A hidden semi-Markov model with explicit state duration for malaria symptoms in children below five years

Abstract

Malaria disease is the primary cause of death and morbidity among children below five years. Globally, children below five years account for 76\% of all malaria deaths; in Africa, which has the highest malaria disease burden worldwide, this percentage increases to 78.1\%. In Kenya, 80\% of all malaria cases are reported among children below five years. Despite detailed literature on malaria disease among children below five years, there is scarce literature on malaria symptoms modeling in children below five years and yet these symptoms are common. Therefore, this study formulated a Hidden Semi-Markov model (HSMM) for malaria symptoms in children below five years. A retrospective study design was used and data was obtained from health records of children below five years at the Vihiga County Teaching and Referral Hospital, Kenya, between the months of January to December 2023. The data comprised of 1281 observations from 300 children below five years, with 2 to 7 repeated observations per child, on fever, vomiting, cough, rigors, sweating, pallor, convulsion, prostration and diarrhoea symptoms. The model assumes that the next state depends only on the current state and that the time spent in a state follows a state-specific duration distribution, with no self-transitions. The Expectation-Maximization approach was used to estimate the model parameters. The findings revealed that 32.55\% (49\% of the children) of all observations were in the severe state, followed by 23.81\% (21\% of the children) healthy, 22.33\% (16\% of the children) moderate and 21.31\% (16\% of the children) mild. Additionally, state duration estimation showed that children tended to remain in the healthy, mild, and moderate states for an average of two days each (1.96, 1.76 and 1.80 days), while the severe state lasted for approximately three days (2.77 days). The findings, especially the short durations in earlier states, highlight how rapidly a child transitions to the severe state, where the risk of mortality is high. This underscores the need for early diagnosis and timely medical intervention to mitigate mortality risks in children. The study is limited by its use of a single hospital, retrospective records, and possible classification errors; thus, prospective multi-hospital studies and external validation are recommended.