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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.

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Acknowledgements

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

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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.

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Correspondence to Auni Aslah Mat Daud.

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The authors declare no conflict of interest.

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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
figure 2

An example of a flow diagram

2.

Formulating differential equations based on the flow diagram:

figure a

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).

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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

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Keywords

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