$ p^{(2)}_f $	$ 0.63 $	Dimensionless	Estimated
$ p^{(3)}_f $	$ 0.61 $	Dimensionless	Estimated
$ p^{(1)}_i $	$ 0.67 $	Dimensionless	Estimated
$ p^{(2)}_i $	$ 0.65 $	Dimensionless	[18]
$ p^{(3)}_i $	$ 0.66 $	Dimensionless	Estimated
$ p^{(1)}_s $	$ 0.57 $	Dimensionless	Estimated
$ p^{(2)}_s $	$ 0.55 $	Dimensionless	[18]
$ p^{(3)}_s $	$ 0.56 $	Dimensionless	Estimated
$ p^{(1)}_r $	$ 2.3 $	Dimensionless	Estimated
$ p^{(2)}_r $	$ 2.5 $	Dimensionless	Estimated
$ p^{(3)}_r $	$ 2.4 $	Dimensionless	Estimated

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	V. Andreasen, The final size of an epidemic and its relation to the basic reproduction number, Bull. Math. Biol., 73 (2011), 2305-2321. doi: 10.1007/s11538-010-9623-3.
[2]	A. C. Campi-Azevedo, P. de Almeida Estevam, J. G. Coelho-Dos-Reis and et al., Subdoses of 17DD yellow fever vaccine elicit equivalent virological/immunological kinetics timeline, BMC Infect. Dis., 14 (2014), 1-12. doi: 10.1186/1471-2334-14-391.
[3]	Z. Chen, K. Liu, X. Liu and Y. Lou, Modelling epidemic with fractional-dose vaccination in response to limited vaccine supply, J. Theor. Biol., 468 (2020), 110085, 10pp. doi: 10.1016/j.jtbi.2019.110085.
[4]	L. Chow, M. Fan and Z. Feng, Dynamics of a multigroup epidemiological model with group-targeted vaccination strategies, J. Theor. Biol., 291 (2011), 56-64. doi: 10.1016/j.jtbi.2011.09.020.
[5]	J. Cui, Y. Zhang and Z. Feng, Influence of non-homogeneous mixing on final epidemic size in a meta-population model, J. Biol. Dyn., 13 (2019), 31-46. doi: 10.1080/17513758.2018.1484186.
[6]	D. Ding and X. Ding, Global stability of multi-group vaccination epidemic models with delays, Nonlinear Anal. Real World Appl., 12 (2011), 1991-1997. doi: 10.1016/j.nonrwa.2010.12.015.
[7]	S. Gandon, M. J. Mackinnon, S. Nee and A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751-755.
[8]	P. Guerin, L. Næss, C. Fogg and et al., Immunogenicity of fractional doses of tetravalent A/C/Y/W135 meningococcal polysaccharide vaccine: Results from a randomized non-inferiority controlled trial in uganda, PLoS Negl. Trop. Dis., 2 (2008), e342. doi: 10.1371/journal.pntd.0000342.
[9]	P. Haldar, P. Agrawal, P. Bhatnagar and et al., Fractional-dose inactivated poliovirus vaccine, India, Bull. World Health Organ., 97 (2019), 328-334. doi: 10.2471/BLT.18.218370.
[10]	J. K. Hale, Ordinary Differential Equations, New York: Robert E. Krieger Publishing Company, Inc., Huntington, 1980.
[11]	M. E. Halloran, C. J. Struchiner and I. M. Longini Jr, Study designs for evaluating different efficacy and effectiveness aspects of vaccines, Am. J. Epidemiol., 146 (1997), 789-803. doi: 10.1093/oxfordjournals.aje.a009196.
[12]	I. F. Hung, Y. Levin, K. K. To and et al., Dose sparing intradermal trivalent influenza (2010/2011) vaccination overcomes reduced immunogenicity of the 2009 H1N1 strain, Vaccine, 30 (2012), 6427-6435. doi: 10.1016/j.vaccine.2012.08.014.
[13]	E. Jonkera, M. van Ravenhorstbs, G. Berbersb and L. Visser, Safety and immunogenicity of fractional dose intradermal injection of two quadrivalent conjugated meningococcal vaccines, Vaccine, 36 (2018), 3727-3732. doi: 10.1016/j.vaccine.2018.05.064.
[14]	U. Joseph, M. Linster, Y. Suzuki and et al., Adaptation of pandemic H2N2 influenza a viruses in humans, J. Virol., 89 (2015), 2442-2447. doi: 10.1128/JVI.02590-14.
[15]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Math. Phys. Eng. Sci., 15 (1927), 700-721.
[16]	V. Künzi, J. M. Klap, M. K. Seiberling and et al., Immunogenicity and safety of low dose virosomal adjuvanted influenza vaccine administered intradermally compared to intramuscular full dose administration, Vaccine, 27 (2009), 3561-3567.
[17]	S. Lee, R. Morales and C. Castillo-Chavez, A note on the use of influenza vaccination strategies when supply is limited, Math. Biosci. Eng., 8 (2011), 171-182. doi: 10.3934/mbe.2011.8.171.
[18]	I. M. Longini, M. E. Halloran, A. Nizam and Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092.
[19]	J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol., 68 (2006), 679-702. doi: 10.1007/s11538-005-9047-7.
[20]	P. Magal, O. Seydi and G. Webb, Final size of an epidemic for a two-group SIR model, SIAM J. Appl. Math., 76 (2016), 2042-2059. doi: 10.1137/16M1065392.
[21]	P. Magal, O. Seydi and G. Webb, Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission, Math. Biosci., 301 (2018), 59-67. doi: 10.1016/j.mbs.2018.03.020.
[22]	R. M. Martins, M. D. Maia, R. H. Farias, L. A. Camacho, M. S. Freire, R. Galler and et al., 7dd yellow fever vaccine: A double blind, randomized clinical trial of immunogenicity and safety on a dose-response study, Hum. Vaccin. Immunother., 9 (2013), 879-888.
[23]	A. J. Mohammed, S. Alawaidy, S. Bawikar and et al., Fractional doses of inactivated poliovirus vaccine in Oman, N. Engl. J. Med., 362 (2010), 2351-2359. doi: 10.1056/NEJMoa0909383.
[24]	J. Mossong, N. Hens, M. Jit and et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med., 5 (2008), e74. doi: 10.1371/journal.pmed.0050074.
[25]	W. Qin, S. Tang and R. A. Cheke, Nonlinear pulse vaccination in an SIR epidemic model with resource limitation, Abstr. Appl. Anal., 2013 (2013), 1-13. doi: 10.1155/2013/670263.
[26]	L. Rass and J. Radclie, Spatial Deterministic Epidemics, Rhode Island: Mathematical Surveys and Monographs, 2003. doi: 10.1090/surv/102.
[27]	Z. B. Reneer and T. M. Ross, H2 influenza viruses: Designing vaccines against future H2 pandemics, Biochem. Soc. Trans., 47 (2019), 251-264. doi: 10.1042/BST20180602.
[28]	S. Resik, A. Tejeda, R. W. Sutter and et al., Priming after a fractional dose of inactivated poliovirus vaccine, N. Engl. J. Med., 368 (2013), 416-424. doi: 10.1056/NEJMoa1202541.
[29]	S. Riley, J. T. Wu and G. M. Leung, Optimizing the dose of pre-pandemic influenza vaccines to reduce the infection attack rate, PLoS Med., 4 (2007), e218. doi: 10.1371/journal.pmed.0040218.
[30]	A. H. Roukens, K. van Halem, A. W. de Visser and L. G. Visser, Long-term protection after fractional-dose yellow fever vaccination: Follow-up study of a randomized, controlled, noninferiority trial, Ann. Intern. Med., 169 (2018), 1761-1765. doi: 10.7326/M18-1529.
[31]	A. H. Roukens, A. C. Vossen, P. J. Bredenbeek, J. T. van Dissel and L. G. Visser, Intradermally administered yellow fever vaccine at reduced dose induces a protective immune response: A randomized controlled non-inferiority trial, PLoS One, 3 (2008), e1993. doi: 10.1371/journal.pone.0001993.
[32]	H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, New York, 1995. doi: 10.1017/CBO9780511530043.
[33]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[34]	J. T. Wu, C. M. Peak, G. M. Leung and M. Lipsitch, Fractional dosing of yellow fever vaccine of extend supply: A modelling study, Lancet, 388 (2016), 2904-2911. doi: 10.1016/S0140-6736(16)31838-4.
[35]	K. N. Wyatt, G. J. Ryan and K. A. Sheerin, Reduced-dose influenza vaccine, Ann. Pharmacother, 40 (2006), 1635-1639. doi: 10.1345/aph.1G645.
[36]	T. Yu, D. Cao and S. Liu, Epidemic model with group mixing: Stability and optimal control based on limited vaccination resources, Commun. Nonlinear Sci. Numer. Simul., 61 (2018), 54-70. doi: 10.1016/j.cnsns.2018.01.011.