A SIRD model applied to COVID-19 dynamics and intervention strategies during the first wave in Kenya

Abstract

The first case of COVID-19 was reported in Kenya in March 2020 and soon after non-pharmaceutical interventions (NPIs) were established to control the spread of the disease. The NPIs consisted, and continue to consist, of mitigation measures followed by a period of relaxation of some of the measures. In this paper, we use a deterministic mathematical model to analyze the dynamics of the disease, during the first wave, and relate it to the intervention measures. In the process, we develop a new method for estimating the disease parameters. Our solutions yield a basic reproduction number, R0 = 2.76, which is consistent with other solutions. The results further show that the initial mitigation reduced disease transmission by 40% while the subsequent relaxation increased transmission by 25%. We also propose a mathematical model on how interventions of known magnitudes collectively affect disease transmission rates. The modelled positivity rate curve compares well with observations. If interventions of unknown magnitudes have occurred, and data is available on the positivity rate, we use the method of planar envelopes around a curve to deduce the modelled positivity rate and the magnitudes of the interventions. Our solutions deduce mitigation and relaxation effects of 42.5% and 26%, respectively; these percentages are close to values obtained by the solution of the SIRD system. Our methods so far apply to a single wave; there is a need to investigate the possibility of extending them to handle multiple waves.

A mathematical model for multiple COVID-19 waves applied to Kenya

“…S(t), I(t), R(t) and D(t) now represent the fractions or proportions of the Susceptible, Infected, Recovered and Dead in the population, at any given time t. The governing differential equations are given as follows (see for example, [16,26]).…”

“…This period serves as the reference timeframe during which the disease spread without any intervention, and hence we refer to it as the baseline period. With the death rate, δ = 0.015, approximated from the Case Fatality Rate (CFR), minimization yielded the following values for the other 2 parameters during the baseline period [26].…”

“…al. [26] developed the model of the first wave in three distinct phases: baseline, mitigation and relaxation. We present a summary here because the methods and the findings are fundamental to developing the models of subsequent waves.…”

“…Compartmental models can also be formulated for more complex problems and the findings can serve as a guide to policymakers, as illustrated by the models for China [23], United Kingdom [5], Ukraine [24] and USA [25]. Models have also been developed that address limited issues in Kenya, for instance, [26][27][28][29].…”

“…The current paper uses the method in [26] together with entirely new concepts, as described hereunder. We noted that the solution of the SIRD system, in the absence of any subsequent interventions, has a computed infection curve which dissipates with time.…”