Proposed Model Change

Background

The initial paper dismissed the significance of migration effects on population models, deeming them inconsequential. While we acknowledge this assertion, we contend that when net migration holds substantial value, the model's failure to encapsulate its impact becomes a concern. Consequently, we propose the inclusion of migration effects to see how that affects the outcomes of the graphs.

This model revision involves modifying both the logistic and exponential models to account for immigration and emigration.

For the exponential model, the revised equation becomes:

dN/dt = aN + I(t) - E(t)

Similarly, the logistic model is adjusted to:

dN/dt = aN(1 - N(t)/K) + I(t) - E(t)

In both cases, I(t) represents the number of immigrants at any time t, and E(t) represents the number of emigrants.

Immigration

Immigration is assumed to depend on the natural propensity of people to migrate into Tanzania, the current population N(t), and a function influenced by economic conditions:

dI/dt = α * N(t) * F(economic_conditions)

In this function, α is the propensity (natural tendency) of people to migrate into the country, N(t) is the population of the country at any time t, and F(economic_conditions) is a function that simplifies the economic condition to the global unemployment rate divided by the number of countries in the world (195). Since it was not easy to get literature values on the value of α, we assumed that α is 0.1%. This simplification is made to make people more inclined to move into Tanzania, rather than using a complex model that considers the unemployment rate in the country.

Emigration

Emigration (E(t)) is modeled based on the propensity of people to move out of Tanzania, the current population N(t), and a function dependent on political stability:

dE/dt = β * N(t) * G(political_stability)

Here, β represents the overall propensity of people to move outside of Tanzania also assumed at 0.05%, N(t) is the population of the country at any time t, and G is a function that depends on political stability. It is assumed that political stability is solely characterized by the stability index. Finally, a constant unemployment rate and political stability index are assumed for modeling.

Evidence

According to the World Population Review (2023), the global unemployment rate is 6.3-6.5%. The worst-case assumption is made, considering the unemployment rate as 6.5%. The stability index for Tanzania ranges from -0.86 in 2003 to 0.09 in 2009, with an average of -0.39 points during the period. The latest value from 2021 is -0.44 points. The mean between the minimum and maximum values is calculated as the stability index: (0.09+−0.86)/2=−0.385

Using Assumptions:

The functions to calculate the influence of economic conditions on immigration and political stability on emigration are defined as follows:

% Function to calculate the influence of economic conditions on immigration

function F_economic_conditions = economicInfluence(unemploymentRate)

F_economic_conditions = exp(-0.25 * unemploymentRate);

end

% Function to calculate the influence of political stability on emigration

function G_political_stability = politicalInfluence(stabilityIndex)

G_political_stability = 1 - exp(-stabilityIndex / 10);

end

Having established the model parameters and assumptions, we proceed to compare the predictions of our modified exponential and logistic models to the actual census data for Tanzania. The modified models, incorporating migration effects, are contrasted against their respective original forms, which excluded migration. This comparison allows us to assess the impact of incorporating migration into the models and evaluate their predictive accuracy.

Outcome After Proposed Model Change

Figure showing the comparison of the logistic model, modified logistic model, and actual census data.

Despite utilizing simplified assumptions due to time limitations, the figure reveals a compelling indication that migration data could exert a substantial impact on population growth in the future. This observation is supported by the closer alignment between the modified logistic model and the actual data compared to the original logistic model, particularly in the initial phase of the graph. This suggests that the modified model, incorporating migration effects, provides a more accurate representation of population dynamics based on our assumptions and reasoning.

The modified logistic model's improved predictive performance can be attributed to its consideration of migration patterns, which play a crucial role in shaping population trajectories. While the impact of migration may not be immediately evident in the initial stages of the graph, its influence is likely to become more pronounced in the long run. This highlights the importance of incorporating migration effects into population modeling exercises to enhance their predictive accuracy and provide a more comprehensive understanding of population trends.

The figure also indicates the potential consequences of neglecting migration effects in population modeling. The original logistic model, which excludes migration, exhibits a divergence from the actual data, particularly in the middle portion of the graph. This discrepancy suggests that failing to account for migration can lead to inaccurate population predictions, potentially hindering effective policymaking and resource allocation.

Figure showing the comparison of the exponential model, modified exponential model, and actual census data.

Similar to the logistic model comparison, the analysis of the exponential model reveals a substantial difference between the original and modified versions, particularly in the context of our assumptions. The graphs illustrate that, for a significant portion of the time period, the modified exponential model provides a more accurate representation of the population dynamics compared to the original model.

This enhanced performance of the modified exponential model can be attributed to its incorporation of migration effects. As migration plays a crucial role in shaping population trajectories, accounting for these effects leads to a more comprehensive and accurate representation of the actual population dynamics.

Code for proposed model change

FURTHER RESEARCH

While our study provides valuable insights into the potential impact of migration on population dynamics in Tanzania, it is important to acknowledge the limitations inherent in our approach. The utilization of estimated values based on assumptions, particularly for parameters such as propensity to migrate and political stability, introduces a degree of uncertainty into our analysis. To fully validate the proposed model modifications and enhance the accuracy of our predictions, further research is essential to quantify these parameters more precisely.

Additionally, certain values were assumed for simplification purposes, such as the constant unemployment rate and political stability index. While these assumptions were necessary to facilitate the initial exploration of the model, future research should investigate methods for dynamically incorporating these variables to better capture their evolving nature and enhance the model's representation of population dynamics.

To address these limitations and further refine the model, we propose the following research directions:

Data Collection and Analysis

Conduct in-depth surveys and gather comprehensive data to accurately quantify the parameters related to migration, such as propensity to migrate, immigration rates, and emigration rates.

Model Validation and Refinement

Conduct rigorous validation exercises, comparing the model's predictions against real-world population data, to refine the model and enhance its accuracy.

Dynamic Parameter Modeling

Develop more sophisticated methods to dynamically incorporate variables such as unemployment rate and political stability index, reflecting their changing trends and their impact on population dynamics.

Collaborative Research Efforts

Foster collaboration among researchers with diverse expertise to conduct comprehensive studies on population dynamics in Tanzania, incorporating migration effects and addressing the identified limitations.

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