Removed (SIR) epidemic model [2, 8] . 25 By fitting the number of infected cases to a sigmoid curve, we implicitly 26 assume that the spread in a particular region is independent of other regions, 27 which contrasts the strong interconnectedness of our modern world. Network-28 based techniques take into account the interaction between different regions, 29 which is due to the movement of people. 30 The interaction can be described by network G with N nodes. Each node i 31 in the network G represents a particular region (country, province, municipality 32 or city), and the link a ij ∈ {0, 1} represents the existence of an interaction from 33 region j to region i, specified by a link weight β ij denoting the infection proba-34 bility from region j to region i. The self-infection probability within a region i 35 is given by β ii , which we expect to be dominant over the other infection proba-36 bilities, because the interaction within a region is stronger than the interaction 37 with other regions. The N × N infection probability matrix B, with elements 38 β ij is, however, unknown and must be derived from past observations of the 39 epidemic. We will address this issue in more detail in Section 2. 40 Throughout this work, we often use "the number of infected cases", which 41 the number of asymptomatic cases cannot be determined on a daily basis, we 48 confine ourselves to the number of reported cases in this work. 49 Many scientific disciplines investigate and forecast the spread of COVID-50 19. Statistical approaches are commonly based on Kalman filtering [10] or 51 consider Bayesian approaches [11] . Network-based approaches consider aero-52 plane networks, daily commute traffic or cell phone traffic [12] . Data scien-53 tists apply machine learning algorithms, like adaptive neuro-fuzzy inference sys-54 tem [13] or Long Short-Term Memory (LSTM) [14] . Mathematicians perform 55 parameter estimation on compartmental models like the Susceptible-Infected- 56 Removed model (SIR) [14, 15] or the Susceptible-Exposed-Infected-Removed 57 (SEIR) model [16] .
::::
Most ::::::::: epidemic ::::::: models :::::::: forecast :::: the :::::::: number :: of :::::::: infected ::::: cases ::: as :: a :::::: point 59 ::::::: forecast ::::::::: (generally: :::: the ::::: mean :: of :: a ::::::::::: distribution) :::::: rather :::: than :: a :::::::: complete ::::::::::: distribution.
:::
All :::::: models ::: in :::: this ::::: work :::: have ::::: been :::::::: designed :: to :::::::: provide ::::: point ::::::::: forecasts, ::: but :::: can 61 ::
be :::::::::: generalised ::: to ::::::: provide :::::::::: prediction :::::::: intervals. :::: We ::::::: discuss :::: this ::::: topic ::::::: further :: in 62 ::::::
Section ::: 2.
The focus of this work is the comparison of a diverse set of methods to fore-64 cast the spread of COVID-19, ranging from fitting closed-form epidemic curves 65 and comprehensive machine-learning algorithms to network-based approaches. 66 We focus on the spread of COVID first COVID-19 case was reported on day k = 1. Nearly :::::: Given :::: that :::::: nearly : all 83 governments report their epidemic data once a day, which explains our choice for 84 ::
we ::::: take a time step of 1 day :: as : a ::::::: natural :::::: choice :::: and :::::::::: investigate ::: the ::::: effect ::: of ::: the 85 :::: time :::: step ::: on ::: the :::::::::: prediction :::::::: accuracy :: in ::::::::: 
and the fraction of susceptible individuals follows as
Here, β ij [k] ≥ 0 denotes the infection probability from region j to region i at 89 time k, and δ i > 0 denotes the curing probability of region i.
:::
The ::::::: spread :: of :::::::::: COVID-19 ::::::: cannot :: be ::::::::: described ::::::: exactly ::: by ::: the :::: SIR ::::::::: equations 91 :::
(1). ::::: The ::::::::::: COVID-19 ::::::::: pandemic ::::::: evolves :: in ::::::::::: continuous ::::: time, :::::::: whereas :::: the :::: SIR 92 ::::: model :::: (1) :::::: evolves ::: in ::::::: discrete ::::: time, ::::: with : a ::::: time :::: step :: of :: 1 :::: day. ::::::::::::: Additionally, ::: the 93 ::: SIR :::::: model ::: (1) ::: is :::::: unable :: to :::::::: describe ::::::::::: phenomena ::: like :::::::: personal :::::: social :::::::::: distancing,
::::::::::
nation-wide :::::::::: lockdowns ::: and :::: the :::::::::: availability :: of :::::::::::: vaccinations. ::::: Each :: of ::::: these :::::: model 95 :::::::::::
assumptions ::::::::: introduces :::::: model :::::: errors. ::::: Prior :: to :::: the ::::::::::: introduction :: of :::::: several :::::::::: forecasting 96 ::::::::::
algorithms, ::: we :::::::: explain :::: how :::::: model ::::::: errors :::: can ::: be ::::: used ::: to :::::: obtain :::::::::: prediction 97 :::::::
intervals ::: for :::: the ::::::::: forecasted :::::::: number :: of ::::::: infected :::::: cases. : 2.1. :::::::: Potential ::::::::::::: generalisation :: to :::::::::: prediction :::::::: intervals 104 ::::::
Before :::::::::: introducing :::: the ::::::: different :::::::::: prediction :::::::: methods, ::: we :::::::::: emphasise :::: that :::: this ::::
work :::::::: focusses :: on :::::::::: short-term :::: point ::::::::: forecasts. :::: The ::::::::: long-term ::::::::: epidemic ::::::::: behaviour : is ::::: very :::::::: random, :::: and ::::::::: providing ::::::: forecast :::::::: intervals :: is :::::::: essential ::: to :::: give :: a :::::::: complete ::::::
picture :: of :::: the ::::::::: long-term :::: viral :::::: spread :::: [20] : . ::::::::: Extending :::: the ::::: point ::::::: forecast :::::::: methods ::
in ::: this ::::: work :: to :::::::::: prediction ::::::: intervals :: is ::::::: outside ::: the ::::: scope :: of :::: this ::::: work. :::::::::::: Nonetheless, ::
we :::::::: consider :: it ::::::: valuable :: to :::::::::::: conceptually :::::: discuss ::: an ::::::::: extension :: of ::: the :::: SIR :::::::: equations ::: (1)
::
to ::::: allow ::: for ::: the :::::::::::: computation :: of ::::::::: prediction ::::::::: intervals. :::: Any :::: real :::::::: epidemic ::::: does ::: not :::::
follow :::: the ::: SIR :::::: model :::: (1) ::::::: exactly. :::::::: Instead, ::: the :::::::: infection ::::: state ::::: I i [k] :::::: evolves ::::: from ::::
time :: k :: to ::::: k + 1 ::: as
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
(4)
:::::
where ::::: w i [k] ::::::: denotes :::: the ::::: model ::::: error :: of :::::: region : i :: at ::::: time :: k, ::: see :::: also ::::::::: Appendix ::: A.
:::
The ::::::::: equations :::: (4) ::: can ::: be ::::: used :: as :: a ::::: basis ::: for ::::::::: prediction :::::::: intervals ::::: with :: a :::::: Monte 106 :::::
Carlo ::::::::: approach. :::::::: Define ::: the :::::: N × 1 :::::: error :::::: vector ::: as ::::::::::::::::::::::: for ::: all :::::: times ::: k.
:::::
Then, :::::: based ::: on :::::::: equation :::: (4), ::: the :::: past :::::::::::: observations ::::::::::: I[1], ..., I[n] :::: and ::: the :::::: errors 109 :::::::::::::::
, ..., w[n − 1], :::::: point ::::::: forecast ::::::::::: algorithms ::::::: provide ::: an ::::::::: estimate :: of :::: the ::::: viral 110 :::: state :::: I[k] ::: at :::::: future ::::: times :::::: k > n. : 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f Journal Pre-proof ::::::::::::
Conceptually, :: a :::::::::: prediction ::::::: interval ::: for :::: the :::::: future ::::: viral ::::: state ::::: I i [k] :::: can ::: be 112 ::::::::
obtained :: by :::: two ::::: steps. :::::: First, :::::: obtain ::::::: random ::::::: samples ::::: from ::: the ::::::::::: distribution :: of ::: the 113 ::::: model :::::: errors ::::::::::::::: w[1], ..., w[n − 1]. :::::::: Second, :: for ::::: every ::::::: sample :: of ::: the :::::: errors ::::::::::::::: w[1], ..., w[n − 1],
::::::
obtain : a ::::: point ::::::: forecast :: of :::: the ::::: future ::::: viral ::::: states ::::: I[k]. :::: The ::::::::: prediction :::::::: intervals ::: for 115 :::
the :::::: future :::: viral ::::: state :::: I[k] :::: can :: be :::::::: obtained ::::: from :::: the :::::::: ensemble :: of ::::: point ::::::::: forecasts. The ::::::: details ::: of ::: the :::::::: outlined :::::::: method ::: for ::::::::: obtaining :::::::::: prediction :::::::: intervals :::: are 118 ::::::
beyond :::: the ::::: scope :: of :::: this :::::: paper. :::: Two :::::::::: particular ::::::::: challenges ::: are ::: the ::::::::::::: determination 119 ::
of ::: the ::::::::::: distribution :: of ::: the :::::: model ::::: errors :::: w[k] :::: and ::: the :::::::::::::: implementation :: of :: a :::::::::::::: computationally 120 ::::::: efficient :::::::: sampling :::::::: method. :
The logistic function is a well-known example of an epidemiological sigmoid curve [3, 7] . We assume the cumulative number of infected cases y i [k] in region i at time k to follow a logistic function:
where y ∞,i is the long-term fraction of infections, K i is the logistic growth rate 123 and t 0,i is the inflection point, also known as the epidemic peak. The parameters 124 y ∞,i , K i and t 0,i are estimated for each region separately using a nonlinear curve 125 fitting procedure, which is explained in Appendix F. Other sigmoid curves, like 126 the Hill function and Gompertz function, are also discussed in Appendix F. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f
which is commonly used in forecasting [32] . Furthermore, we quantify the Percentage Error (PE)
for every region i and time k to investigate over-and underestimations. We representatives of the spread of COVID-19, let alone general infectious diseases.
Rather, these regions illustrate the strengths and weaknesses of our methods. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 
As a second case study, we regard the spread of COVID-19 in the Nether- 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
We have compared the prediction accuracy of eight algorithms to forecast the 310 spread of COVID-19. We summarise the results in Table 2 . The error in Table   311 2 is obtained by averaging over all sMAPE forecast errors for forecast horizons 312 between one and six days. Fitting a sigmoid curve, like the logistic function, we mention that the sMAPE error penalises over-and underestimates of the 344 true number of infected cases equally. If policymakers : In :: a ::::::::: practical ::::::: setting,
::::
such :: as :::: the ::::::: current :::::::::: COVID-19 :::::::::: pandemic, :::::::::::: policymakers ::::: might : prefer to antici-346 pate to a worst-case scenario, then : . ::: In :::: that ::::: case, : an asymmetric error metric 347 :::: that :::::::: penalises ::::::::::::::: underestimations ::::: more ::::::::::: significantly ::::: than :::::::::::::: overestimations : may be 348 more suitable. 349  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f