MPH MSc Robert J Kolesar
email: robertkolesar@outlook.com

Martine Audibert
email: martine.audibert@uca.fr

Catherine Araujo Bonjean

Keywords: Vaccines, Immunizations, Developing countries, Financing, Mortality reduction JEL Codes I, I1, I00, H, H43

We completed a retrospective multivariate panel and longitudinal trend study to evaluate the effect of Gavi Vaccine Alliance grants on vaccine-preventable disease (VPD) postneonatal mortality. Feasible Generalized Least Squares (fGLS) regression analysis was used to examine the association between VPD post-neonatal mortality rates and Gavi funding. We also applied segmented regression analysis to assess the structural trend in VPD post-neonatal mortality rates, the impact of Gavi funding, and trend changes following Gavi support. We separately tested a composite VPD mortality rate and five vaccinepreventable mortality rates: pertussis, meningitis, measles, diarrhea, and pneumonia (lowerrespiratory infection) as dependent variables. All 77 countries eligible for Gavi assistance from 2000 to 2014 were included in the study. To isolate the effect of Gavi funding in our primary model, we controlled for known and likely predictors of child mortality. Among other factors, Gavi investment and corruption control/system efficiency are important elements to reduce vaccine-preventable mortality rates. For every $1 per capita invested by the Gavi Vaccine Alliance, there are highly statistically significant effects-decreasing the vaccine-preventable disease post-neonatal mortality rate by 2.535 per 1,000 live births. We also found Gavi investments to be significantly associated with reductions in four vaccine preventable disease-specific rates: pertussis, meningitis, diarrhea, and pneumonia. Furthermore, we found Gavi support to be significantly associated with both immediate impacts and trend changes on vaccine-preventable mortality rates. We conclude that Gavi investments in developing country immunization programs have measurably contributed to reductions in post-neonatal VPD mortality rates. Furthermore, we found evidence of a longer term catalytic effect of Gavi funding with accelerated reductions in the trend for post-neonatal meningitis, diarrhea, and pneumonia mortality rates.

Introduction

The United Nations Development Goals aim to end preventable child death by 2030. [START_REF]Sustainable Development Goals: 17 Goals to Transform Our World[END_REF] The World Health Organization (WHO) estimates that immunization averts at least two to three million deaths every year in all age groups. [START_REF]Global Immunization Data[END_REF] Furthermore, vaccinations are among the most cost-effective public health interventions. [START_REF] Who | World Bank. State of the world's vaccines and immunization[END_REF][START_REF] Horton | Cost-Effectiveness of Interventions for Reproductive, Maternal, Neonatal, and Child Health[END_REF] The Gavi Vaccine Alliance is an international public-private partnership that aims to create equal access to new and under-used vaccines for children living in the world's poorest countries. [START_REF] The | About Gavi, the Vaccine Alliance[END_REF] Since its inception in 2000, the Gavi Alliance has become a major health development donor: providing more than $9.43 billion (current US$) to 77 countries from 2000 to 2014. [START_REF] Dieleman | Development assistance for health: past trends, associations, and the future of international financial flows for health[END_REF] Gavi is a public-private partnership that aims to (1) accelerate equitable uptake and coverage of vaccines; (2) increase effectiveness and efficiency of immunization delivery as an integrated part of strengthened health systems; (3) improve sustainability of national immunization programs; and, (4) shape markets for vaccines and other immunization products. [START_REF] The | About Gavi, the Vaccine Alliance[END_REF] Figure 1 illustrates the reduction in post-neonatal (children 1-59 months of age) mortality rates (per 1,000 live births) for Vaccine Preventable Disease (VPD) overall, and its component mortality rates for five specific vaccine-preventable diseases: pertussis, meningitis, measles, diarrhea, and pneumonia among the 77 countries that received any Gavi assistance from 2000 to 2014. The average annual post-neonatal VPD decrease from 2000 to 2014 is 1.67 per 1,000 live births; the average annual GAVI funding increase over the period is $102.4 million. The Gavi Alliance focuses its financial support on the world's poorest countries with national incomes determining eligibility. From 2000 to 2006, the Gross National Income (GNI) per capita eligibility threshold was US$ 1,000 (based on 1998 World Bank data). Seventy-four countries were initially eligible to apply for Gavi support. Since 2000, three additional countries have become eligible for and received support: Timor Leste (2002), Kiribati (2007), and South Sudan (2011). Presently, countries are eligible to apply for Gavi financing when their Gross National Income (GNI) per capita is below or equal to US$ 1,580 on average over the past three years. Among the 77 countries that have received Gavi financial support, 31 are currently classified as low-income economies, 38 as lower-middle income and eight as upper-middle income [START_REF]Sustainable Development Goals: 17 Goals to Transform Our World[END_REF] (per 2016 World Bank income groupings). Figure 2 Over the period 2000-2014, the average total disbursement by country is $122.5 million (current US dollars), with range of $713,511 (Kiribati) to $928,030,656 (Pakistan). Ten countries have received a combined total of $4.969 billion representing 53% of total disbursements: Pakistan, Ethiopia, Democratic Republic of the Congo, Nigeria, Bangladesh, Kenya, Tanzania, Sudan, Uganda, and Ghana.

The total average per capita investment by Gavi from 2000 to 2014 is $5.11 with the highest per capita investment being $14.70 (The Gambia) and the lowest being $0.05 (China). Reflective of Gavi's focus on the world's poorest, the total average per capita assistance is highest among countries currently classified as low-income ($6.71), compared to per capita investments in lower-middle income ($4.57) and upper-middle income ($1.47) countries. Figure 3 shows the total Gavi investment per capita from 2000 to 2014 by country. Countries are color-graded by income group with dark blue corresponding to lowincome countries, medium blue corresponding to lower-middle income countries, and light blue corresponding to upper-middle income countries.

Econometric models

We examine the relationship between longitudinal vaccine-preventable post-neonatal mortality rates and Gavi Alliance disbursements among all 77 countries ever eligible to receive Gavi grants. We apply an approach similar to that used by Yan, Korenromp, and Bendavid to evaluate mortality changes after grants from the Global Fund to Fight AIDS, tuberculosis and malaria. [START_REF] Yan | Mortality changes after grants from the Global Fund to Fight AIDS, tuberculosis and malaria: an econometric analysis from 1995 to 2010[END_REF] To enable estimation in the presence of heteroscedasticity and correlated error structure, we used feasible Generalized Least Squares (fGLS) regression for panel data as our primary model. [START_REF] Kennedy | A Guide to Econometrics 6E[END_REF][START_REF] Angrist | Mostly Harmless Econometrics: An Empiricist's Companion[END_REF][START_REF] Statacorp | Stata Longitudinal-data/Panel-data Reference Manual[END_REF] To isolate the effect of Gavi support in this model, we included known and likely predictors of child mortality described in the literature as control covariates. [START_REF] Kovacs | Deconstructing the differences: A comparison of GBD 2010 and CHERG's approach to estimating the mortality burden of diarrhea, pneumonia, and their etiologies[END_REF][START_REF] Alkema | Child Mortality Estimation: A Comparison of UN IGME and IHME Estimates of Levels and Trends in Under-Five Mortality Rates and Deaths[END_REF][START_REF] Wang | More Health Expenditure, Better Economic Performance? Empirical Evidence from OECD Countries[END_REF][START_REF] Williamson | Foreign Aid and Human Development: The Impact of Foreign Aid to the Health Sector[END_REF][START_REF] Wang | Global, regional, and national levels of neonatal, infant, and under-5 mortality during 1990-2013: a systematic analysis for the Global Burden of Disease Study 2013[END_REF][START_REF] Liu | Regional, and National Causes of Child Mortality: An Updated Systematic Analysis for 2010 with Time Trends Since[END_REF][START_REF] Muldoon | Health system determinants of infant, child and maternal mortality: A cross-sectional study of UN member countries[END_REF] We also applied a segmented regression model for longitudinal data to assess the structural trend, the immediate impact of Gavi funding, and a change in the trend following Gavi support. [START_REF] Lagarde | How to do (or not to do). Assessing the impact of a policy change with routine longitudinal data[END_REF] Our primary model includes a time trend variable to separately account for mortality changes over time.

We use corruption control score as a proxy variable for system efficiency.

The fGLS regression model is expressed as an equation below.

𝑀𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑡 = 𝛼 𝑖 + 𝛽 1 𝑇𝑖𝑚𝑒 𝑖 + 𝛽 2 𝐺𝑎𝑣𝑖_𝑎𝑠𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒_𝑝𝑐 𝑖𝑡 + 𝛽 [START_REF] Who | World Bank. State of the world's vaccines and immunization[END_REF] Where Mortality it is the crude post-neonatal mortality rates for country i in year t, Time is a time trend covariate taking on the value t in the t th year, Gavi_assistance_pc it is the per-capita disbursement from the Gavi Alliance (lagged one year), Other_Health_pc it is all other non-Gavi health expenditure per capita (lagged), ANC_access it is the proportion of pregnant women accessing antenatal care at least once, Water_access it is the proportion of the rural population with access to water, GNI_pc it is the natural log of the Gross National Income (GNI) per capita, Primary it is the proportion of girls attending primary school and represents maternal education, Birth_rate it is the birth rate, Corruption it is the corruption control score. Epsilon is the error term which includes both between-entity error (u it ) and within-entity error (v it ).

For the segmented regression, post-neonatal mortality data was temporally realigned with Time 0 representing the first year of Gavi funding for each country, regardless of the calendar year. We used Ordinary Least Squares regression with robust standard errors to independently assess the effects immediately following the initiation of Gavi support and the change in the trend for each dependent variable.

Statistical testing

All statistical analysis was completed using Stata 14.1. For the fGLS regression, we tested the model for heteroscedasticity in panel data using the Likelihood-ratio test (p<0.000). We also detected serial correlation using the Wooldridge test for serial correlation in panel-data models (p-value<0.000). [START_REF] Drukker | Testing for serial correlation in linear panel-data models[END_REF] We addressed these issues by using the feasible Generalized Least Squares (fGLS) regression method which enables estimation in the presence of heteroscedasticity and correlated error structure. [START_REF] Bartels | Beyond "Fixed Versus Random Effects[END_REF][START_REF] Kohler | Data Analysis Using Stata[END_REF] For the segmented regression, we calculated Durban Watson (dw) statistics to test for the presence of first-order auto-correlation using a 1% level of significance to indicate not to reject the null hypothesis of autocorrelated errors. The results for positive first-order correlation related to the Measles post-neonatal mortality rate as a dependent variable were inconclusive. Likewise, the result was inconclusive for negative first-order correlation relative to Pertussis and diarrhea mortality rates. To err conservatively, we used the Prais-Winsten estimator as the basis for our results for these three dependent variables.

Variations on the model

We completed multiple secondary analyses for sensitivity testing of our results. First, we repeated the primary analysis using standard panel regression with complete case analysis using random effects (to address serial correlation) and robust standard errors (to address heteroscedasticity). This analysis (not shown) yielded similar associations, but less significant results. Second, we explored additional explanatory/control variables, testing the effect of supplementary covariates using a manual backwards stepwise regression process. We tested health workforce density, urban population percent, sanitation coverage, and political stability. We also tested adult HIV prevalence, child malaria, and malnutrition rates in the model as possible factors that could be correlated with increased mortality. This could be due to either increased immuno-vulnerability or the economic burden resulting in reduced access to health services associated with each health condition. We interacted antenatal care (ANC) access with all other health expenditure to control for the effect of health system capacity, in this case capacity to render services available as a function of resources invested. These variables were excluded from the final model due to concerns with excessive data missingness (in the case of health workforce density); or, lack of statistical significance and/or inconsistency across dependent variables. Third, we varied time-lags from 1-year to 2-4 years, with similar results. Fourth, we ran separate regressions on each dependent mortality variable by current country income group. To control for differences potentially associated with a change in income group classification, we included a discrete variable to account for the number of years the country was reclassified to a higher income group. An F-test for nested models indicated that this variable did not contribute information, thus we excluded the variable from the final fGLS model. late/early and/or repeat grades. By contrast, the birth rate decreases among countries in higher income groups.

In order to minimize the loss of observations resulting from one missing value in a country-year observation, we completed Multiple Imputation using chained equations (MICE) with predictive mean matching to the five nearest neighbors. [START_REF] White | Multiple imputation using chained equations: Issues and guidance for practice[END_REF][START_REF] Statacorp | Stata Multiple Imputation Reference Manual[END_REF] This approach enabled the estimation of missing values for six variables: all other health expenditure per capital, ANC access, water access (rural), GNI per capita, girls' primary school completion, and corruption control. The proportion of missing data that was imputed is shown in Table 1. Missing data for these variables was considered Missing At Random (MAR) as missingness is attributed to multi-year national survey cycles or lack of reporting due to either political instability or system inefficiency. We addressed the later by including political stability and corruption iterations by simulating from a (approximate) Bayesian posterior predictive distribution of the missing data. [START_REF] Angrist | Mostly Harmless Econometrics: An Empiricist's Companion[END_REF]

Results

Regression results from the primary model are presented in Table 2. The coefficients represent the direction and magnitude of the change in mortality rates associated with a one-unit increase of the corresponding independent variable, controlling for all other covariates in the model. P-values are shown in parentheses below each coefficient. In relation to the fGLS regression, we found a significant Gavi effect, reducing the post-neonatal mortality rates for VPD overall; and, four of the five disease-specific rates. For every $1 per capita invested by Gavi, there are highly statistically significant effects (p<0.000) -decreasing the post-neonatal the mortality rate for VPD by 2.535 per 1,000 live births; for meningitis by 0.569; and, for pneumonia by 1.547. We also found a statistically significant (p<0.05) Gavi effect on post-neonatal mortality for pertussis (0.028 rate reduction); and, diarrhea (0.387 rate reduction). No statistically significant relationship was detected between Gavi funding and the measles mortality rate. We found highly statistically significant (p<0.05) associations between our composite VPD rate and seven of the nine explanatory variables. The time trend and ANC access variables had statistically significant associations across all six dependent mortality rate variables. We found Gavi investment per capita to be significantly associated with reduction for five of the six vaccine-preventable disease mortality rates. We also found significant associations for five mortality rates with increased girls primary school completion. We also found an increase in the birth rate to be significantly associated with an increase in (five of the six) mortality rates.

Finally, we found corruption control (used as a proxy for health system efficiency) to have a highly statistically significant effect on reducing the overall VPD rate as well as for diarrhea and pneumonia mortality rates. A one-point increase in the corruption control score is associated with a reduction in the VPD rate by 3.303 (p<0.006). We did not find any association between Gavi funding and Measles mortality rates.

Using segmented regression, we found the VPD mortality rate decreased by 4.624 per 1,000 live births (p<0.018) directly following the initiation of Gavi support with mortality rate reductions for Measles (by 2.0665) and diarrhea (by 1.231). In addition to the immediate effect, we also observed statistically significant reductions in the post-neonatal mortality rate trends for the overall VPD mortality rate of 1.973; and, for three disease-specific rates: Meningitis (by 0.3252), diarrhea (by 0.6197), and pneumonia (by 0.8538). No underlying structural trends were detected. Figure 4 illustrates the mean change and fitted results for overall VPD and disease specific post-neonatal mortality rates relative to the initiation year of Gavi support.

General

Vaccination is widely recognized as one of the most cost-effective public health interventions. [START_REF] Cleland | Maternal Education and Child Survival in Developing Countries: The search for pathways of influence[END_REF] However, the effective administration and delivery of vaccination services, particularly in developing country contexts, can present many challenges. The Gavi Alliance conducts routine monitoring as well as rigorous independent country-specific evaluations to enhance program effectiveness. To our knowledge, this study is the first econometric evaluation of the overall post-neonatal mortality impact of Gavi's $9.43 billion investment (from 2000 to 2014) among all 77 countries that have received assistance. As development assistance for health resources are limited, public health program planners, and especially donors, increasingly seek accountability for results and evaluation data for strategic investment decision making. [START_REF] Dieleman | Development assistance for health: past trends, associations, and the future of international financial flows for health[END_REF] Our findings provide evidence that Gavi investments in developing country immunization programs have measurably contributed to reductions in post-neonatal VPD mortality rates. We note the variation of the strength and magnitude of the relationship between Gavi funding and post-neonatal mortality for each VPD. Particularly, our final model did not detect a statistically significant association for measles. It is important to note that the factors impacting on post-neonatal mortality, particularly measles, diarrhea, and pneumonia are complicated and dynamic. For example, there is evidence linking malaria in pregnancy to Intrauterine Growth Restriction which increases risk of stunting-a proximal risk factor for post-neonatal measles. [START_REF] Hopkins | LiST The Lives Saved Tool-software that predicts survival of mothers and children[END_REF] Although we explored the inclusion of several additional explanatory variables in our model (see Variations in the model section), several were not retained. We took the approach of fitting one model to all six dependent variables to improve consistency and comparability of the results. Thus, our final model is not specific to any disease-specific factor, but rather the composite VPD rate and the five disease specific rates. Thus, we hypothesize that the effect differences of any one explanatory variable across the six dependent mortality rate variables may be attributable to disease-specific epidemiological and/or environmental factors that were not captured in our model.

The results related to coefficients for other factors associated with child mortality are consistent with other published literature. For example, Cleland and van Ginneken found that for every additional year of a mother's education corresponds with a 7-9% reduction in child mortality. This is primarily attributable to economic advantages associated with education, but may also be explained by the mother's increased use of preventative and curative health services as well as domestic health practices. [START_REF] Cleland | Maternal Education and Child Survival in Developing Countries: The search for pathways of influence[END_REF] And, reduced fertility can reduce child mortality through increasing the birth interval, thus augmenting birth weight. [START_REF] Cleland | Family planning: the unfinished agenda[END_REF][START_REF] Davanzo | The Effects of Birth Spacing on Infant and Child Mortality, Pregnancy Outcomes, and Maternal Morbidity and Mortality in Matlab, Bangladesh[END_REF][START_REF] Rutstein | Effects of preceding birth intervals on neonatal, infant and under-five years mortality and nutritional status in developing countries: evidence from the demographic and health surveys[END_REF] In addition, smaller family size may increase household income per capita, thus enabling improved investment in health and nutrition.

We used the corruption control score as a proxy variable for system efficiency and found significant associations for reduced VPD post-neonatal mortality, and three disease-specific rates. Our results are consistent with Muldoon et al. who found that having a less corrupt government as a protective determinant for infant, child, and maternal mortality rates. [START_REF] Muldoon | Health system determinants of infant, child and maternal mortality: A cross-sectional study of UN member countries[END_REF] This finding underscores the importance of system issues on service provision.

Finally, our segmented regression model provides evidence that Gavi funding may have a longer term catalytic effect by accelerating trend reductions for three disease-specific post-neonatal rates: meningitis, diarrhea, and pneumonia. This would suggest that, in some cases, Gavi investments have an effect beyond the immediate impact.

Strengths and limitations

It is important to note that financial support from the Gavi Alliance is highly country specific: countries submit applications and provide technical justification including strategic priorities, country context, and disease epidemiology. To explore the possibility of assessing the association between disease specific funding and the relevant post-neonatal mortality rate we reviewed country and year funding earmarks by vaccine-preventable disease. However, we determined that the number of country observations and level of funding specified for any one specific disease would present significant methodological challenges. In many cases, financial support was provided for health systems strengthening which is intended to have a broader effect. We believe that modeling a composite VPD indicator and all Gavi funding in addition to each disease specific rate by country and year enabled the analysis of a more generalized effect of Gavi funding.

Our analysis did not incorporate data prior to 2000 as mortality rate estimates were not available from the source. The inclusion of additional prior years would have strengthened the segmented regression estimates. In the case of health workforce density, extensive missing data precluded use of these data in our analysis. Furthermore, as we did not find strong evidence of abrupt changes in mean mortality rate trends, our segmented regression model did not incorporate any temporal variables to control for events such as natural disasters or economic crisis.

Figure 1 .

 1 Figure 1. Annual post-neonatal mortality rates for vaccine-preventable diseases and Gavi investment from 2000-2014 in 77 Gavi-supported countries

  illustrates the geographic distribution of countries that have received Gavi financing, color-graded by total disbursements (in 2016 US dollars) from 2000 to 2014.

Figure 2 .

 2 Figure 2. Countries receiving Gavi funding, color-graded by total disbursements between 2000 and 2014, from lowest (lightest) to highest (darkest)

Figure 3 .

 3 Figure 3. Total (2000-2014) average Gavi investment per capita (in current USD) by country, colorgraded by income group

Figure 4 .

 4 Figure 4. Mean change and fitted results for overall Vaccine Preventable Disease and disease specific post-neonatal mortality rates relative to the initiation of Gavi support

  𝐴𝑙𝑙_𝑜𝑡ℎ𝑒𝑟_𝐻𝑒𝑎𝑙𝑡ℎ_𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒_𝑝𝑐 𝑖𝑡 + 𝛽 4 𝐴𝑁𝐶_𝑎𝑐𝑐𝑒𝑠𝑠 𝑖𝑡 + 𝛽 5 𝑊𝑎𝑡𝑒𝑟_𝑎𝑐𝑐𝑒𝑠𝑠 𝑖𝑡 + 𝛽 6 𝑙𝑛_𝐺𝑁𝐼 𝑖𝑡 + 𝛽 7 𝑃𝑟𝑖𝑚𝑎𝑟𝑦 𝑖𝑡 + 𝛽 8 𝐵𝑖𝑟𝑡ℎ_𝑟𝑎𝑡𝑒 𝑖𝑡 + 𝛽 9 𝐶𝑜𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛 𝑖𝑡 + 𝜀 𝑖𝑡 𝜀 𝑖𝑡 = 𝑢 𝑖 + 𝜈 𝑖𝑡

Table 1 . Descriptive characteristics of countries by income group

 1 (as a proxy for efficiency) covariates in our imputation equation. We also included sanitation coverage in the imputation model to inform estimates of missing data for water access. All other regression equation variables were included in the MI model. We generated 20 imputations with 200

		All countries	Low-income	Lower-middle income Upper-middle income	% imputed
	Number of countries		77		31		38		8
	Gavi assistance per capita	$	5.11	$	6.71	$	4.57	$	1.47	0%
	(mean)	$	4.86	$	6.62	$	4.16	$	0.99
	(median)
	All other Health Expenditure per capita
	(mean)	$	993.06	$	351.98	$	1,075.30	$	3,086.56	3.96%
	(median)	$	649.52	$	374.87	$	945.42	$	2,669.88
	Antenatal care access									17.40%
	(mean)		79.8%		73.8%		82.3%		91.3%
	(median)		84.1%		80.8%		88.9%		95.5%
	Water access (rural)									2.25%
	(mean)		62.4%		54.4%		67.8%		67.1%
	(median)		64.1%		52.3%		68.9%		74.5%
	Gross National Income per capita									8.31%
	(mean)	$	1,108.25	$	436.76	$	1,253.73	$	2,851.38
	(median)	$	892.67	$	422.00	$	1,047.00	$	3,182.00
	Primary school completion rate (girls)									23.98%
	(mean)		83.6%		70.4%		89.3%		103.4%
	(median)		87.7%		69.6%		95.6%		103.9%
	Birth rate									0%
	(mean)		31.66		38.99		28.25		19.43
	(median)		35.87		39.68		29.04		14.88
	Control of corruption									8%
	(mean)		-0.77		-0.89		-0.70		-0.68
	(median)		-0.77		-0.88		-0.74		-0.59

control

Table 2 . Panel regression results for vaccine-preventable disease (VPD) and disease-specific post- neonatal mortality rates in 77 Gavi-supported countries from 2000-2014

 2

	VARIABLES	VPD	Pertussis Meningitis	Measles	Diarrhea Pneumonia
	Time trend	-0.764***	0.015***	-0.044**	-0.172***	-0.389***	-0.234***
		(0.000)	(0.000)	(0.013)	(0.006)	(0.000)	(0.000)
	Gavi investment per capita +	-2.535***	-0.028**	-0.569***	0.016	-0.387**	-1.547***
		(0.009)	(0.022)	(0.000)	(0.965)	(0.026)	(0.000)
	All other health expenditure +	0.024	-0.000**	0.004*	0.015	0.012***	0.001
		(0.185)	(0.044)	(0.085)	(0.241)	(0.000)	(0.816)
	Antenatal care access	-0.177***	-0.005***	-0.024***	-0.031**	-0.044***	-0.078***
		(0.000)	(0.000)	(0.000)	(0.025)	(0.000)	(0.000)
	Water access (rural)	-0.136***	-0.003***	-0.008	-0.031	-0.034***	-0.061***
		(0.002)	(0.000)	(0.220)	(0.235)	(0.001)	(0.000)
	GNI per capita (logged)	-1.255	-0.044***	-0.641***	-1.107***	0.279	0.309
		(0.223)	(0.000)	(0.000)	(0.009)	(0.203)	(0.243)
	Primary school (girls)	-0.168***	-0.001***	-0.018***	-0.029**	-0.061***	-0.062***
		(0.000)	(0.002)	(0.000)	(0.026)	(0.000)	(0.000)
	Birth rate	0.891***	0.024***	0.083***	0.026	0.370***	0.384***
		(0.000)	(0.000)	(0.000)	(0.631)	(0.000)	(0.000)
	Corruption control	-3.303***	0.028*	-0.199	-0.140	-1.371***	-1.431***
		(0.006)	(0.051)	(0.101)	(0.788)	(0.000)	(0.000)
	Constant	44.536***	0.879***	7.778***	16.015***	6.980***	13.540***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.002)	(0.000)
	Observations	1,078	1,078	1,078	1,078	1,078	1,078
	Countries	77	77	77	77	77	77

Table 3 . Results of segmented linear regression model by Vaccine Preventable Disease rates

 3

	Variable	VPD	Pertussis		Meningitis	Measles		Diarrhea		Pneumonia
				Prais-Winsten			Prais-Winsten		Prais-Winsten
	Mean baseline rate	38.92*** (0.000)	0.7741*** (0.000)	0.7675*** (0.000)	3.5385*** (0.000)	5.9672*** (0.000)	5.1769*** (0.001)	12.9331*** (0.000)	12.9651*** (0.000)	15.7139*** (0.000)
	Structural trend, independent of Gavi	0.4762	0.0056	0.0002	0.1452	-0.0996	-0.3294	0.1756	0.1121	0.2495
	support	(0.409)	(0.677)	(0.986)	(0.278)	(0.495)	(0.310)	(0.687)	(0.718)	(0.498)
	Immediate impact on mean mortality	-4.624**	0.0059	0.01296	-0.085	-3.2844***	-2.0065*	-1.116	-1.231**	-0.1448
	rate following Gavi support	(0.018)	(0.658)	(0.335)	(0.700)	(0.001)	(0.071)	(0.117)	(0.048)	(0.777)
	Change in trend following Gavi	-1.937***	-0.0143	-0.0082	-0.3252**	-0.0526	0.1329	-0.6910	-0.6197*	-0.8538**
	support	(0.005)	(0.296)	(0.295)	(0.024)	(0.774)	(0.725)	(0.128)	(0.059)	(0.032)
	Durban Watson statistic	2.223	2.76	1.80		1.378	1.937	2.348	1.587	2.447
	(p-values in parentheses)	*** p<0.01, ** p<0.05, * p<0.10

Études et Documents n° 7, CERDI, 2017

Low-income economies are defined as those with a GNI per capita, calculated using the World Bank Atlas method, of $1,045 or less in

2014; lower middle-income economies are those with a GNI per capita of more than $1,045 but less than $4,125 and upper-middle-income countries are those with a GNI per capita of $4,125 to $12,736 33.

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/This study was initiated as part of an internship with the Gavi Alliance [agreement number 19303]. The sponsor provided comments on the preliminary data analysis. All final data analysis, interpretation of findings, and preparation of this manuscript was at the full discretion of the authors. Submission of the article for publication was the sole decision of the authors.

We thank Hope Johnson from the Gavi Alliance for review and comment on the initial data analysis.

Data

The dataset was constructed using publicly accessible information. We retrieved disease-specific mortality data from the World Health Organization (WHO) Global Health Observatory; and, financial data for development assistance for health from the Institute for Health Metrics and Evaluation (IHME). [START_REF]Global Health Observatory data repository[END_REF][START_REF]Global Health Data Exchange[END_REF] All other health and development data was retrieved from the World Bank and the University of Gothenburg's Quality of Government Basic dataset. [START_REF]Data, Indicators[END_REF][START_REF] Dahlberg | The Quality of Government Basic Dataset[END_REF] We generated a composite Vaccine Preventable Disease (VPD) rate by summing the individual rates for each of the five disease-specific rates. Gavi assistance, all other (non-Gavi) public health expenditure, and Gross National Income (GNI) are all expressed as per capita in 2016 US dollars. In addition, Gavi assistance, all other (non-Gavi) public health expenditure variables were lagged by one-year to account for the delay from the time resources are dispersed to their earliest potential health impact. GNI per capita was transformed to natural log to reduce skewness. In relation to Antenatal care (ANC) access and Girls' primary completion variables, for countries with at least two data points, missing values between these points were interpolated as it is reasonable to expect linear changes over relatively short periods.

Table 1 shows the mean and median values for all independent variables included in the primary model.

Values are provided for all 77 countries included in the analyses as well as by current income group classification. For comparative purposes the table shows totals for Gavi assistance and all other (non-Gavi) health investment (both per capita) for the period (2000-2014). As noted above, the total average Gavi-assistance per capita is higher among low-income countries. By contrast, all other health expenditure per capita increases dramatically by income group. ANC access, rural water access, and girls' primary school completion increase among countries in higher income groups. Girls' primary school completion rates can exceed 100% as over-aged and under-aged children who enter primary school


Michel Philippe
email: philippe.michel@ec-lyon.fr

Optimal proliferation rate in a cell division model

Keywords: cell division, long time asymptotic, eigenvalue, min-max, variation, asymmetry AMS class. No: 35P05, 92B05, 93B60

We consider a size structured cell population model where a mother cell gives birth to two daughter cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a min-max principle and a differentiation principle to find the variation of the first eigenvalue with respect to a parameter of asymmetry of the cell division. We prove that the symmetrical division is not always the best fitted division, i.e., the Maltusian parameter may be not optimal.

Introduction

General models of cell division are known for a long time. Although the most classical case is division into two equal new cells, it is now well established that this is not always the case (see [START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF][START_REF] Kirkwood | The evolution of ageing and longevity[END_REF][START_REF] Partridge | Optimality, mutation and the evolution of ageing[END_REF][START_REF] Ackermann | Senescence in a bacterium with asymmetric division[END_REF][START_REF] Nystrom | Aging in bacteria[END_REF][START_REF] Sinclair | Paradigms and pitfalls of yeast longevity research[END_REF]). In particular in hematopoiesis (see [START_REF] Brummendorf | Asymmetric cell divisions sustain long-term hematopoiesis from single-sorted human fetal liver cells[END_REF][START_REF] Huang | Symmetry of Initial Cell Divisions Among Primitive Hematopoietic Progenitors Is Independent of Ontogenic Age and Regulatory Molecules[END_REF] or [START_REF] Sugimoto | A Cell Kinetic and Cytological Study on the Asymmetric Cell Division of Thymic Lymphoblasts of the Embryonic Rat, Development[END_REF] for thymic lymphoblasts), in the large class of budding yeasts (see [START_REF] Jazwinski | Growing old: Metabolic control and yeast aging[END_REF][START_REF] Breitenbach | Yeast as a model for ageing and apoptosis research[END_REF][START_REF] Barker | Replicative ageing in the fission yeast Schizosaccharomyces pombe[END_REF][START_REF] Qu | Coordination of cell growth and cell division: a mathematical modeling study[END_REF]), E.Coli or for some other bacteria [START_REF] Dens | Cell division theory and individual-based modeling of microbial lag: part I. The theory of cell division[END_REF] like Physcomitrella protoplast, division is not always symmetric and a mother cell can give birth to a bigger and a smaller cells. The goal of the present paper is to propose a possible explaination for the existence of different way of cell division (symmetric or asymmetric) based on adaptive dynamic. In some cases, the symmetric division is not the best fitted. The natural model to study it is a cell division model (see [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF]) in which the density of cells n(t, y) is structured by their size y and the evolution is described by the master equation

∂ ∂t n(t, y) + ∂ ∂y n(t, y) + B(y)n(t, y) = 1 σ B( y σ )n(t, y σ ) + 1 1 -σ B( y 1 -σ )n(t, y 1 -σ ), (1.1)
where a cell of size y gives birth to a cell of size yσ and another one of size (1 -σ)y, with σ ∈]0, 1[. A similar model also arises to describe fragmentation in physics [START_REF] Fournier | Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition[END_REF][START_REF] Laurençot | Steady states for a fragmentation equation with size diffusion[END_REF] and the growth term ∂ y n arises after rescalling [START_REF] Bertoin | Asymptotic laws for nonconservative self-similar fragmentations[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. The division part of this equation can be understood as follows. The cells of size y are produced from the cells of size y/σ (resp. y/(1 -σ)). The number of cells which size belongs to [y, y + δy] that appear after the division are δy σ B( y σ )n(t, y σ ) (resp. δy 1-σ B( y 1-σ )n(t, y 1-σ )). We have then the population balance equation (1.1). We know from the general theory (see [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]) that the asymptotic behavior of such an evolution equation is given by the rate λ, the eigenvalue of 1 the following eigenproblem

                     ∂ ∂y N (y) + [λ + B(y)]N (y) = 1 σ B( y σ )N ( y σ ) + 1 1 -σ B( y 1 -σ )N ( y 1 -σ ),
-∂ ∂y φ(y) + λφ(y) = B(y) φ(σy) + φ((1 -σ)y) -φ(y) , N (y = 0) = 0, N, φ ≥ 0, ∞ 0 N (y)φ(y)dy = 1, φ(0) = 1.

(1.2)

More precisely, we know from general relative entropy method [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF], that there exists a constant C only depending on the initial condition n(0, .) such that : -CN (.)e λt is a solution to (1.1), -φe -λt is a solution to the backward equation of (1.1), or similarly, φ is a solution to the dual eigenproblem, -finally, we have the convergence in the weighted space L 1 ([0, ∞[, φdy) with the norm f := | f (y) | φ(y)dy, lim t→∞ n(t, .)e -λt = C N (.) in L 1 ([0, ∞[, φdy).

Consequently, the dynamics of the density n(t, y) is strongly related to the eigenvalue in (1.2) that gives an invasive parameter of the population (and thus the fitness of the population). We use a min-max method [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to non linear propagation phenomena[END_REF][START_REF] Horn | MatrixAnalysis[END_REF] and a differentiation method to study the variation of the first eigenvalue λ with respect to the asymmetry parameter σ. We prove that when a cell divides early then the asymmetric division gives a better growth, i.e., a larger first eigenvalue. On the contrary, for late division, asymmetric division is better.

This paper is organized as follows. Firstly, in Section 2, we give the main results on the variation of the invasive parameter (first eigenvalue in (1.2)) with respect to the asymmetry of the division. We show that the convexity of the dual eigenfunction φ is related to this problem of variation. Then, in Section 3, we show that a min-max method and a differentiation method can be used to study the variation of the invasive parameter (first eigenvalue) with respect to the parameter of cell division asymmetry σ in (1.1). In Section 4, we consider birth rate B such that we obtain directly the convexity of the dual eigenfunction. In Section 5, we extend the results found in Section 2 to more general cell division models. Finally, in the last section, we give some simple applications of the main methods.

Main results

The purpose of this section is to show that the symmetric division is not necessarily the best fitted division. We study two different cases, when the birth rate B has a compact support separated from

y = 0, Supp B ⊂ [a, b], a > b/2, (2.1)
or Supp B contains y = 0. In this case we suppose that

Supp B = [0, b], λ(σ) λ(σ) -B(0) B(0) < B (y) B(y) ≤ 0, ∀y ∈]0, bσ 0 [, ∀σ ∈ [1/2, σ 0 [, (2.2)
with σ 0 ∈ [1/2, 1[, λ(σ) the first eigenvalue λ in (1.
2) where σ is the asymmetric parameter. We notice that B(y) = 1 [0,b] satisfies assumption (2.2). This condition means that the cells begin to divide early (at size y = 0) and the birth rate decreases. Theorem 2.1 Assume that Supp B is compact and (2.1) holds. Then

d dσ λ(σ) ≤ 0, ∀σ ∈ [1/2, a/b[. (2.3)
Moreover, we have

sup σ∈]1-a/b,a/b[ λ(σ) = λ(1 -a/b). Theorem 2.2 Assume that b > 0, σ 0 ∈ [1/2,
λ(σ) = λ(1/2).
Proof of Theorems 2.1-2.2 First, we will recall some results on the existence of a solution to the eigenproblem. Then, we differentiate the first eigenvalue λ(σ) with respect to the parameter σ to study the variation of λ(σ). We notice that the variation of the first eigenvalue is directly linked to the properties of the dual eigenfunction φ σ to be concave or convex. Finally, we prove that under the assumptions of Theorems 2.1-2.2, we can directly prove the convexity or concavity of the dual eigenfunction.

Step 1. Existence and uniqueness of the solution to eigenproblem (1.

2) The eigenproblem (1.2) associated to the cell division model can be written with more compact notations

L * σ (φ σ ) = λ(σ)φ σ , L σ (N σ ) = λ(σ)N σ , ∞ 0 N σ (y)φ σ (y)dy = 1, (2.5)
where

L 1 w ([0, ∞[) := L 1 ([0, ∞[, (1 + B(y))(1 + y)dy), L σ (f ) := - ∂f ∂y (.) -B(.)f (t, .) + 1 σ B( . σ )f ( . σ ) + 1 1 -σ B( . 1 -σ )f ( . 1 -σ ), f ∈ L 1 w ([0, ∞[), (2.6)
and the dual operator,

L * σ (g) := ∂g ∂y (.) -B(.)g(t, .) + B(.) g(.σ) + g .(1 -σ) , g ∈ L 1 w ([0, ∞[) * . (2.7)
Remark. We notice that it is enough to consider σ ∈ [1/2, 1[, since, using the symmetry of (1.2), we have λ(σ) = λ(1 -σ).

Here, we cannot expect to find an exact solution as in the McKendrick-VonFoerster model for instance. Nevertheless, we have the following lemma. Lemma 2.3 Under the assumption (2.1) (resp. (2.2)), there exists a solution (N σ , λ(σ), φ σ ) to (2.5)-(2.7). Moreover, we have

∀σ ∈]0, 1[, ∃C σ < ∞ :            N σ (y) ≤ C σ e -λ(σ)y , φ σ (y) ≤ C σ (1 + y), | ∂ ∂y φ σ (y) |≤ C σ (B(y) + 1)(1 + y). (2.8)
We refer to [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF] for the proof of this result.

Step 2. The differentiation method. Lemma 2.4 The function σ → λ(σ) is well defined on ]0, 1[ and is differentiable. Moreover, we have

∂ ∂σ λ(σ) = ∞ 0 B(y) ∂ ∂y φ σ (σy) - ∂ ∂y φ σ ((1 -σ)y) N σ dy.
(2.9)

We prove this lemma in Section 3.1 (as an application of Lemma 3.2) and using the same method, we find :

Lemma 2.5 The following equality holds :

( ∂ 2 ∂σ 2 λ(σ)) σ=1/2 = 2 ∞ 0 B(y)y 2 ∂ 2 ∂y 2 φ 1/2 (y/2)N 1/2 (y)dy.
Proof. We use Lemma 2.4 and the same method to prove that ∂ 2 ∂σ 2 λ(σ) exists and satisfies the above equality.

2

Now, we can give a direct corollary of Lemma 2.4. We suppose that

σ 0 ∈ [1/2, 1[, (2.10)
K = Supp B ⊂ [0, ∞[. (2.11)
Corollary 2.6 Assume that φ σ0 is a convex (resp. concave) function on [1 -σ 0 , σ 0 ].ConvHull(K).

Then we find that

( ∂ ∂σ λ σ ) σ=σ0 ≥ 0 (resp.( ∂ ∂σ λ σ ) σ=σ0 ≤ 0).
Proof. Indeed, using Lemma 2.4 and that φ σ0 is a convex function, we find that ∂ ∂y φ σ0 is increasing. Since σ 0 ≥ (1 -σ 0 ), we have ( ∂ ∂σ λ σ ) σ=σ0 ≥ 0. 2 Remark. The strict convexity (resp. strict concavity) of φ σ0 on K implies ∂ ∂σ λ > 0 (resp. ∂ ∂σ λ < 0).

Thus, we reduce the problem of the variation of λ(σ) with respect to σ to a problem on φ σ . In the next section we prove that under some assumptions on B, the dual eigenfunction φ σ is a convex or concave function.

Step 3. Convexity of the dual eigenfunction φ σ When the birth rate has a compact support, the following result holds.

Theorem 2.7 Assume that (2.1) (resp. (2.2) with σ 0 ∈ [1/2, 1[) holds. Then for all σ ∈]1 -a/b, a/b[ (resp. σ ∈ [1/2, σ 0 ]), φ σ is a strictly convex (resp. strictly concave) function.
We prove Theorem 2.7 in section 4.

Step 4. Conclusion We can now prove the main Theorems 2.1-2.2 on the variation of the eigenvalue λ(σ) with respect to the asymmetry parameter σ. Indeed, using Corollary 2.6 and the results proved in Section 2, we get directly Theorems 2.1-2.2. 2

Main methods

Formally, we have a family of linear bounded operators (and their dual operators) which depend on a parameter p ∈ I, where

I ⊂] -∞, ∞[ is an interval, L p : B → C, L * p : C * → B * , (3.1)
B and C are Banach spaces of real functions (in duality ., . with B * and C * ) such that

∀p ∈ I ∃(N p , λ(p), φ p ) ∈ B + ×] -∞, ∞[×C * + : L p (N p ) = λ(p)N p , L * p (φ p ) = λ(p)φ p , (3.2)
where

B + (resp C * + ) is the positive cone of B (resp C * ) and ∀p ∈ I φ p , N p = 1. (3.3)
We will study the variation of λ(p) with respect to p. We notice that the variation of N p , φ p in Banach spaces of real functions makes the problem rather complex.

Our purpose is to find the variation of the first eigenvalue in a general eigenvalue problem (3.1)-(3.3). To study the variation of the first eigenvalue, we develop two points of view, one which favors the min-max principle and the other which uses the differentiation of the eigenvalue (and eigenproblem). Even if the second one needs the operators in the eigenproblem to be more regular, it gives better results in some problems as the cell division one.

Min-max method: We have the following principle Lemma 3.1 Assume that there exists a solution (N p , λ(p), φ p ) to (3.1)-(3.3). Then we have for all p ∈ I,

λ(p) = max f ∈Λ min g∈Λ * g, L p f g, f , (3.4)
where Λ ⊂ B + which satisfies N p ∈ Λ, φ p ∈ Λ * ⊂ C * + and g, f > 0 for all f ∈ Λ and g ∈ Λ * . Moreover, if L p is monotone with respect to p ∈ I then so is λ(p). More precisely, we have

∀p ≤ q ∈ J, ∀(f, g) ∈ Λ × Λ * g, L p (f ) ≤ g, L q (f ) =⇒ λ(p) ≤ λ(q). (3.5)
Finally, if L p is continuous with respect to p,

∀(f, g) ∈ Λ × Λ * lim p→q | g, L p (f ) -L q (f ) |= 0, (3.6)
then λ(p) is continuous with respect to p.

Differentiation method :

We have the following principle Lemma 3.2 Assume, there exists (N p , λ(p), φ p ) solution to (3.1)-(3.3) and ∀q ∈ J, lim

p→q∈ J N p = N q , strongly in B, (3.7
)

∀q ∈ J ∃ ∂ p L * p | p=q ∈ L(C * , B * ) : ∀f ∈ Λ, lim p→q∈ J L * p -L * q p -q φ q , f = ∂ p L * q | p=q φ q , f . (3.8)
Then p → λ(p) is differentiable and we have for all q ∈ J ⊂ I,

d dp λ (q) = ∂ p L * q | p=q φ q , N q . (3.9)
Moreover, under the assumptions of Lemma 2.4, we have (2.9).

Proof of Lemma 3.1

Since N p ∈ Λ and L p N p = λ(p)N p then we have

∀g ∈ Λ * , λ(p) = g, L p N p g, N p .
Hence we obtain the inequality

λ(p) ≤ max f ∈Λ min g∈Λ * g, L p f g, f .
Now, assume there exists f 0 ∈ Λ such that

λ(p) < min g∈Λ * g, L p f 0 g, f 0 .
In particular, this inequality holds for g = φ p and so we have

λ(p) < φ p , L p f 0 φ p , f 0 = L * p φ p , f 0 φ p , f 0 = λ(p),
which is impossible. Thus (3.4) is satisfied. Now, assume that L p is increasing in the sense of (3.5).

Then we have the inequalities g, L p (f ) ≤ g, L q (f ) , and (using (3.4)),

min g∈Λ * g, L p f g, f ≤ min g∈Λ * g, L q f g, f ≤ λ(q).
In particular, for f = N p , we find :

λ(p) = min g∈Λ * g, L p N p g, N p ≤ min g∈Λ * g, L q N p g, N p ≤ λ(q).
Finally, if L p is continuous with respect to p, then using the min-max property, we have,

g, L p f g, f ≤ g, L q f g, f + (p, q),
where (p, q) → p→q 0. Thus, we obtain that

min g∈Λ * g, L p f g, f ≤ min g∈Λ * g, L q f g, f + (p, q) ≤ λ(q) + (p, q),
and so λ(p) ≤ λ(q) + (p, q).

Similarly, λ(q) ≤ λ(p) + 1 (p, q), where 1 (p, q) → p→q 0. Therefore, we prove that λ(p) is continuous with respect to p. 2

Proof of Lemma 3.2

For all q, p ∈ J, we have : λ q = L * q φ q , N q . Thus, we find

λ q -λ p = L * q φ q , N q -L * p φ p , N p . (3.10)
We obtain, using the normalization (3.3), i.e. φ q , N q = φ p , N p = 1, and

(φ s , N s ) ∈ C * × (B C), λ q -λ p = L * q φ q -L * p φ q , N p + L * q φ q , N q -N p -L * p φ p -L * p φ q , N p ,
that gives, using (1.1), (3.2), and the duality between L q and L * q :

λ q -λ p = L * q φ q -L * p φ q , N p + λ q φ q , N q -N p -λ p φ p -φ q , N p .

Therefore, using the normalization (3.3), we find

λ q -λ p = L * q φ q -L * p φ q , N p + λ q φ q , N q -N p -λ p φ q , N q -N p , So, we obtain (λ q -λ p ) 1 -φ q , N p -N q = L * q φ q -L * p φ q , N p . ( 3
1 -φ q , N p -N q = 1, lim p→q L * q φ q -L * p φ q , N p q -p = ∂ p L * q | p=q φ q , N p ,
and so λ(p) is differentiable and by passing to the limit, we find (3.9).

We prove Lemma 2.4. We have (see the equation (3.11)),

(λ(σ)-λ(σ-)) 1- ∞ 0 φ σ-(y)-φ σ (y) N σ-(y)dy = ∞ 0 L * σ (φ σ )(y)-L * σ-(φ σ )(y) N σ-(y)dy.
Using (2.7), we have

(L * σ -L * σ-)(g) = B(y) g(yσ) -g(yσ -y ) + g y(1 -σ) -g(y(1 -σ) + y ) .
Thus we find

(λ(σ) -λ(σ -)) 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy = ∞ 0 B(y) φ σ (yσ) -φ σ (yσ -y ) + φ σ y(1 -σ) -φ σ (y(1 -σ) + y ) N σ-dy,
which can be written also as

(λ(σ) -λ(σ -)) 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy = ∞ 0 B(y) yσ y(σ-) ∂ ∂y φ σ (s)ds + y(1-σ) y(1-σ+ ) ∂ ∂y φ σ (s)ds N σ-dy.
Finally, dividing by , we have

λ(σ) -λ(σ -) = ∞ 0 B(y) yσ y(σ-) ∂ ∂y φ σ (s) ds + y(1-σ) y(1-σ+ ) ∂ ∂y φ σ (s) ds N σ-dy 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy . ( 3
.12) Thus, using [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF], Lemma 2.3 and the Lebesgue dominated convergence theorem, we pass to the limit in (3.12) as → 0. Therefore the function λ(σ) is differentiable. Moreover, passing to the limit, we obtain (2.9). 2

Proof of the convexity of the dual eigenfunction

In this section, we are interested in global conditions on the birth rate B such that the dual function φ σ (.) is convex (resp. concave) on the real line. We prove here Theorem 2.7.

I -Assume that (2.1) holds. Using (1.2), we have,

φ σ (y) = Cte e λ(σ)y , ∀y ∈ [0, a],
where φ σ is convex on [0, σb] ⊂ [0, a] for all σ such that σb < a.

II -Assume that (2.2) holds. Then we first prove the concavity of the dual eigenfunction in a neighborhood of y = 0 and we extend the property of concavity to y ∈ [0, σ 0 b].

We notice that φ σ (0) > 0, ∂φσ ∂y (0) = (λ σ -B(0))φ σ (0) < 0 since B is decreasing (and non constant) and λ σ < B(0) = sup B. Moreover, assumption (2.2) implies

∂ 2 φ σ ∂y 2 (0) < 0. Using - ∂φ σ ∂y + (B(y) + λ σ )φ σ (y) = B(y) φ σ (σy) + φ σ ((1 -σ)y) , y ≥ 0,
we find,

∂ 2 φ σ ∂y 2 = λ σ ∂ ∂y φ σ (y) - B (y) B(y) - ∂ ∂y φ σ + λ σ φ σ (y) -B(y) σ ∂ ∂y φ σ (σy) + (1 -σ) ∂ ∂y φ σ ((1 -σ)y) - ∂ ∂y φ σ (y) , y ≥ 0.
Now, assume that there exists 0 < α < bσ 0 such that

∂ 2 φ σ ∂y 2 (y) < 0, y ∈ [0, α[, ∂ 2 φ σ ∂y 2 (α) = 0, (4.1)
and

∂φ σ ∂y (y) < 0, y ∈ [0, α[. (4.2)
Then we have

-B(y) σ ∂ ∂y φ σ (σy) + (1 -σ) ∂ ∂y φ σ ((1 -σ)y) - ∂ ∂y φ σ (y) < 0, y ∈ [0, α],
and

∂φ σ ∂y (y) < ∂φ σ ∂y (0) = λ σ -B(0) φ σ (0) < 0, y ∈ [0, α], 0 < φ σ (y) < φ σ (0), y ∈ [0, α].
Assumption (2.2) implies

λ σ + B (y) B(y) ≥ λ σ + λ σ (λ σ -B(0)) B(0) = λ 2 σ B(0) ≥ 0, ∀y ∈ [0, bσ 0 ].
Thus we obtain

∂ 2 φ σ ∂y 2 (y) < λ σ + B (y) B(y) (λ σ -B(0)) - B (y) B(y) λ σ φ σ (0), y ∈ [0, α], ∂ 2 φ σ ∂y 2 (y) < λ σ λ σ -B(0) B(0) - B (y) B(y) φ σ (0)B(0), y ∈ [0, α],
and finally, we have

∂ 2 φ σ ∂y 2 (y) < 0, y ∈ [0, α].
Therefore, the strict inequalities (4.1) and (4.2) are satisfied on [0, bσ 0 ] and φ σ is a concave function on [0, σ 0 b] for all σ ∈ [1/2, σ 0 ]. 2

Extension to a more general cell division model

In this section we extend the main results to a more general model of cell division. More precisely, in Section 5.1, we generalize the results to the homogeneous cell division (see [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF]) and in Section 5.2, we allow cells to have non constant rate of division.

Homogeneous cell division

A cell of size y may give birth to a cell of size ys, s ∈ [0, 1], with "probability" Θ(s) where

Θ(s) = Θ(1 -s), (5.1)
1 0 sΘ(s)ds = 1, 1 0 Θ(s)ds = 2, (5.2) 1 η Θ(s)ds ≤ 1, for η ∈]0, 1[. ( 5
Again, we prove, using General Relative Entropy method that the asymptotic behavior of such an evolution equation is given by the following eigenproblem (5.6)

                       ∂ ∂y N (
Then we have λ Θ ≥ λ sym , (resp. λ Θ ≤ λ sym ).

Thus, the less (resp. the 'best') fitted division is the symmetric division.

Proof of Theorem 5.2 Under assumption (2.1) (resp. (2.2)), we directly obtain that the dual eigenfunctions are convex (resp. concave) on

ConvHull(Supp Θ).Supp B := {ry, r ∈ ConvHull(Supp Θ), y ∈ Supp B}, for all Θ that satisfies (5.1)-( 5.3) and (5.6). We use a min-max principle to conclude the proof of the theorem.

Step 1. Min-max method: Let (5.8)

We have the min-max lemma (see Section 3). Lemma 5.3 Assume that there exists (N, λ, φ) solution to (5.5). Then we have for all Θ,

λ Θ = max f ∈Λ min g∈Λ * g, Lf g, f , (5.9)
where φ ∈ Λ * ⊂ D om (L * ) and N ∈ Λ ⊂ D om (L).

Step 2. Application to the homogeneous cell division: Let (5.12)

L sym (f ) := - ∂f ∂y (.) -B(.)f (t, .) + 4B(2.)f (2.), ( 5
Proof. We have shown that for all Θ satisfying (5.1)-( 5.3) and (5.6), and B satisfying (2.1) (resp. 2.2)), the dual functions φ (and in particular φ sym ) are convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. Thus, we can choose Λ * as the set of positive convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. We have

L * (f ) -L * sym (f ) = B(.) 1 0 Θ(s) f (s.) -f (./2) ds, ∀f ∈ Λ * ,
which can be rewritten as

L * (f ) -L * sym (f ) = 2B(.) 1 1/2 Θ(s) f (s.) + f ((1 -s).) 2 -f (./2) ds, ∀f ∈ Λ * .
Since f ∈ Λ * is a convex (resp. concave) function on ConvHull(Supp Θ).Supp B, then we have

f (s.)+f ((1-s).) 2
-f (./2) ≥ 0 (resp. ≤ 0), and

L * (f ) -L * sym (f ) ≥ 0, ∀f ∈ Λ * , (resp. ≤ 0
). Thus, using the min-max Lemma 5.3, we have λ(Θ) ≥ λ sym (resp. λ(Θ) ≤ λ sym ). 2

Cell division model with non constant speed rate

In this section we consider the cell division equation

∂ ∂t n(t, y) + ∂ ∂y V (y)n(t, y) + B(y)n(t, y) = 1 0 Θ(s)B(y/s)n(t, y/s)ds/s, (5.13)
where the birth rate satisfies Supp B(y) ⊂ [a, b] and the speed rate V (y) = y µ . As above, we prove, using General Relative Entropy method that the asymptotic behavior of such an evolution equation is given by the following eigenproblem Here, we only have to prove the convexity (resp. the concavity) of the dual eigenfunction in (5.14). First, we recall results on the existence of a solution to the eigenproblem. Then, we use the differentiation method in order to find the variation of the invasive parameter with respect to Θ by proving that the solution φ Θ to the dual problem is convex (or concave).

                       ∂ ∂y (V (
Step 1. Existence and uniqueness of the solution to the eigenproblem (5.14). We have the following lemma. Lemma 5.6 Under the assumption µ > -1, there exists a solution (N σ , λ(σ), φ σ ) to (2.5). We refer to [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF] for the proof of this result.

Step 2. Corollary of the min-max method Using the proof of Theorem 5.2 and Lemmas 5.3 and 5.4, we have the following lemma.

Lemma 5.7 If φ σ is a convex (resp. concave) function on [1/2 -η, 1/2 + η].Supp B = σSupp B with Θ = δ σ + δ 1-σ for σ ∈ [1/2 -η, 1/2 + η], then ∀ 1/2 -η < σ 0 < σ 1 < 1/2 + η, λ σ0 < λ σ1 (resp. > λ σ1 ).
Moreover, if we assume that the probabilities Θ satisfy (5.1)-( 5.3) and (5.6), and the dual eigenfunction φ is convex (resp. concave) on ConvHull(Supp Θ).Supp B, then we have

inf Θ λ Θ = λ sym (resp. sup Θ λ Θ = λ sym ).
Proof. Indeed, we have,

L * σ0 (f ) -L * σ1 (f ) = B(.) f (σ 0 .) + f ((1 -σ 0 ).) -f (σ 1 .) -f ((1 -σ 1 ).) ds, ∀f ∈ Λ * ,
where Λ * is the set of positive convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. Since 1/2 -η < σ 0 < σ 1 < 1/2 + η, then we find that the constant t given by the equality

t = σ 0 + σ 1 -1 2σ 1 -1 ∈]0, 1[, satisfies σ 0 = tσ 1 + (1 -t)σ 1 and 1 -σ 0 = tσ 1 + (1 -t)σ 1 . We have f (σ 0 y) + f ((1 -σ 0 )y) -f (σ 1 y) -f ((1 -σ 1 )y) = f (σ 0 y) -tf (σ 1 y) -(1 -t)f ((1 -σ 1 )y) + f ((1 -σ 0 )y) -tf ((1 -σ 1 )y) -(1 -t)f (σ 1 y) ,
with f positive convex (resp. concave) function and t ∈]0, 1[. Thus, we find that

L * σ0 (f ) -L * σ1 (f ) ≤ 0 (resp. ≥ 0), ∀f ∈ Λ * .
We conclude using the extension of the min-max Lemma 5.3. Next, we assume that for all Θ satisfying (5.1)-(5.3) and (5.6), the dual functions φ (and in particular φ sym ) are convex (resp. concave) functions on ConvHull(Supp Θ).Supp B = [1 -σ, σ].Supp B. Thus, we can choose, in Lemma 5.3, Λ * as the set of positive convex (resp. concave) functions on [1 -σ, σ].Supp B. We have

L * (f ) -L * sym (f ) = B(.) 1 0 Θ(s) f (s.) -f (./2) ds, ∀f ∈ Λ * ,
which can be rewritten as

L * (f ) -L * sym (f ) = 2B(.) 1 1/2 Θ(s) f (s.) + f ((1 -s).) 2 -f (./2) ds, ∀f ∈ Λ * .
Since f ∈ Λ is a positive convex (resp. concave) function, then we have f (s.)+f ((1-s).)

2 -f (./2) ≥ 0 (resp. ≤ 0), L * (f ) -L * sym (f ) ≥ 0 (resp. ≤ 0), ∀f ∈ Λ * .
Thus, using the min-max Lemma 5.3, we have λ(Θ) ≥ λ sym (resp. λ(Θ) ≤ λ sym ) .

2

Step 3. Concavity of the dual eigenfunction We have

φ (y) = λ 2 V (y) 2 1 -V /λ , y ∈ Supp B,
with V (y) = y µ and Supp B = [a, b]. Therefore, we have clearly that assumptions (5.16) or (5.17) lead to φ (y) > 0 on ConvHull(Supp Θ).Supp B and condition (5.15) leads to φ (y) < 0 on ConvHull(Supp Θ).Supp B.

Step 4. Conclusion Thus, using Lemma 5.7 and step 3, we prove the theorem. 2

6 Simple application of the differentiation and the min-max methods

In this section, in order to apply both methods, we focus our attention on a simple model of McKendrick-VonFoerster in which cells are structured by their age (see for instance [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF][START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF][START_REF] Webb | Theory of non-linear age-dependent population dynamics[END_REF]). The density of the population of age a at time t is denoted by n(t, a) and follows the renewal equation

         ∂ ∂t n(t, a) + ∂ ∂a n(t, a) + d(a)n(t, a) = 0, n(t, 0) = ∞ 0 B(a)n(t, a)da, (6.1)
where B is the growth rate and d the death rate. As above, the GRE gives the asymptotic behavior of n(t, .) ∼ cst N (.)e λt , where (N, λ, φ) is a solution to the eigenproblem The relevance of this model is its simplicity, contrary to the cell division model (1.1). We can easily compute (N, λ, φ). Indeed, if there exists λ such that Thus we have the following lemma. 6.1 Evolution of the invasive parameter λ with respect to the growth of the death rate d or the birth rate B.

                     ∂ ∂a N (
The aim of this part is to study the evolution of the first eigenvalue λ when the death rate (resp. the birth rate) increases.

Growth of death rate. Let n 1 (resp. n 2 ) be the density of the population satisfying (6.1) with the death rate d = d 1 (resp. d = d 2 ) that verifies the assumptions of Lemma 6.1.

Let λ 1 (resp. λ 2 ) be the invasive parameter, that is the eigenvalue, associated to the first population n 1 (resp. the second population n 2 ) Lemma 6.2 Assume that 0 ≤ d 1 ≤ d 2 . Then λ 2 ≤ λ 1 . Thus, the population n 2 is less fitted than the population n 1 .

Proof. Here we can use both methods. Indeed, let

L 1 (f ) = -[ ∂ ∂a f (a) + d 1 (a)f (a)] + δ a=0 ∞ 0 B(a )f (a )da , L 2 (f ) = -[ ∂ ∂a f (a) + d 2 (a)f (a)] + δ a=0 ∞ 0 B(a )f (a )da , then L 1 (f ) = L 2 (f ) + (d 2 -d 1 )(a)f (a) ≥ L 2 (
f ) for all f ≥ 0 and thus, using Lemma 3.2, we obtain that λ 1 ≥ λ 2 . 2 Remark. The differentiation method could be used to prove the lemma. Indeed, let

L * p (f ) = -[- ∂ ∂a f (a) + d p (a)f (a)] + B(a)f (0), with d p = pd 2 + (1 -p)d 1 and p ∈ [0, 1]. Then L * p (f ) -L * q (f ) = d q (a) -d p (a) f (a) = (p -q)(d 1 -d 2 ),
and so (3.8) is satisfied. Moreover, the implicit form of λ(p) and the explicit form of N p imply directly condition (3.7). Therefore, we have,

d dp λ(p) = ∞ 0 (d 1 -d 2 )(a)φ p (a)N p (a)da ≤ 0,
and we find λ

2 = λ(1) ≤ λ(0) = λ 1 .
Growth of birth rate. Let n 1 (resp. n 2 ) be the density of the population satisfying (6.1) with the birth rate B = B 1 (resp. B = B 2 ) that verifies the assumptions of Lemma 6.1. Let λ 1 (resp. λ 2 ) be the invasive parameter, i.e., eigenvalue, associated to the first population n 1 (resp. the second population n 2 ) Lemma 6.3 Assume that 0 ≤ B 1 ≤ B 2 . Then λ 1 ≤ λ 2 . Thus, the population n 2 is better fitted than the population n 1 .

Proof. Here we can use both methods. Indeed, let

L 1 (f ) = -[ ∂ ∂a f (a) + d(a)f (a)] + δ a=0 ∞ 0 B 1 (a )f (a )da , L 2 (f ) = -[ ∂ ∂a f (a) + d(a)f (a)] + δ a=0 ∞ 0 B 2 (a )f (a )da . Then L 1 (f ) = L 2 (f ) + δ a=0 ∞ 0 (B 1 (a ) -B 2 (a ))f (a )da ≤ L 2 (
f ) for all f ≥ 0 and thus, using Lemma 3.2, we obtain that λ 1 ≤ λ 2 .

2 Remark. The differentiation method could be used to prove the lemma. Indeed, let We notice that both examples give the same results using the min-max method or the differentiation method. Here, we give an example which is more difficult to study with the min-max method.

L * p (f ) = -[- ∂ ∂a f (
6.2 Evolution of the invasive parameter λ with respect to a complex variation of the death rate d.

Child Vs Oldness death. Let n p be the density of the population satisfying (6.1) with the death rate d p (y) = 1 y∈[p,p+1] d(y) that verifies the assumptions of Lemma 6.1. More precisely, the death rate d p has its support on the interval [p, p + 1], which means that for small p, only childs die and for large p, only old individuals die. Let λ(p) be the invasive parameter, that is the eigenvalue, associated to the population n p . In this part, we assume that

Supp B = [α, β] ⊂ [0, ∞[, (6.5
)

and d ∈ C 1 ([0, ∞[) decreases on [0, β -1]. (6.6)
Then we have the following result.

Lemma 6.4 Assume that (6.5) and (6.6) are satisfied. Then λ(p) is increasing, that is the juvenile death rate is less fitted than 'oldness' death rate.

Proof. Here we can use only the differentiation method. Indeed, let Therefore, we obtain that λ(p) is increasing. 2

L * p (f ) = -[- ∂ ∂a f (
We notice that the min-max method is not well adapted when the growth of the operators L p with respect to p is hard to prove. As for instance in the McKendrick-Von Foerster model when the death rates cannot be compared. The question of knowing the variation of the invasive parameter with respect to the asymmetry parameter leads to the same remark and therefore, we first use the differentiation method.

Discussion

In this paper we have studied the variation of the Malthusian growth rate of a cell population due to the variability of the symmetry of the cell division during mitosis [START_REF] Huang | Symmetry of Initial Cell Divisions Among Primitive Hematopoietic Progenitors Is Independent of Ontogenic Age and Regulatory Molecules[END_REF]. We show that under the assumption of a constant speed rate, equal mitosis is not necessarily the division giving the better growth rate. More precisely, when the cells divide lately, that is for large size relatively to the average size of a dividing cell, then equal mitosis is not the best fitted division. When cells can divide early, then equal mitosis is the best fitted division.

Obviously, this model is a simple one and the conclusion depends on the assumptions such as : -there is no interaction between cells, no competition between species with different asymmetry parameters, -the speed rate is constant (or a power of the size) which cannot be the case in general, -there is no apoptosis in the model.

Nevertheless, the tools we have used to study this problem, that is the differentiation method and the min-max method, seem to be well fitted to study the variation of the Malthusian growth rate with respect to a parameter for various problems (see Section 6).

An interesting question to study is the competition between two species of cells (with two different cell divisions) as for instance for normal and tumoral (due to mutation) cells.

0 N( 5 . 5 ) 5 . 1

 05551 y) + [λ + B(y)]N (y) = φ(ys)ds -φ(y) , N (y = 0) = 0, N, φ ≥ 0, ∞ (y)φ(y)dy = 1, φ(0) = 1. Definition We call λ Θ the first eigenvalue associated to eigenproblem (5.5) and λ sym the eigenvalue associated to Θ(s) = δ s=1/2 . Then we have the following theorem. Theorem 5.2 Assume that (2.1) (resp. (2.2)) holds and ConvHull(Supp Θ) := Convex Hull(Supp Θ) ⊂ [1 -a/b, a/b] (resp.[1 -σ 0 , σ 0 ]).

1 0Θ

 1 operator, L * (f ) := ∂f ∂y (.) -B(.)f (t, .) + B(.) (s)f (.s)ds -f (.) .

. 10 )Lemma 5 . 4

 1054 and the dual operator,L * sym (f ) := ∂f ∂y (.) -B(.)f (t, .) + B(.) 2f (./2) -f (.) . (5.11) Assuming (2.1) (resp. 2.2)), we have for all Θ satisfying (5.1)-(5.3) and (5.6), λ Θ ≥ λ sym (resp ≤ λ sym ).

1 0Θ 1 0Θ 5

 115 y)N (y)) + [λ + B(y)]N (y) = (s)B(y/s)N (y/s)ds/s, y ≥ 0, -V (y) ∂ ∂y φ(y) + [λ + B(y)]φ(y) = B(y) (s)φ(ys)ds, y ≥ 0,N (0) = 0, N ≥ 0, φ ≥ 0, N(y)φ(y)dy = 1.(5.14)If λ sym = 0 and one of these conditions holds 0 < µ < 1 and 0 < λ sym < µa µ-1 , (5.15)µ > 1 and λ sym > µa µ-1 , Assume that Θ satisfies (5.1)-(5.3) and (5.6), and µ satisfies (5.15) (resp. (5.16) or (5.17)). Then we have sup Θ λ Θ ≤ λ sym (resp. ≥ λ sym ).(5.18)

  a) + d(a)N (a) + λN (a) = 0, N (0) = ) + d(a)φ(a) + λφ(a) = B(a)φ(0),

∞ 0 B 0 B

 00 (a )e -a 0 (d(s)+λ)ds da = 1) + d(a)f (a)] + δ a=0 ∞ (a )f (a )da , L * (g) = -[-∂ ∂a g(a) + d(a)g(a)] + g(0)B(a).

Lemma 6 . 1 0 B

 610 Assume that B, d ∈ L ∞ ([0, ∞[, [0, ∞[) and Supp B ⊂ [0, A] is compact.Then we have the existence of a solution (N, λ, φ) to eigenproblem (6.2). Moreover, we have,L(f ) : L 2 ([0, A]) → M([0, A]), L * (f ) : C 0 0 ([0, ∞[) → L 2 ([0, ∞[),where M([0, A]) is the set of a bounded measure on [0, A] andN ∈ C 0 b ([0, A]), N ≤ 1, Supp φ ⊂ [0, A], φ ≤ A(s)dse λA . Therefore (3.1)-(3.3) are satisfied for B = L 2 and C = C 0 0 ([0, ∞[).

0 (B 2 -

 02 a) + d(a)f (a)] + B p (a)f (0), with B p = pB 2 + (1 -p)B 1 and p ∈ [0, 1]. Then (as above) B 1 )(a)φ p (0)N p (a)da ≥ 0, and we find λ 2 = λ(1) ≥ λ(0) = λ 1 .

  a) + d p (a)f (a)] + B(a)f (0). Then for d p (y) = 1 y∈[p,p+1] d(y) = 1 p∈[y-1,y] d(y) we have -∂ ∂p d p (y) = -δ p=y-1 d(p + 1) + δ p=y d(p), and so, we find d dp λ(p) = d(p)φ p (p)N p (p) -d(p + 1)φ p (p + 1)N p (p + 1). Therefore, using (6.4),we obtain, d dp λ(p) = -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da + d(p) ∞ p B(a )e -a 0 (d(s)+λ)ds da , and so, we have, d dp λ(p) = d(p) -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da + d(p) p+1 p B(a )e -a 0 (d(s)+λ)ds da . Since d increases (see (6.6)), then d(p) -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da ≥ 0 and d dp λ(p) ≥ 0.