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. 2021;11(1):1.
doi: 10.1186/s13362-020-00097-x. Epub 2021 Jan 5.

An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19

Karunia Putra Wijaya  1 , Naleen Ganegoda  2 , Yashika Jayathunga  1 , Thomas Götz  1 , Moritz Schäfer  1 , Peter Heidrich  1
Affiliations

An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19

Karunia Putra Wijaya et al. J Math Ind. 2021.

Abstract

This paper stresses its base contribution on a new SIR-type model including direct and fomite transmission as well as the effect of distinct household structures. The model derivation is modulated by several mechanistic processes inherent from typical airborne diseases. The notion of minimum contact radius is included in the direct transmission, facilitating the arguments on physical distancing. As fomite transmission heavily relates to former-trace of sneezes, the vector field of the system naturally contains an integral kernel with time delay indicating the contribution of undetected and non-quarantined asymptomatic cases in accumulating the historical contamination of surfaces. We then increase the complexity by including the different transmission routines within and between households. For airborne diseases, within-household interactions play a significant role in the propagation of the disease rendering countrywide effect. Two steps were taken to include the effect of household structure. The first step subdivides the entire compartments (susceptible, exposed, asymptomatic, symptomatic, recovered, death) into the household level and different infection rates for the direct transmission within and between households were distinguished. Under predefined conditions and assumptions, the governing system on household level can be raised to the community level. The second step then raises the governing system to the country level, where the final state variables estimate the total individuals from all compartments in the country. Two key attributes related to the household structure (number of local households and number of household members) effectively classify countries to be of low or high risk in terms of effective disease propagation. The basic reproductive number is calculated and its biological meaning is invoked properly. The numerical methods for solving the DIDE-system and the parameter estimation problem were mentioned. Our optimal model solutions are in quite good agreement with datasets of COVID-19 active cases and related deaths from Germany and Sri Lanka in early infection, allowing us to hypothesize several unobservable situations in the two countries. Focusing on extending minimum contact radius and reducing the intensity of individual activities, we were able to synthesize the key parameters telling what to practice.

Keywords: COVID-19; Direct transmission; Fomite transmission; Mathematical model; Parameter estimation.

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Conflict of interest statement

Competing interestsThe authors declare that they have no competing interests.

Figures

Figure 1
Figure 1
Timeline of highlighted actions in Germany (above) and Sri Lanka (below) related to the model fitting. The actions are G1 (the first “extraordinary urgency” on allotment of personal hygiene and warning against chaos), G2 (restriction on public movement), G3 (contact ban), S1 (school closure), and S2 (public curfew)
Figure 2
Figure 2
Fitting results using data for Germany. The magenta curve on the top panel shows the initial condition picked only at three points representing three days. Vertical dashed lines encode the commencements of “extraordinary urgency”, restriction on public movement, and contact ban
Figure 3
Figure 3
Fitting results using data for Sri Lanka. Vertical dashed lines encode the commencements of school closure and public curfew, respectively
Figure 4
Figure 4
Sum of the predicted active cases for the next 10d (April 12–21, 2020) in Germany under variations of r and R. The value r=3 leads to the reduction to 98.31%, R≈1/4,1/3,1/2,3/4 to 98.73%, 98.86%, 99.15%, and 99.57%, respectively
Figure 5
Figure 5
For the next 30d (April 12–May 11, 2020) in Germany, r=3 leads to the reduction to 95.59%, R≈1/4,1/3,1/2,3/4 to 96.65%, 96.99%, 97.73%, and 98.85%, respectively
Figure 6
Figure 6
Sum of the predicted active cases for the next 10d (April 12–21, 2020) in Sri Lanka under variation of r and R. The value r=4.5 leads to the reduction to 78.90%, R≈1/4,1/3,1/2,3/4 to 93%, 93.73%,95.3%, and 97.63%, respectively
Figure 7
Figure 7
For the next 30d (April 12–May 11, 2020) in Sri Lanka, r=4.5 leads to the reduction to 78.90%, R≈1/4,1/3,1/2,3/4 to 79.19%, 81.2%, 85.64%, and 92.56%, respectively
Figure 8
Figure 8
Two dummy countries of equivalent number of entire households, with different household structures and associated risk. Different localizations are bordered by thin lines; rectangles and small dots encode households and their members, respectively
Figure 9
Figure 9
Numerical solutions of (16) using Pouzet-type SSPRK3 and different M’s calculated on a computer with the specification: Mac OSX 10.14, Processor 2.2 GHz, RAM 8 GB, Matlab 2015b. CT stands for computation time. A short observation indicates that the error accumulate as time grows
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