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UDC 519.1, 51-76 Вестник СП6ГУ. Прикладная математика. Информатика... 2022. Т. 18. Вып. 1 MSC 92D30, 94С15
Quality criteria for control of epidemic process
S. E. Miheev1, V. S. Mikheev2
1 St Petersburg State University, 7-9, Universitetskaya nab., St Petersburg, 199034, Russian Federation
2 Kansas State University, 1228, av. Martin Luther King, Manhattan, KS 66506, USA
For citation: Miheev S. E., Mikheev V. S. Quality criteria for control of epidemic process. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2022, vol. 18, iss. 1, pp. 149-162. https://doi.org/10.21638/11701/spbul0.2022.112
The total number of those infected at the end of an epidemic and the maximum number of infected during an epidemic are considered as two quality criteria for control by delayed isolation of the SIR- and SIRS-type infections. The temporal Barabasi — Albert graph is used to model the contacts between individuals. Simulations are run to estimate optimal delays.
Keywords: delayed isolation, SIR, SIRS, control of epidemics, temporal Barabasi — Albert graph, total number of infected, maximum number of infected, temporal network.
1. Introduction. How can we control the epidemics of type Susceptible ^Infected ^ Recovered (SIR) or Susceptible ^Infected ^Recovered ^ Susceptible (SIRS)?
In March 2020, many nations around the world reacted to the spread of COVID-19 through policies of social isolation to help to stop the pandemic and minimize the damage. Isolation typically involves voluntary and non-voluntary restrictions on social interactions. Simply speaking, in-person contacts are limited to the closest circles. Notice that in papers published before 2020, "isolation" usually applied only to the infected individuals. These measures have obvious economic repercussions. So, isolation has its own price. This economic criterion will certainly be studied extensively in the future. In this work, however, we analyze the effect of the isolation on the spread of the disease. There have already been some examples of this kind of study.
Brownian Motion. One good animation of the Brownian motion approach can be found in fl] and the theoretical support can found in [2].
Differential Equations. The classic differential equations approach is explained well in [3]. The problem of epidemic control using "prevention" (similar to the notion "isolation" in this paper) in percolation model with SIR differential equations considered in [4-7].
The model SIR on differential equations is very popular and as a result many researchers associate the name SIR with it. Nevertheless, here by SIR and SIRS we understand no more than just the concept of transitioning between the states S, I, R.
Both Brownian motion and differential equations are ongoing directions in research. However, they don't fit well to simulate the isolation of all population. So, we will focus on other models.
Contact Networks. We apply the network (graph) approach. The agent-based approach for SIS (Susceptible ^Infected ^ Susceptible) and SIR diseases was developed in [8, 9].
© St Petersburg State University, 2022 https://doi.org/10.21638/11701/spbul0.2022.112
So far we don't see papers considering isolation of all population as method to control the epidemic on a network. However, there are some good works on a problem of epidemic control with isolation of infected people. In [10], authors consider SIS on Erdos — Reni and Barabasi — Albert graphs. Isolation of infected on multilayer network with SIR is considered in [11]. SIS model (no immunity after illness) and multilayer network aren't applicable to COVID-19 and influenza.
We consider SIR and SIRS process on a temporal graph. This graph is built according to Barabasi — Albert preferential attachment mechanism [12], which is known to be one of the best to reflect social connections. Then we modify it by making the edges temporal, meaning they available only at a specific times. The theory of temporal graphs can be found in [13]. Then we run this SIR simulation with 2, 3 and 4 week restriction on communication with "popular" nodes (ones with the degree equal or greater than 5), starting with different delays from the beginning of the epidemic process. The result is presented as plots of the difference of infected and recovered. The Section 5, repeats the experiment for SIRS model. The Python computer code can be found in an open access at [14].
Disclaimer. Of course, our work is inspired by unprecedented event of COVID-19 outbreak. However, this paper should not be seen as a direct recommendation to control this epidemic or any other specific epidemic. We make a lot of assumptions about parameters. We try to make it close to the reality, but the data has been changing every week. Instead, we suggest seeing our work as one small step into the direction of using agent-based model on temporal networks for the mentioned above problem.
2. Epidemic model and parameters. We introduce the epidemic model, which consists of three parts: population model, epidemic process and isolation. Also, we explain the choice of parameters.
2.1. Population model. A population is presented as a graph, where nodes are the individuals and edges correspond to the contacts between them of such a type that a considered disease can be transferred. Barabasi — Albert structure of this graph is one of the assumptions of this paper. In order to keep the narrative simple, we don't consider other types of graphs here. However, our code in [14] can be easily changed to Erdos — Reni or any other known type of graphs, and everyone is welcome to do it.
Fig. 1. Histogram of the distribution of node degrees in Barabasi graph, n = 100, m = 3
Barabasi — Albert graph of size n is built by a consecutive augmenting process [12]. After initiation of the graph G with m0 nodes and no edges, one connects a new node x to G with m < m0 edges in such a way that the probability that x is connected to a e G is equal to the degree of a. So, "popular" nodes grow faster (Fig. 1). In this work
m = m0 = 3.
G n G
graph by assigning the times of availability to the edges (see an example in Figures 2 and 3). Edges connect the nodes only in the assigned days of availability.
[2] [2,4]
®— [i] —(»)— [3] —(7)
Fig. 2. Example of a temporal graph
Fig. 3. Corresponding time sequence of static graphs for the
For instance, nodes 5 and a are connected by the edge (s, a) at the time 1, the nodes b and f are connected by the edge (b, f) at times 2 and 4. We assume time to be discrete, measured in days by natural numbers.
If two nodes x,y e G are connected by an edge with the assigned day of availability t, we say x and y are temporal neighbors at t. The number of temporal neighbors of a node x e G at t is called temporal degree of x at t. In Subsection 2.3 (Isolation) we will describe how we assign availability moments to the edges of the graph.
n
can simulate an epidemic with the SIR process (Fig. 4). If an infectious node at a day t has an available edge to a susceptible node a, then a gets infected with probability p.
a
Fig. 4- Transition between the states of a node in SIR process
All recovered assumed to be immune to secondary infection during the considered G
infectious if and only if they have the status "infected". It may be asked why we don't have a status "dead". Despite how cruel it sounds, for the spread of the disease, dead and recovered play the same role because they can't get infected no more. After G is initiated with i infected nodes at the day zero (Figures 5, 6), every day each infectious node infects with denoted by p probability each of its temporal neighbors with the status "susceptible", meaning changing the temporal neighbor's status to "infected". The duration of status
"infected" is 13 days, i.e. average duration of the infectious conditions [15, 16] after which
p
p
should be equal to the reproductive number of the infection, i. e. how many new people on average get infected from one infectious. The reproductive number of COVID-19 is an open question. Different statistical methods give quite different results [17]. Here we assume that COVID-19 has reproductive number 3.6. The average temporal degree of 5000-node Barabasi-Graph is about 5.99. So, we set p = 0.046 « 3.6/(5.99 • 13). Again, these parameters should not be seen as a statement. They are just assumptions. Our goal
is to present the approach rather than an actual solution. Everyone is welcome to change p
Fig. 1. Infected 15-node graph pig. 8. Barabasi 100-node graph
after 15 days (blue nodes after 15 days (blue nodes
represent recovered or dead) represent recovered or dead)
Since the random number generator is involved, the infection process may go differently even on the same graph and same initially infected nodes. We will address this issue in Section 3.
2.3. Isolation. One inconvenience of classic epidemic models based on differential equations or Brownian motion is that it is hard to conduct experiments with isolation. Temporal graphs allow us to do it relatively easy. First, for simplicity, we give one availability time moment per day to each edge of G during [ti ,tT], meaning that an individual maintains each of their contact daily. (In the future works, someone can try to model more complex distributions of schedules on the graph, for example, make some edges available on weekdays and others only on the weekends.) Then we enforce "No gathering 5 or more people" rule from day ts to day tf as following: every node x e G in the time interval [ts,tf ] C [t1, tT] disconnects from its neighbor which has temporal degree 5 or more. Since
G
lar" nodes stop being "popular" and keep just few connections. The time of the beginning of isolation ts we call isolation delay. Why do we choose "No gathering 5 or more people"? State of Kansas, USA, issued "Stay at Home" order [18] prohibiting mass gathering more than 10 people. However, since all the places people could gather were closed, in fact, no gatherings were happening at all. So, we assume that average number of contacts per day-was less than 5 for one person.
Someone could say that isolating "popular" nodes isn't the same that "no gathering". Well, we think it is equivalent because gathering in a group of k people at day t means that the temporal degree of each person in this group greater or equal k. You may see
k
k
gathering", they lower their temporal degree and the temporal degree of the professor.
3. Quality criteria. Epidemics bring complex damage to society. Individuals going through the illness work less and, in severe cases, may die. So, it makes sense to consider measures against the infection and study the effectiveness of them.
Let Q be the space of events, causing randomness of epidemic processes.
Let t, ts, l, w), further referred as hectic function, be the number of currently infected at a day t of the epidemic with the delay of isolation ts, the length of isolation l and random event w e Q.
As we mentioned before, the infection processes we run have random nature. Thus, to be able to make conclusions, we shall talk about expectation and variation. So, we will repeat each experiment multiple times and compute the average and standard deviation. The value E[p(t, ts,l, w)] will be estimated as f (t, ts, l), the average hectic function in the finite number of experiments.
Let F{ot(ts) be the total number of people getting sick during the epidemic, i.e. Ftot(ts) = J2i=i f (t,ts,l)/ds. Here tend is the last day of the epidemic and ds is duration of the infectious status. In all numerical experiments of this paper we adopt ds = 13.
We choose minimality of Ftot (ts) to be the first quality criterion of the actions against the epidemic.
Now let's think about lethal cases. Their number depends on the total number of infected but also depends on the available spots in hospitals. If too many ill people in severe conditions simultaneously come for medical service, there may not be enough resources to help them. Thus, another approach would be to minimize the maximum number of infected during the epidemic period (so-called "flattening"). So, the second quality criterion to be minimized is Fmax(ts) = maxt f (t,ts,l).
We consider isolation 3B cl Sin gle tool to reduce both Ftot(ts) and Fmax(ts). The isolation has two parameters to control: its duration l and the starting time of it ts. The
l
make experiments with a few fixed values of l. So, we place l to the upper indices of both quality criteria and when it is fixed we name them objective functions.
In the majority of cases, the cost of isolation does not depend on ts. This allows
^tot and Fmax
l
Problem 1. Minimize the objective function
to perform optimization of F{ot and Fmax over ts ignoring the cost of the isolation for a
^end ^end
Ftot (ts) =Y, f (t,ts,l)/ds = Y^E[<P(t,ts,l,u)]/ds.
t=l t=l
Problem 2. Minimize the objective function
Flmax(ts) = max f (t,ts,l) = maxE[y(t, ts,l, w)].
The experiments conducted in the Section 4 are meant to address the two problems above but, of course, may give some other food for thought.
4. Experiments and observations. We conducted experiments on 5000-node temporal Barabasi — Albert graph with no isolation and with the isolation of lengths l = 14,21 and 42 days with delays 7,8,..., 14 days. However, for simplicity of narrative, here we provide plots only for experiments with the delays t = 7,14 Mid t*s. The later is the one which gives a minimal value for Flax.
Each simulation is, in fact, a random process and so, we run it multiple times with
l
and ts. This applies us with estimates f of hectic function f. Thus, we obtain estimates Ft'ot(ts) and Fmax(ts) of objective functions Ft'ot (ts) and Flax(ts).
Estimates F{ot (ts) for l = 14, 21,42 (2, 3, 6 weeks) and delays ts = 7,8,..., 15, to, where to means "No isolation", are shown in Table 1. Table 2 contains estimates Fmax(ts). In Tables 1 and 2 the isolation duration £ is in weeks, its delay ts is in days.
Table 1. Estimated expectations of total numbers of infected Ftot
""""""-\I3elay 1 7 8 9 10 11 12 13 14 15 оо
2 4299 4259 4189 4131 4061 4033 3991 3972 3987 4453
3 4278 4189 4121 4019 3952 3879 3839 3831 3856 4453
6 4110 4011 3823 3653 3522 3425 3427 3518 3644 4453
Table 2. Estimated expectations of maximum numbers of infected F
""""""-\IDelay 1 7 8 9 10 11 12 13 14 15 оо
2 2428 2227 1933 1645 1540 1714 2025 2215 2431 3217
3 2286 1895 1632 1259 1481 1785 1996 2219 2442 3217
6 1605 1264 982 1270 1492 1749 2006 2212 2399 3217
Observe that the minimum of Ft'ot(is) have been obtained on a 14th day delay for 2-, 3-isolation and on 12th day delay for 6-week isolation. The minimum of imax(is) has been obtained with delay equal 11, 10 and 9 for 2-, 3- and 6-week isolation, respectively.
On Figures 9 and 10 one can see the graphics of F7,t10t(is) and Fmax(is) for ts running from 1 to 20.
Fig. 9. Values of F^Ot for isolation delays from 1 to 20
Fig. 10. Values of F^tx for isolation delays from 1 to 20
f
l h ts in the experiments brings us to the following observation. With big values of tsj the function f has a single local maximum ("first wave"). With sufficiently small tsj the second local maximum ("second wave") appears in a later time of epidemic. With the further
ts
magnitude of the first wave is getting smaller monotonically as well (Figures 11 and 12). At some point (ts < 12 for l = 14, ts < 11 for l = 21,...), the first wave exceeds the second one.
Fig. 11. Estimated hectic function on 2-week isolation with its standard deviation on the top narrow plot
One obvious conclusion from that: When the time is discretized by 1-day step, there exists a pair of delays (ts,ts + 1) such that, for tsj the second wave is greater than the first wave, and otherwise for ts + 1. We have found such pairs for different l experimentally. As anticipated it appears that the maximum magnitude of both waves is minimum for one of those delays from the pair than for any other possible delay. For the convenience of comparison, the graphs of simulations for the same length of isolation are combined in one plot. Thus, we obtained three plots in Figures 11 and 12 for l = 14, 21, 42, respectively.
On 10 000-node graph, the results are similar. For 6-week isolation, one may see the values of the objective functions in Table 3 and the estimated hectic function with its standard deviation on the Fig. 13.
Fig. 12. Top pair a: estimated hectic function for 3-week isolation with its std. div. and Bottom pair b: estimated hectic function on 6-week isolation with its std. div.
on lOK-node graph, delay in days
F 7 8 9 10 11 12 13 14 15 oo
rtot 8306 8187 7869 7488 7211 6979 6871 6958 7208 8908
F42 max 3490 3019 2126 2197 2659 3244 3698 4229 4645 6426
Fig. 13. lOK-nodes, estimated hectic function for 6-week isolation and std. div.
In lOK-population just like in the 5K-population case, 6-week isolation brings in big sensitivity of the objective function F^x f°r the isolation delay approaching the optimal
42
42
point (9 days). Its values increases by 42% (29% for 5K-population) with the delay decrease by just one day. The decrease of the delay by two days causes the 64 % (63 % for 5K-population) increase of
If one doesn't notice the scale on y-axis of Fig. 12, b and Fig. 13, they may think that these are the same pictures. Indeed, they are very similar. That shows that the results are well-scalable on lOK-population and, perhaps, on even bigger populations. Unfortunately, limited computer power did not let us to experiment with more than lOK-nodes.
5. SIRS. Several cases of people being infected second time several months after they recovered from COVID-19 were widely announced in the news. However, up to this moment, there is no study of the length of the obtained immunity for COVID-19. Here we suggest to look how the described above isolation experiments would look if we assume exponential decay of the immunity: I = 2-t/C, where t is the number of days since recovery and C, the half-life period, is the parameter as said before we don't really know yet. Then each recovered node contacting an infectious node in the graph will be infected with probability (1 — I)p instead of being forever protected like it was in SIR model. We set C to be 365 (a year) and 548 = 365 + 183 (a year and half). That means the immunity is 2-1 = 0.5 in a year or a year and a half, respectively.
With the described above modification, we have obtained the results, which are shown on Figures 14-19.
Fig. 14. 5K-nodes, estimated hectic function on 2-week isolation, half-life period C = 365
The first conclusion from conducted numerical experiments about the epidemic with
100%
100%
we will see that they coincide totally until the appearance of the first recovered. And then,
C
end of the last wave (if second wave doesn't exist, the first one is the last).
Thus, the beginnings of graphics in Fig. 19, in fact, are the quadruple horizontally compressed graphics in Figures 12 and 13. Similar observation can be done about graphics of mean values of hectic functions in Figures 11 and 14, 15. The same compression transforms the graphics with the delays 7, 14 days and "No isolation" in the Fig. 12 (top) to the beginnings of the graphics with the same delays in Figures 16 and 17.
Boundedness by time of immunity gives infection a chance for a long life in a population. For the convenience of the following narrative, we need to split the presence of infection in population into 3 time intervals: a) epidemic, rapid growth and decrease of
3000 2500 2000 1500 1000 500 0
of infe cted and heal id
i
I А i * * Ti*:
г * ¡ i : - — No isi olation
II уц Ь * rrom / aay - from 11th day + from 14th day
Т * т * 1 \ i
ч —
400 Days
Fig. 15. 5K-nodes, estimated hectic function on 2-week isolation, half-life period C = 548
200 о
Diff.c
3000 2500 2000 1500 1000 500
j infec £ teds ind healei
i
Л
t • + i * ¡ + 1 J. * - — No is olation
+ i * i/W * * from 7 day - from 10 th day + from 14th day
Jr \ —
50
100
150
200
250
300
350
400 Days
Fig. 16. 5K-nodes, estimated hectic function on 3-week isolation, half-life period C = 365
Std. div.
of inf< cted and heal ed
i
ji ai ñ
f + * 1 * i: ! +• *
----- No isolation * from 7 th day - from 10th day + from 14th day
+ J* * < 11* ÍS * *
Jr \ —
0 50 100 150 200 250 300 350 400
Days
Fig. 17. 5K-nodes, estimated hectic function on 3-week isolation, half-life period C = 548
infected (waves); b) depression phase, the time interval with very low number of infected, follows epidemic and has about the same length; c) steady state, time interval with small waves, follows depression. Obviously, these intervals don't have strict borders. Sick indi-
Fig. 18. 5K-nodes, estimated hectic function on 6-week isolation, half-life period C = 365
Fig. 19. 5K-nodes, estimated hectic function on 6-week isolation, half-life period C = 548
viduals in epidemic are first-time-infected mostly. Most of the infected of depression phase and steady state are second-timers.
We observe that the greater the magnitude of the last epidemic wave, the higher the probability that infection disappears in depression phase.
Let's demonstrate the behavior of hectic function in depression phase and steady state on the example of "No isolation" experiment on 5 000-population with half-life period C = 365.
So, we run 30 random processes. In all of them, we observe a single wave with almost the same magnitude, which exceeds the magnitudes of the waves in the experiments with isolation. However, in the depression phase and steady state, the numbers of infected are much lower with no isolation than with any significant (affecting the epidemic) isolation. The end of epidemic lands approximately on 60th day. At 101st day, the infection disappears in 20 processes (2/3). In the processes with the still present infection, the numbers of
infected are 47, 24, 20, 16, 3, 3, 3, 2, 1, 1. Later last 7 processes nulled quickly: on
th th th th th th th th
121st day — 7th, on 129th day — 9th, on 142nd day — 4th. The remaining three processes th
approximately.
In the experiments with 2-week isolation delayed by 7 days, with half-life C = 365 the number of nulled processes at 101st day had been significantly smaller (5 from 30). Later, in depression phase lasting 106 days, this number has grown up to 23. At steady-state beginning at 207th day and observed till 1095th day, oscillations of hectic function lay between 2 and 8 %.
It appears that if we assume the half-life period to be greater than a year then there is a chance that the infection continues to oppress the population.
Then with some oscillations, the estimated hectic function of survived infections has %C than a year and a half, then, in any case, there is a day when the infection disappear from the population. Note that the last observation was obtained on relatively small populations (5000-10 000). It may not be true for larger populations.
6. Conclusion and recommendations. The experiments we conducted for SIR show that the objectives to decrease total number of infected and to decrease maximum number of infected (Problems 1 and 2 stated in Section 3) are not antagonistic to each other. Firstly, the values of both criteria improve when the delay increases starting from zero. Secondly, they get worse when the delay becomes significantly large. Thirdly, a good compromise can be found because the optimal values are achieved at closely spaced delays. For the isolation with length from 2 to 4 weeks, that compromise is the delay between 7 and 14 days.
The experiments with SIRS show that short half-life period of immunity leads to possible existence of infection in the endemic state for a long time. Then the changes of the number of infected have a wavy behavior with magnitude significantly less (around 5-8% from the population) than in the epidemic (from 20 to 70% depending on the isolation measures).
In some realizations of the random epidemic process with long enough half-life period
C
C
that the chance that the infection leaves the population is higher when the last epidemic wave has greater magnitude.
Alas, to compute the recommendation when to start an isolation at a certain day of epidemic, besides the optimal delay topt, we need to know the day of the beginning of epidemic, which is usually unknown. Fortunately, we know n, the size of population. Also, /, the amount of infected at the current day ttoday can be estimated. Thus, we can run "No isolation" simulation on n-node graph to the day t1, when the number of infected in the simulation is equal to /.Then ttoday —11 + topt is the time to start the isolation.
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Received: January 20, 2021.
Accepted: February 01, 2022.
Authors' information:
Sergey E. Miheev — Dr. Sci. in Physics and Mathematics, Associate Professor; him2@mail.ru
Vikenty S. Mikheev — PhD in Physics and Mathematics; vikentym@ksu.edu
Критерии качества для управления эпидемическими процессами
С. Е. Михеев1, В. С. Михеев2
1 Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7-9
2
KS 66506, Манхэттен, авеню Мартина Лютера Кинга, 1228
Для цитирования: Miheev S. Е., Mikheev V. S. Quality criteria for control of epidemic process // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2022. Т. 18. Вып. 1. С. 149-162. https://doi.org/10.21638/11701/spbul0.2022.112
Общее количество инфицированных до конца эпидемии и максимальное их число рассматриваются как два критерия качества для контроля эпидемиями типов SIR и SIRS посредством задержки введения изоляции. Временной граф Варабаси — Альберта используется для моделирования контактов между индивидуумами. Проведены численные эксперименты для оценки оптимальных, согласно данным критериям, задержек. Ключевые слова: задержка введения изоляции, SIR, SIRS, управление эпидемией, временной граф Варабаси — Альберта, общее число инфицированных, максимальное число инфицированных, временная сеть.
Контактная информация:
Михеев Сергей Евгеньевич — д-р физ.-мат. наук, доц.; him2@mail.ru Михеев Викентий Сергеевич — канд. физ.-мат. наук; vikentym@ksu.edu
