Abstract
Malaria remains a serious and potentially fatal vector-borne disease, consistently ranking among the world’s deadliest infections. This study presents a systematic review of age-structured malaria transmission models. Articles were sourced from PubMed, Google Scholar, and the Research Gate Library, resulting in the identification and inclusion of eleven papers in the review. The findings highlight that children under the age of five are more susceptible to malaria than adults, due to their still-developing immune systems. The highest rates of morbidity and mortality are seen in youngsters, pregnant women, and people with impaired immune systems, making age structure a critical factor in the spread of malaria within populations. Personal protection and vector control are key strategies in reducing the transmission of malaria in communities. The study also suggests that the use of fractional operators in modeling could offer new insights into the dynamics of malaria transmission and potential control strategies.
1 Introduction
Malaria remains one of the world’s deadliest vector-borne diseases, posing a serious and potentially fatal threat, particularly in tropical and subtropical regions (1). This infectious disease is a significant global cause of morbidity and mortality, often associated with poverty and contributing to the inhibition of economic growth in affected areas (2). Malaria is especially devastating in endemic regions, where it disproportionately impacts children under five, pregnant women, and non-immune adults. According to the World Health Organization (WHO) (3), there were an estimated 249 million cases of malaria and 608,000 deaths worldwide. Children under the age of five accounted for 76% of all malaria-related deaths. Furthermore, 94% of cases and 95% of deaths occurred in the African region, with children under five representing 78% of the fatalities in this region (3).
Malaria is caused by Plasmodium parasites, a group of protozoa transmitted primarily through the bites of infected female Anopheles mosquitoes (4). Five species of Plasmodium infect humans, with Plasmodium falciparum being the most prevalent and lethal, particularly in Africa and Southeast Asia (5–7). Symptoms of malaria typically appear within weeks of infection and include fever, chills, sweating, vomiting, abdominal pain, rapid heartbeat, and headaches. Some species of the parasite can remain dormant in the liver, causing relapses months or even years after the initial infection (8). Early diagnosis and treatment are critical for survival, and malaria control strategies, such as insecticide-treated bed nets, anti-malarial medications, and indoor residual spraying, have proven effective in reducing transmission (9–11). While newborns receive temporary immunity from their mothers (12), a multifaceted approach is essential to control the spread of malaria and work toward the ultimate goal of eradicating this preventable disease.
Climatic variability, particularly temperature and rainfall, significantly influences malaria transmission dynamics. Temperature affects both mosquito and parasite behavior (13–17). Higher temperatures lead to more frequent mosquito feeding due to quicker blood digestion (13), but also shorten mosquito lifespan (14). Furthermore, the maturation period of malaria parasites within the mosquito decreases with rising temperatures, from nineteen days at 22°C to eight days at 30°C (16). Temperature fluctuations can also affect malaria transmission rates, either lowering or speeding them up (17). Rainfall plays a crucial role in mosquito breeding and survival (18, 19). While warm, moist conditions in the tropics support stable transmission, excessive rainfall can negatively impact mosquito breeding by flushing away breeding sites (19). Conversely, moderate rainfall can provide suitable breeding habitats for mosquitoes, increasing larval populations (7). These factors demonstrate the complex interplay between climate and malaria transmission, highlighting the need to consider climate variability in public health strategies.
Malaria is preventable and curable, though no single prevention method exists. Various control strategies, including insecticide treated nets (ITNs), treatment of infected individuals, and adulticide, can reduce transmission. Optimal control theory has been applied to identify effective combinations of these strategies, with studies demonstrating the potential for disease eradication using four control strategies such as LLINs, treatment of symptomatic and asymptomatic infections, and IRS (20). A study (21) the cost-effectiveness of optimal control strategies for malaria, considering partial immunity and protected travelers, and incorporating ITNs, prophylaxis, treatment, and vector control.
Vaccination is an effective way of preventing and controlling the prevalence of infectious diseases (22). Recent breakthroughs in malaria vaccines include the WHO’s October 2021 recommendation for broad use of RTS and S/AS01 (the first malaria vaccine) in children from moderate to high malaria transmission. In October 2023, the R21/Matrix-M (R21) malaria vaccine became the second vaccine recommended by WHO to prevent malaria in children living in areas of risk (3).
Recently, mathematical models have become increasingly important for understanding infectious disease dynamics (23–25) and remain a powerful tool for controlling infectious diseases like malaria (26–32). Deterministic and stochastic models are used to study transmission mechanisms, design control strategies, and forecast outbreaks (26–32). Early models by Ross and Macdonald highlighted the importance of mosquito control, with Ross suggesting eradication is possible by reducing mosquito populations below a certain threshold (29). Macdonald’s model, incorporating an exposed class in the mosquito population, showed that reducing mosquito numbers has limited impact in areas with intense transmission (28). These models introduced the concept of the basic reproduction number which represents the average number of new infections caused by one infected individual (26).
The vulnerability of children under five years to malaria, due to their lack of immunity, makes age structure a significant factor in transmission dynamics (1, 33). Recent models incorporating age-structure have provided insights into control strategies (20, 34–37). Models have highlighted the importance of targeting asymptomatic carriers and utilizing various control measures, including mosquito nets with long-lasting treatment, indoor residual spraying, and the screening and care of both symptomatic and asymptomatic people (20). Other studies have shown that increasing mosquito lifespan and birth rate can lead to higher infection and mortality rates (34). The incorporation of age-structure into models has also been applied to other infectious diseases, such as SARS (38), and has been used to explore the effectiveness of optimal control strategies (39). Further research exploring nonlinear incidence and infection age in malaria transmission has revealed complex interactions that impact the stability of disease states (40).
Fractional calculus, a generalization of classical calculus, has emerged as a valuable tool for modeling real world problems, including infectious disease dynamics (41–47). The most common definitions of fractional derivatives, such as Caputo-fractional and Riemann-Liouville, utilize power decay and derivatives as kernels (41, 45). However, the Atangana-Baleanu and Caputo-Fabrizio operators, with non-singular kernels and non-power law distributions, offer superior modeling capabilities for complex systems, including infectious disease models (42, 43, 46). The Atangana-Baleanu derivative, in particular, is well-suited for modeling real world problems due to its non-singular and non-local kernel properties (42).
Fractional order calculus is increasingly popular for its ability to capture memory effects and hereditary properties present in biological systems, which are not adequately represented by integer-order derivatives (41, 43, 48–50). Models incorporating fractional order derivatives have been developed for malaria transmission, considering factors such as vaccination, anti-malarial drugs, and mosquito control strategies (51–55) and the references therein. This systematic review focuses on age-structured malaria transmission models published between 2019 and 2024.
2 Methods
2.1 Study design
A document analysis review design method was employed in the study.
2.2 Search strategy
A systematic literature search was carried out using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines (56). This search was carried out by using the PubMed, Google Scholar, and Research Gate databases from January 2019 and May 30, 2024. We utilized various keywords and Boolean operators in a systematic manner, including “age-structured” OR “class-age structured,” as well as “malaria” AND “model*.” The Mendeley software was employed to categorize and remove duplicate references (see Figure 1).
Figure 1
PRISMA flow diagram.
2.3 Inclusion and exclusion criteria
The study used the following inclusion criteria to determine study eligibility:
Deterministic modeling approach.
Articles published in English.
Full text of studies available.
Latest published articles from 2019–2024.
Studies indexed by PubMed, Google Scholar, and Research Gate.
Studies meeting the following criteria were excluded from the review:
Stochastic modeling approach
Article is not published in English
The full text of the study is not available.
Articles published before 2019.
Studies not indexed in PubMed, Google Scholar, and Research Gate.
Text book or book series is not included.
3 Results
By taking into account all of the aforementioned inclusion and exclusion criteria, a database search engine on age-structured malaria models found 1,097 (one thousand ninety-seven) published articles. Following that, a total of 51 (fifty-one) duplicate research articles were eliminated out of a total of 1,097 (one thousand ninety-seven) that were searched. The titles and abstracts of 1,046 (one thousand forty-six) studies were then used to filter them. Ultimately, eleven of the 35 (thirty-five) articles that we had read through to the end were deemed suitable for this systematic review. The included studies are classified as age-structured malaria transmission models, optimal control and cost effectiveness of age-structured malaria transmission models, and fractional order of age-structured malaria transmission models, which are listed in the following Tables 1–3.
Table 1
| Year | Author | Models | Aims/objectives of the study | Finding/Conclusion |
|---|---|---|---|---|
| 2023 | Seck and Ivorra (60) |
|
Investigated the existence, stability, and implications of disease-free and endemic equilibria.
Conducted numerical simulations to visualize the impact of age structure on malaria transmission dynamics under various scenarios. |
|
| 2022 | Tchoumi et al. (66) |
|
Investigated the existence and stability of equilibria.
Conducted numerical simulations that visualize malaria transmission dynamics within the two group structure under various scenarios, |
|
| 2021 | Kalula et al. (62) |
|
Investigated the impact of asymptomatic carriers on the overall malaria transmission dynamics, emphasizing their contribution to disease persistence and spread within the population.
Conducted sensitivity analyses on key model parameters. Performed numerical simulations to visualize malaria transmission dynamics under various scenarios. |
|
| 2019 | Azu-Tungmah et al. (64) | SIS model for the human population SI model for the mosquitoes population |
Established the existence and stability of equilibria within the model.
Performed sensitivity analyses on model parameters related to infants and pregnant women. |
|
| 2019 | Guo et al. (63) |
|
Investigated the global stability of the model and established the existence and uniqueness of equilibria. Conducted numerical simulations to visualize the dynamics of malaria transmission across different age groups under different scenarios. |
|
Age-structured malaria transmission models.
Table 2
| Year | Author | Models | Aims/objectives of the study | Finding/Conclusion |
|---|---|---|---|---|
| 2023 | Kalula et al. (59) |
|
Applied optimal control strategies aimed at minimizing malaria transmission and disease burden.
Conducted numerical simulations to evaluate the dynamics of malaria transmission under different intervention scenarios and temperature conditions, aiding in understanding the potential outcomes of various control strategies. |
|
| 2022 | Tchoumi et al. (67) |
|
Investigated the existence and stability of equilibria.
Applied optimal control strategies aimed at minimizing malaria transmission and disease burden. Conducted numerical simulations to visualize and analyze the impact of different vaccination strategies and coverage levels on malaria transmission dynamics. |
|
| 2021 | Wang et al. (61) |
|
Performed a global stability analysis of the model.
Applied an optimal control problem aimed at minimizing malaria transmission and maximizing health outcomes through the strategic deployment of vaccination and management of relapse cases. Implemented numerical simulations to investigate the dynamical behavior of the model under various scenarios. |
|
| 2019 | Azu-Tungmah et al. (65) |
|
Investigated the existence and stability of disease-free and endemic equilibrium.
Performed sensitivity analyses on key model parameters. Conducted numerical simulations to visualize the impact of age structure and vulnerabilities of children and pregnant women on malaria transmission dynamics under various intervention scenarios. Assessed the cost-effectiveness of implemented control strategies. |
|
Optimal control and cost effectiveness of an age-structured malaria transmission model.
Table 3
| Year | Author | Models | Aims/objectives of the study | Finding/Conclusion |
|---|---|---|---|---|
| 2024 | Gizaw and Deressa (58) | SIS model for the human population SEI model for the mosquito population. |
Investigated the existence and stability of equilibria within the age-structured model using both classical and ABC fractional operators.
Conducted sensitivity analyses on key parameters to identify the factors most significantly affecting malaria transmission dynamics, and performing numerical simulations to visualize and compare these dynamics under various intervention scenarios using both classical and fractional modeling approaches. |
|
| 2024 | Menbiko and Deressa (57) | Model of SPIR for the human population SI model for the of mosquito population. |
Investigated the existence and stability of equilibria.
Implemented numerical simulations to visualize transmission dynamics under the influence of age structure and fractional derivatives, and compared these results with classical models. |
|
Fractional order on age-structured malaria transmission model.
4 Discussion
In order to investigate the dynamics of malaria transmission, Menbiko and Deressa (57) used a mathematical model that included age structure and made use of the Atangana Baleanu fractional derivative. The author included classical and Atangana Baleanu fractional operators in the transmission model details section. The author employed a system of ordinary differential equations along with the SPIR model for the human population and the SI model for the mosquito population. The stability of the endemic and disease-free equilibriums was examined, and the basic reproduction number was calculated. Additionally, numerical simulations are run to investigate the model’s behavior for various values of the fractional order alpha. The outcome showed that the endemic spread slows down when the value of α decreases from 1.
In order to comprehend the dynamics of malaria transmission, Gizaw and Deressa (58) investigated mathematical models using both fractional order and classical (integer order) derivatives. By adding classical and ABC fractional operators, the author expanded on “Analysis of an age structured malaria model incorporating infants and pregnant women.” The author employed a system of ordinary differential equations along with the SIS model for the human population and the SEI model for the mosquito population. The basic reproduction number was determined, and the stability of both disease free and endemic equilibrium was explored. Using numerical simulations, the behavior of the model for various values of the fractional order alpha was investigated. The findings showed that when the value α decreases from one, the spread of the endemic grows slower.
According to Kalula et al. (59), an age-structured deterministic model for malaria transmission that includes symptomatic carriers, temperature variability, and its optimal control and effectiveness was studied. The model was analyzed when temperature-dependent parameters are kept constant and a system of ordinary differential equations is used. The basic reproduction number of the model was determined, as was the stability of both disease-free and endemic equilibrium.
According to the findings of a sensitivity study, the most sensitive criteria for the basic reproduction number are the rates of mosquito bites and deaths. Furthermore, it has been noted that factors pertaining to individuals under the age of five hold greater significance than those relating to those aged five and above. The recovery rate of carriers who do not exhibit symptoms is also comparatively more significant than the recovery rate of persons who do exhibit symptoms. As a result, efforts to lower mosquito bite rates and raise mosquito mortality rates have a bigger effect on lowering the spread of disease.
Controlling the condition in children and carriers who exhibit symptoms will also have a significant impact. Numerical simulations of the optimal control problem indicate that treating symptomatics, treating asymptomatic carriers, implementing long-lasting insecticide nets, and insecticide spraying is the most effective strategy for reducing the number of infected individuals. The cost-effectiveness analysis also shows that treating symptomatic patients, identifying and treating asymptomatic carriers, and applying pesticide spraying are the most cost-effective ways to stop the spread of malaria in scenarios where resources are limited.
Seck and Ivorra (60), studied a mathematical model to understand how malaria spreads within a population, specifically considering the influence of age on disease transmission. The author employed a system of ordinary differential equations along with the LSEI model for the mosquito population and the SEIRS model for the human population. The next-generation matrix approach was used to calculate the fundamental reproduction number, and the stability of the endemic and disease-free equilibrium was examined.
Tchoumi et al. (2022), conducted a theoretical analysis and formulation of a two group malaria model structured by age, wherein individuals under five years old receive vaccine. The author employed a system of ordinary differential equations, the SVEI and SEIR models in the human population, and the SEI model in the mosquito population. The next generation matrix approach was utilized to calculate the basic reproduction number. When the disease induced death rate in both human groups is zero, the disease free equilibrium is globally asymptotically stable; otherwise, backward bifurcation may happen. The outcomes of the numerical simulation showed that the three intervention measures personal protection, treatment, and vaccination of children under five should be implemented concurrently in order to effectively combat the malaria outbreak in a community.
Tchoumi et al. (66), studied how malaria spread within a population that was divided into two distinct groups. They used the SEI(R)S model for the human population, the SEI model for the mosquito population, and a system of ordinary differential equations. The next generation matrix approach was used to calculate the basic reproduction number, and equilibrium stability was explored. On the other hand, when an endemic equilibrium coexists with a disease free equilibrium, the model may display the phenomena of backward bifurcation under specific circumstances. According to the numerical result, there is a greater sensitivity in the contact rate between mosquitoes and people under five years old than there is between mosquitoes and people over five. Lastly, increasing the effort to combat malaria in children under five has a higher effect on reducing the disease’s spread throughout the population, which may result in better progress toward the disease’s eventual mitigation and eradication.
Wang et al. (61), examined how to understand and manage malaria transmission using a mathematical model that integrates age structure, immunization, and the occurrence of relapse. The author employed a system of ordinary and partial differential equations, the SVIR model for the human population, and the SI model for the mosquito population. The basic reproduction was calculated, and the disease free equilibrium was both locally and globally asymptotically stable, otherwise unstable, because the basic reproduction number is smaller than unity. According to the numerical result, controlling the disease in combination with insecticide spraying will be more successful.
According to Kalula et al. (62), malaria spreads in a population, considering the influence of age, infected immigrants, and asymptomatic carriers. The author used the SEI(A)R model for the human population, the SEI model for the mosquito population, and a system of ordinary differential equations. The basic reproduction number was computed, and the stability of the disease-free and endemic equilibrium was discussed. Besides, the sensitivity of the basic reproduction number shows that children’s class parameters are more sensitive than those of adults. In the presence of infected immigrants, the model does not admit a disease-free equilibrium. The numerical results indicate that the asymptomatic carriers have more impacts than the infected immigrants on malaria dynamics.
Using a mathematical model, Guo et al. (63) investigated the long-term (global) effects of preventive measures on malaria transmission while taking into account the distinct vulnerabilities of different age groups. The author used the SPEIR model for the human population and the SEI model for the mosquito populations, and the model was described as a system of both ordinary and partial differential equations. The author clearly explained well-posedness and uniform persistence. The basic reproduction number was computed, and both local and global stability were discussed. Finally, the author concluded that the basic reproduction number decreased as the rate of prevention increased; that is, increasing the degree of prevention could lead to malaria becoming extinct.
Azu-Tungmah et al. (64, 65) investigated how the prevalence of malaria in humans varies with age and gender. The author did an extension of “Analysis of an age-structured malaria transmission model” or Wang et al. (36) by incorporating an infectious compartment for pregnant women and avoiding the recovered compartment from Addawe and Lope’s model. The author used the SIS model for the human population, the SI model for the mosquito population, and a system of ordinary differential equations. The basic reproduction number confirmed that the disease-free and endemic equilibrium is both locally and globally stable. Sensitivity analysis has proved that malaria can be controlled or eliminated if the following parameters are controlled: biting rates, recruitment rate, density-dependent natural mortality rate for mosquitoes, and clinical recovery rates for humans.
The most effective and efficient methods of preventing malaria transmission were investigated by Azu-Tungmah et al. (64, 65), with a particular emphasis on the special vulnerability of pregnant women and children under five. By including optimal control, the author expanded on “Analysis of an age structured malaria model incorporating children under five years and pregnant women.” For the human population, the author employed the SIS model, and for the mosquito population, the SI model. The system of ordinary differential equations was used to define the model. The next generation approach was used to determine the effective reproduction number. The impact of the controls was examined, and it was found that each of the four controls had a positive effect.
5 Conclusion
This systematic review examined the transmission dynamics of malaria in age-structured populations, revealing the crucial role of age in shaping disease spread. Children under five, due to their weaker immune responses, are significantly more susceptible to malaria than adults. The highest rates of morbidity and mortality are seen in youngsters, pregnant women, and people with impaired immune systems. The distribution of age groups plays a crucial role in the spread of malaria within communities, with different age groups contributing uniquely to transmission patterns. Moreover, effective control strategies including personal protection methods, targeted treatment regimens, and vector control interventions have demonstrated a substantial impact in reducing malaria transmission. The application of fractional operators in mathematical models offers further insights into the complexities of malaria transmission and may provide new approaches for optimizing control strategies. Overall, this review underscores the intricate relationship between age structure, demographic factors, and control measures in shaping malaria transmission dynamics. These findings provide valuable guidance for public health efforts aimed at minimizing the global malaria burden.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
DD: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. TK: Conceptualization, Methodology, Supervision, Validation, Writing – review & editing. CD: Conceptualization, Methodology, Resources, Supervision, Validation, Writing – review & editing.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Acknowledgments
The authors would like to thank the reviewers and editors for their valuable comments and suggestions which improved the quality of the paper.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Gen AI was used in the creation of this manuscript.
Publisher’s note
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Summary
Keywords
malaria, deterministic model, age-structure, control strategies, systematic review
Citation
Dinsa DW, Keno TD and Deressa CT (2024) A systematic review of age-structured malaria transmission models (2019–2024). Front. Appl. Math. Stat. 10:1512390. doi: 10.3389/fams.2024.1512390
Received
16 October 2024
Accepted
03 December 2024
Published
18 December 2024
Volume
10 - 2024
Edited by
Hua Nie, Shaanxi Normal University, China
Reviewed by
Pankaj Tiwari, University of Kalyani, India
Zhiguo Wang, Shaanxi Normal University, China
Updates
Copyright
© 2024 Dinsa, Keno and Deressa.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Dechasa Wegi Dinsa, dech2003@gmail.com
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