Formulating a logistic growth model to forecase population counts across counties in Liberia from 2021 to 2031

Introduction

Accurate population forecasting is essential for effective national planning, particularly in ensuring equitable resource allocation and promoting sustainable development across all counties in Liberia. In its 2023 publication, the Liberia Institute of Statistics and Geo-Information Services (LISGIS) reveals the results of the national population housing census (NPHC), which utilized the medium fertility scenario to project population counts from 2008 to 2020. These projections suggest that the annual growth rate for each county from 2009 to 2020 remains nearly constant over time. Notable, most traditional demographic studies in Liberia continue to rely on methodologies that reflect a deterministic modeling assumption, as opposed to more adaptive dynamic or stochastic frameworks. For instance, population forecasting has historically relied on deterministic approaches used in Liberia, which assume fixed parameters and do not account for uncertainty or variability in demographic behavior. One of the most frequently applied methods is the exponential growth model, which assumes a constant growth rate and projects population using the formula as: 𝑁(𝑡) = 𝑁(0)𝑒 𝑟𝑡 .

Due to its mathematical simplicity, the linear growth method is widely used for implementation (LISGIS 2023) [2] . This approach assumes a steady annual increase in population, making it suitable for short-term projections where fluctuations in fertility or migration rates are minimal (Smath, et. al., 2013) [4] .

However, more sophisticated methods such as the cohortcomponent method for projecting population (CCMPP) are considered the gold standard in demographic forecasting globally but remain underutilized in Liberia due to limited access to detailed age and sex-disaggregated data, (United Nations Department of Economic and Social Affairs (UN DESA), 2017) [8] . According to UN DESA (2017), CCMPP is not a complete projection method on its own, as it requires the components of change to be projected in advance. Rather, it is an application of matrix algebra that enables demographers to calculate the effects of the assumed future patterns of fertility, mortality, and international migration on a population at some given point in the future (Preston, et. al., 2001; Whelpton, 1936) [3,11] . The medium fertility scenario, modeled after the United Nations framework has been prominently employed by the Liberia Institute of Statistics and Geo-Information Services (LISGIS) in national reports, including the National Population Housing Census (NPHC). This method assumes a moderate fertility trajectory and typically applies a fixed growth rate assumption extrapolated from past census intervals to estimate population counts at the county level, thereby reinforcing a deterministic approach to county-level population projection (LISGIS 2023) [2] . Additionally, standard tools are built into Microsoft Excel, such as interpolation and extrapolation for sectoral ministries and county planning offices for estimating annual population counts. This approach is called the interpolation and extrapolation spreadsheet model, while the deterministic methods that provide a baseline for planning lack the flexibility to respond to abrupt changes in migration, fertility, or health outcomes. As such, there is a growing advocacy within the research community in Liberia to adopt more dynamic or stochastic models, such as logistic growth models, Bayesian inference, or Monte Carlo simulations, to better capture Liberia's changing demographic realities and improve the accuracy of long-term population forecasts (Verhulst 1838, Tsoularis and Wallace 2002, Allen 2003) [9,5,1] . In Liberia, given the limitations of traditional deterministic approaches and the scarcity of detailed demographic data, this study proposes a logistic growth modeling framework to forecast population trends across counties in Liberia from 2021 to 2031. By incorporating time-dependent growth rates and dynamic carrying capacities, the logistic growth model aims to better reflect changing demographic patterns and resource constraints. This approach not only provides a more nuanced understanding of population expansion but also serves as a valuable tool for long-term policy planning and development. Ultimately, it seeks to enhance the accuracy and relevance of population forecasts in Liberia's evolving socioeconomic context.

Materials and Methods

Study Design

The logistic growth function was adopted for the use of a quantitative modeling approach to model and forecast countylevel population dynamics in Liberia and further investigates whether the population growth rates used in Liberia's official county-level population projections remain constant or vary over time from the national population housing census (NPHC) (2008). The analysis of this study focuses on medium fertility scenario projections published by the Liberia Institute of Statistics and Geo-Information Services (LISGIS) for the years 2008 to 2020 across all 15 counties. It compares yearover-year growth rates and fits a logistic curve to each county's population series. The study aims to determine if the underlying growth assumptions reflect changing demographic patterns or adhere to static growth estimates.

Data Source

This study utilizes a secondary dataset obtained from the Liberia Socioeconomic Statistical Bulletin: projected population by county, 2008-2020 (medium fertility scenario) published by LISGIS (2023). The projections are based on medium fertility assumptions and provide a basis for analysing temporal growth trends [2] .

Logistic Growth Equation

Let 𝑖 denote each county in Liberia. Using a logistic differential equation framework, this study derives a logistic population model tailored to Liberia's demographic structure. The logistic growth equation in its general form is given by:

Where the time-dependent growth rate 𝑟 𝑖 (𝑡) is defined as:

Substitute ( 2) into (1) and obtained this:

In this formulation, 𝑁 𝑖 (𝑡) represents the population at time 𝑡 for each county 𝑖, 𝑟 𝑖 (𝑡) is the instantaneous population growth rate at time 𝑡 specific to each county, 𝐾 𝑖 is the carrying capacity and 𝑟 𝑖𝑜 is the initial population growth rate specific to each county. The equilibrium solution of the logistic growth model occurs when the rate of change of the population is zero, i.e.

To integrate both sides, the study applied separation of variables:

Using partial fraction decomposition, the left side becomes:

Integrating both sides, yields: Solving for 𝑟 𝑖0 , the equation becomes:

This study assumes that the initial growth rate is adjusted by taking the absolute value and scaling by a factor by 1000, such that

This adjustment ensures that the growth rate remains positive and is proximately scaled for interpretation in the model. Applying rule of logarithmic, the solution simplifies to the logistic equation in exponential form:

To determine 𝐶, the researcher substitute 𝑁 0 𝑖 (𝑡) (initial population) yielding as:

Substituting Equation (4) into Equation ( 3) and simplifying gives the study explicit logistic solution:

Finally, by multiplying the numerator and denominator by

Carrying Capacity 𝑲 𝒊

𝒕

Tsoularis and Wallace (2002) [5] illustrated that the logistic growth model parameter 𝐾 𝑖 denotes the carrying capacity, representing the maximum sustainable population for a given county. Population growth is constrained by available resources and infrastructure, where carrying capacity serves as a theoretical upper limit. However, this capacity is not necessarily fixed and may evolve over time in response to various socioeconomic and environmental factors. For example, infrastructure expansion, education systems, urban development, and improvement of health care can increase a county's capacity to support a larger population. Conversely, environmental degradation or economic stagnation can reduce the effective carrying capacity. Thus, this study is applying logistic models to long-term population forecasting (2021 to 2031). It is often necessary to account for potential changes in 𝐾 𝑖 rather than treating it as a static value (United Nations 2022) [7] .

To estimate carrying capacity 𝐾 𝑖 , the study adopts a compound growth rate 𝑔 𝑖 approach using projection data from 2008 to 2020. Specifically, the projected carrying capacity is modeled as:

Where the compound growth rate is computed as:

𝐾 𝑖 2008 ×10

This process allows the model to reflect incremental changes in carrying capacity over time, consistent with the realities of demographic and infrastructure development.

Results

Discussions

The proposed logistic growth model produced significant results, where the performance of the model accounts for time-dependent changes in growth rates and carrying capacities. The study provides a more realistic basis for forecasting population dynamics across counties. This allows policymakers to better anticipate demands for essential services, such as healthcare, education, housing, and employment. Additionally, the model offers a framework that can be adapted and updated as new demographic data becomes available, thereby supporting evidence-based decision-making and promoting sustainable development. Moreover, it encourages a shift away from static, deterministic assumptions toward more dynamic modeling practices in demographic research in Liberia.

Conclusion

Existing practices of demographic forecasting in Liberia have been enhanced by applying a logistic growth model in this study. Unlike conventional deterministic approaches that rely on fixed growth rates, the proposed model accounts for the dynamic nature of population growth by incorporating both time-dependent growth rates and flexible carrying capacities. The analysis demonstrates that population growth in Liberia is not uniform and that growth rates decline over time as counties approach their respective carrying capacities because of the logistic growth behaviour. This finding not only aligns with biological and environmental growth principles but also mirrors the infrastructural and economic constraints present in many counties.

Along with supporting growth principles, these projections offer valuable insights for policymakers, suggesting where population pressures may intensify and where strategic investments in infrastructure, health, and education are most needed. Ultimately, the study reinforces the effectiveness of logistic models in long-term population forecasting while advocating for their inclusion in Liberia's planning and national statistical frameworks to improve Liberia's development initiatives.

References

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