Mathematical modeling and analysis of anemia during pregnancy and postpartum

Abstract

Anemia is a significant public health problem worldwide especially among pregnant women in low- and middle-income countries. In this study, a mathematical model of the population dynamics of anemia during pregnancy and postpartum is constructed. In the modeling process, four independent variables have been considered: (1) the numbers of nonpregnant nonanemic women, (2) anemic nonpregnant women, (3) anemic pregnant or postpartum women and (4) anemic pregnant or postpartum women with complications. The mathematical model is governed by a system of first-order ordinary differential equations. The stability analysis of the model is conducted using Routh–Hurwitz criteria. There is one nonnegative equilibrium point which is asymptotically stable. The equilibrium point obtained indicates the influential parameters that can be controlled to minimize the number of patients at each stage. The proposed model can be employed to forecast the future incidence and prevalence of the disease and appraise intervention programs.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Allen LH (2000) Anemia and iron deficiency: effects on pregnancy outcome. Am J Clin Nutr 71:1280S-1284S. https://doi.org/10.1093/ajcn/71.5.1280s

    CAS  Article  PubMed  Google Scholar 

  2. Allen LJS (2006) Introduction to mathematics biology. Pears Education, Slough

    Google Scholar 

  3. Arnold DL, Williams MA, Miller RS, Qiu C, Sorensen TK (2009) Iron deficiency anemia, cigarette smoking and risk of abruption placentae. J Obstet Gynaecol Res 35:446–452. https://doi.org/10.1097/01.ogx.0000361376.67139.7d

    CAS  Article  PubMed  Google Scholar 

  4. Beard JL, Hendricks MK, Perez EM et al (2005) Maternal iron deficiency anemia affects postpartum emotions and cognition. J Nutr 135:267–272

    CAS  Article  Google Scholar 

  5. Brabin JB, Hakimi M, Pelletier D (2001) An analysis of anemia and pregnancy-related maternal mortality. J Nutr 131:604S-615S

    CAS  Article  Google Scholar 

  6. Colomer J, Colomer C, Gutierrez D et al (1990) Anemia during pregnancy as a risk factor for infant iron deficiency: report from the Valencia Infant Anemia Cohort (VIAC) study. Paediatr Perinat Epidemiol 4:196–204

    CAS  Article  Google Scholar 

  7. Frass KA (2015) Postpartum hemorrhage is related to the hemoglobin levels at labor: observational study. Alexandria J Med 51:333–337. https://doi.org/10.1016/j.ajme.2014.12.002

    Article  Google Scholar 

  8. Garnett GP (2002) An introduction to mathematical models in sexually transmitted disease epidemiology. Sex Transm Inf 78:7–12

    CAS  Article  Google Scholar 

  9. Haas JD, Brownlie TI (2015) Iron Deficiency and reduced work capacity: a critical review of the research to determine a causal relationship. J Nutr 131:676S-688S

    Article  Google Scholar 

  10. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653. https://doi.org/10.1137/S0036144500371907

    Article  Google Scholar 

  11. Jackson S (1995) Estimated resident population by age and sex in statistical local areas: Victoria. Australian Bureau of Statistics, Canberra

    Google Scholar 

  12. Kassebaum NJ, Jasrasaria R, Naghavi M et al (2014) A systematic analysis of global anemia burden from 1990 to 2010. Blood 123:615–624. https://doi.org/10.1016/S0140-6736(13)61326-4

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  13. Kavle JA, Stoltzfus RJ, Witter F, Tielsch JM, Khalfan SS, Caulfield LE (2008) Association between anemia during pregnancy and blood loss among women with vaginal births in Pemba Island, Zanzibar. Tanzania. J Health Popul Nutr 26:232–240

    PubMed  PubMed Central  Google Scholar 

  14. Kochanek KD, Murphy SL, Xu J, Tejada-Vera B (2016) Deaths: final data for 2014. National vital statistics reports 65, pp 1–122. https://www.cdc.gov/nchs/data/nvsr/nvsr65/nvsr65_04.pdf. Accessed 17 May 2016

  15. Kopp M (2011) Equilibria and stability analysis. Universitet Wien. http://www.mabs.at/koop/teaching/modelling2011/files/stab.pdf. Accessed 15 May 2016

  16. Lichtenberg FR (2005) Importation and innovation. Columbia University and National Bureau of Economic Research. https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/2062/Importationandinnovation2005-04-19.pdf. Accessed 25 May 2016

  17. Mat Daud AA (2019) Mathematical modeling and stability analysis of population dynamics. In: Dynamical systems, bifurcation analysis and applications. DySBA 2018. Springer Proceedings in Mathematics & Statistics, Springer, vol 295, pp 3–13

  18. Mat Daud AA (2020) Some issues on the mathematical modeling of population dynamics using differential equations. Int J Math Comput Sci 15(2):501–513

    Google Scholar 

  19. Mat Daud AA, Toh CQ, Saidun S (2019) A mathematical model to study the population dynamics of hypertensive disorders during pregnancy. J Int Math 22(4):433–450. https://doi.org/10.1080/09720502.2019.1632025

    Article  Google Scholar 

  20. Mat Daud AA, Toh CQ, Saidun S (2020) Development and analysis of a mathematical model for the population dynamics of Diabetes Mellitus during pregnancy. Math Models Comput Simul 12(4):620–630

    Article  Google Scholar 

  21. Ministry of Health Malaysia (2012) Health indicator 2012. Ministry of Health Malaysia, Putrajaya. http://www.moh.gov.my/images/gallery/publications/md/hi/hi_2012.pdf. Accessed 25 May 2016

  22. Ministry of Health Malaysia (2013) Health facts 2013. Ministry of Health Malaysia, Putrajaya. http://www.moh.gov.my/images/gallery/publications/HEALTH%20FACTS%202013.pdf. Accessed 25 May 2016

  23. Nair M, Choudhury MK, Choudhury SS, Kakoty SD, Sarma UC, Webster P, Knight M (2016) Association between maternal anemia and pregnancy outcomes: a cohort study in Assam, India. BMJ Glob Health 1:e000026. https://doi.org/10.1136/bmjgh-2015-000026

    Article  PubMed  PubMed Central  Google Scholar 

  24. National Center for Health Statistics (2016). Anemia or iron deficiency. Center for Disease Control. http://www.cdc.gov/nchs/fastats/anemia.htm. Accessed 17 May 2016

  25. Nik Abdul Rashid NR, Rashid HH, Mohamed NM, Mohd R, Mohamed Z (2015) Report on the confidential enquiries into maternal death in Malaysia 2009–2011. Family Health Development Division Ministry of Health Malaysia, Putrajaya

    Google Scholar 

  26. Perez EM, Hendricks MK, Beard JL et al (2005) Mother-infant interactions and infant development are altered by maternal iron deficiency anemia. J Nutr 135:850–855

    CAS  Article  Google Scholar 

  27. Ravichandran J (2011) MDSR: The role of government agencies and professional bodies. MDSR Action Network. http://mdsr-action.net/wp-content/uploads/2015/11/MDSR_professional-bodies_Malaysia_Ravichandran.pdf. Accessed 18 May 2016

  28. Ravichandran J, Karalasingam SD (2015) National Obstetrics Registry 3rd report of National Obstetrics Registry 2011–2012. Clinical Research Centre (CRC) and Ministry of Health Malaysia, Kuala Lumpur. http://www.acrm.org.my/nor/doc/reports/NOR_REPORT_2012.pdf. Accessed 25 May 2016

  29. Stevens GA, Finucane MM, De-Regil LM et al (2013) Global, regional, and national trends in haemoglobin concentration and prevalence of total and severe anemia in children and pregnant and non-pregnant women for 1995–2011: a systematic analysis of population-representative data. Lancet Glob Health 1:e16–e25. https://doi.org/10.1016/S2214-109X(13)70001-9

    Article  PubMed  PubMed Central  Google Scholar 

  30. Stoltzfus RJ, Mullany L, Black RE (2015) Iron deficiency anemia. In: Ezzati M, Lopez AD, Rodgers A, Murray CJL (eds) Comparative quantification of health risks: global and regional burden of disease attributable to selected major risk factors. World Health Organization, Geneva

    Google Scholar 

  31. Suppiah WR, Abdul Rahman N, Mohamad Ashray FS, Lim HC, Zakaria N (2014) Statistics on women, family and community Malaysia 2014. Ministry of Women, Family and Community Development, Kuala Lumpur

    Google Scholar 

  32. Tey NP, Ng ST, Yew SY (2015) Proximate determinant of fertility in Peninsular Malaysia. Asia Pac J Public Health 24:495–505

    Article  Google Scholar 

  33. World Health Organization (2015) The global prevalence of anemia in 2011. World Health Organization, Geneva. http://apps.who.int/iris/bitstream/10665/177094/1/9789241564960_eng.pdf?ua=1&ua=1. Accessed 25 May 2016

  34. World Health Organization (2014) Global nutrition targets 2025: Anemia policy brief. World Health Organization, Geneva. http://apps.who.int/iris/bitstream/10665/148556/1/WHO_NMH_NHD_14.4_eng.pdf?ua=1. Accessed 23 May 2016

Download references

Acknowledgements

This research was funded by The Ministry of Education, Government of Malaysia under the Research Acculturation Grant Scheme (RAGS 57108).

Author information

Affiliations

Authors

Contributions

AAMD and Salilah Saidun contributed to conceptualization; AAMD helped with methodology, funding acquisition and resources and supervision; AAMD and CQT contributed to formal analysis and investigation; AAMD, CQT, and SS helped with writing—original draft preparation; and AAMD and SS contributed to writing—review and editing.

Corresponding author

Correspondence to Auni Aslah Mat Daud.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1: Formulating the governing equations of the model

The state variables are depicted by the boxes in the flow diagram, while the arrows illustrate the movement of people between different states in the population. The flows shown as arrows are calculated using the terms on the right-hand side of the equations: a flow pointing out of a box is taken away from a state variable and is a negative term, while a flow pointing into a box is added and is a positive term. Some constant parameters are introduced adjacent to the arrows to represent the proportionality rates of the flows. An illustration of an example of a flow diagram is shown in Fig. 

Fig. 2
figure2

An example of a flow diagram

2.

Formulating differential equations based on the flow diagram:

figurea

See Table 3.

Table 3 Description of the parameters in the model

Appendix 2: The computation of equilibrium point

Consider a mathematical model governed by a system of differential equations:

$$\begin{aligned} & \frac{{{\text{d}}x_{1} }}{{{\text{d}}t}} = f_{1} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & \frac{{{\text{d}}x_{2} }}{{{\text{d}}t}} = f_{2} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & \quad \quad \quad \quad \vdots \\ & \frac{{{\text{d}}x_{n} }}{{{\text{d}}t}} = f_{n} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & f_{1} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ & f_{2} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ \vdots \\ f_{n} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ \end{aligned}$$

The equilibrium point \(\left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right)\) is obtained by solving the equations above simultaneously or using any methods of solving the system of equations in algebra.

Appendix 3: The stability analysis using Routh–Hurwitz criteria

In this study, we will study a multiple variables model with continuous time. Therefore, the stability analysis is performed using the following steps:

  • Step 1 Calculate a Jacobian matrix

$$J = \left( {\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{1} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{1} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ {\frac{{\partial f_{2} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{2} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{2} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\frac{{\partial f_{n} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{n} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{n} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ \end{array} } \right)$$

where \(\frac{{\partial f_{i} }}{{\partial x_{j} }}\left( {x_{1} , x_{2} \ldots ,x_{n} } \right)\) is the partial derivative of \(f_{i}\) with respect to its variable, \(x_{j}\) (\(i,j = 1, 2, \ldots , n\)).

  • Step 2 Find the Jacobian matrix

The Jacobian matrix is evaluated at the equilibrium values, \(\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}\). A local stability matrix, \(\hat{J} = \left. J \right|_{{x_{1} = \widehat{{x_{1} }}, x_{2} = \widehat{{x_{2} }}, \ldots , x_{n} = \widehat{{x_{n} }}}}\) is obtained. Then, find the characteristic polynomial using \(\det \left( {\hat{J} - \lambda I} \right) = 0\), where \(I\) is the identity matrix, and rewrite in the following form:

$$P\left( \lambda \right) = \lambda^{n} + a_{1} \lambda { }^{n - 1} + \ldots + a_{n - 1} \lambda + a_{n}$$

with real coefficients \(a_{i}\) for \(i = 1, 2, \ldots , n.\)

  • Step 3 Use Routh–Hurwitz criteria

The \(n\) Hurwitz matrices are defined as follows:

$$\begin{aligned} & H_{1} = \left( {a_{1} } \right),\;H_{2} = \left( {\begin{array}{*{20}c} {a_{1} } & 1 \\ {a_{3} } & {a_{2} } \\ \end{array} } \right),\;{\text{and}} \\ & H_{n} = \left( {\begin{array}{*{20}c} {a_{1} } & 1 & 0 & 0 & \cdots & 0 \\ {a_{3} } & {a_{2} } & {a_{1} } & 1 & \cdots & 0 \\ {a_{5} } & {a_{4} } & {a_{3} } & {a_{2} } & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & {a_{n} } \\ \end{array} } \right) \\ \end{aligned}$$

Note that if \(j > n\), then \(a_{j} = 0.\)

If and only if all \(\det H_{j} > 0\) with \(j = 1, 2, \ldots ,n\), then \(P\left( \lambda \right)\) has roots that are negative or have negative real part and hence the equilibrium point is said to be asymptotically stable.

The Routh–Hurwitz criteria for polynomials of degree, n = 4 are:

$$a_{1} > 0, a_{3} > 0,a_{4} > 0,\quad {\text{and}}\quad a_{1} a_{2} a_{3} > a_{3}^{2} + a_{1}^{2} a_{4} .$$

Appendix 4: Proof of Theorem 1

Given that the initial conditions \({A\left(0\right)=A}_{0}, { B\left(0\right)=B}_{0}, {C\left(0\right)=C}_{0}\) and \({D\left(0\right)=D}_{0}\) are nonnegative. It is clear from Eq. (1) that

$$\frac{{{\text{d}}A}}{{{\text{d}}t}} + \left[ {\xi + \mu + \varphi + \alpha } \right]A\left( t \right) \ge 0,$$

so that

$$\frac{{\text{d}}}{{{\text{d}}t}}\left[ {A\left( t \right)\exp \left( {\xi + \mu + \varphi + \alpha } \right)t} \right] \ge 0.$$
(13)

Integrating (13) with respect to \(t\) gives

$$A\left( t \right) \ge A\left( 0 \right)\exp \left[ { - \left( {\xi + \mu + \varphi + \alpha } \right)t} \right] > 0, \quad \forall \,t \ge 0.$$

Similarly, it can be shown that \(B\left( t \right) > 0,C\left( t \right) > 0\), \(D\left( t \right) > 0\) for all time \(t > 0\). This completes the proof.

It is crucial to note that Eqs. (1)–(4) will be analyzed in a feasible region \(D\) given by

$$D = \left\{ {\left( {A,B,C,D} \right) \in R_{ + }^{4} :A + B + C + D = N} \right\},$$

which can be easily verified to be positively invariant with respect to Eqs. (1)–(4), In what follows, the model is epidemiologically and mathematically well posed in D (see Hethcote 2000).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mat Daud, A.A., Toh, C.Q. & Saidun, S. Mathematical modeling and analysis of anemia during pregnancy and postpartum. Theory Biosci. 140, 87–95 (2021). https://doi.org/10.1007/s12064-020-00334-2

Download citation

Keywords

  • Anemia
  • Pregnancy
  • Postpartum
  • Mathematical modeling
  • Differential equation
  • Routh–Hurwitz criteria