# Global prediction model of the COVID-19 pandemic with characteristic multiple peaks and local fluctuations | BMC Medical Research Methodology

The spread of COVID-19 is influenced synthetically by natural and human factors. In order to ensure the accuracy and scientificity of the time series prediction model, it is necessary to integrate the influence of these factors. Therefore, SEACRcircle and IRcircle The proposed model is based on the SEIR model, and the Logistics function is used to measure the infection rate under the influence of various factors. Meanwhile, the impact on human intervention was modeled using NO2 concentration outliers.

### Infectious disease model based on long time series characteristics

#### SEACRcircle model

The classical dynamic model translates the problem of changing the number of infected people into a mathematical differential equation. Among them, the SEIR and SIR models are the most classic. However, the premise for using the dynamic model is that population movements in and out are not taken into account. Moreover, the dynamic model is a positive one-way population transformation and usually does not return from omitted humans (R) to susceptible humans (S), because the model assumes that people who die or acquire antibodies will not become infected again. However, this is the opposite of COVID-19 infection, and the traditional dynamic model does not work for COVID-19 with long time series characteristics and multiple peaks.

SEACRcircle The model is proposed by improving the classical model, which has a more complete dynamic mechanism. It divides the population into vulnerable (S), exposed (E), asymptomatic (A), infected (I), detected (C), and removed (R) man. At the start of the outbreak, all but one or a few infected people who migrated were susceptible (S). When they are in effective contact with an infected case, they are called exposed humans (E), did not immediately show symptoms. After the incubation period, some exposed humans show clinical symptoms and then become symptomatic infected humans (I) and the rest are still asymptomatic but infectious, and are called asymptomatic infected humans (A). Furthermore, there are two ways for infected humans (A and I) to exit the transmission system. One way is isolation through testing and they are called detected infected humans (C). The other way is through immunization, treatment, and death and they are called excluded humans (R). Eventually, recovered people will be infected again due to the effectiveness of vaccines and virus mutations, which can achieve a closed-loop mechanism of infectious disease transmission to adapt to the COVID-19 pandemic. As shown in the following Equation. (1, 2, 3, 4, 5, 6) and Figure 2.

$${E}_t={E}_1+\sum \limits_{j=1}^{t-1}\left(\beta {S}_j-\sigma {E}_j\right)$$

(2)

$${A}_t={A}_1+\sum \limits_{j=1}^{t-1}\left(v\sigma {E}_j-\left(\theta +{\gamma}_A\right ){A}_j\right)$$

(3)

$${I}_t={I}_1+\sum \limits_{j=1}^{t-1}\left(\left(1-v \right)\sigma {E}_j-\left(\varphi +{\gamma}_I+{d}_I\right){I}_j\right)$$

(4)

$${C}_t={C}_1+\sum \limits_{j=1}^{t-1}\left(\theta {A}_j+\varphi {I}_j-\left({\gamma}_C+ {d}_C\right){C}_j\right)$$

(5)

$${R}_t={R}_1+\sum \limits_{j=1}^{t-1}\left({\gamma}_C{C}_j+{\gamma}_A{A}_j+{\gamma }_I{I}_j-\eta {R}_j\right)$$

(6)

Where is the effective transmission rate; is the degree of progression from an exposed state to an infectious state; is the recovery rate; d is the number of deaths; and is the detection rate of asymptomatic and symptomatic infected cases, respectively; is the new infectious human fraction that is asymptomatic; is the proportion of recovered humans who are likely to be infected again.

#### IRcircle model

Many things are unknown in the epidemiology of infectious diseases, especially in the face of sudden outbreaks of new infectious diseases, such as the number of people exposed, the number of people infected with no symptoms, and the latent period of infectivity. The actual data reported are only the number of people confirmed, people died, and people recovered. However, the number of latent people is unknown. In this case, it is inaccurate that the dynamic model parameters are estimated using only the number of confirmed, dead, and recovered persons as validation data, which will bring some ambiguity and uncertainty to the prediction results. Okuonghae [17]Alberti [23] and Cao [24] also shows that there is great uncertainty in using the initial sample data to predict unknown parameters. Therefore, the population is simply divided into infected humans and expelled humans. The humans who were transferred contained the dead and the recovered. As shown in the following Equation. (7 and 8). Although the model is simple, it maintains the dynamic mechanism of infectious disease and the significance of the epidemiological parameters of the model parameters.

$${I}_t={I}_1+\sum \limits_{j=1}^{t-1}\left(\beta {I}_j-\gamma {I}_j\right)$$

(7)

$${R}_t={R}_1+\sum \limits_{j=1}^{t-1}\gamma {I}_j$$

(8)

### Improved dynamic model parameters under the influence of many factors

The classical dynamic model is the ideal transmission of infectious disease, and the prediction results usually show a smooth and standard normal curve. However, the spread of infectious diseases is influenced by various factors, and the infection curve is irregular and has large fluctuations, asymmetry, and several peaks. Therefore, the dynamic model needs to consider the influence of many factors.

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#### Infection rate model based on periodic logistic function

Although many factors have different effects on infectious disease, they can be associated with changes in infection rates in dynamic models. For example, human intervention is to reduce infection rates, and influenza virus infection rates show seasonal characteristics. In the classical dynamic model, considered as a constant, which can only be applied when the infectious disease is in an ideal state of spread. At the start of the outbreak, COVID-19 was free of transmission and the infection rate was relatively high. As the number of infected people continues to increase, interventions will begin; the infection rate will continue to decrease after some time. When the intensity of intervention decreases, COVID-19 can spread again and infection rates will continue to rise again. This process is very similar to the Logistics growth function in mathematics, as shown in Fig. 3, and its form is shown in Eq. 9 below. Parameters of the Logistics function were estimated by using genetic arithmetic for the solution of the estimated parameters from the appropriate epidemic data.

$$\beta =\left\{\begin{array}{l}\kern0.5em {p}_1+\frac{p_2}{1+\exp \left(1+{p}_3\ast \left({ p}_4-t\right)\right)},\kern0.5em \mathrm{declining}\ \mathrm{stage}\\ {}\kern0.5em {p}_1+\frac{p_5}{1+\exp \left(1+{p}_6\ast \left(t-{p}_7\right)\right)},\kern0.5em \mathrm{rising}\ \mathrm{stage}\end{array}\right .$$

(9)

Where t is day. During a period of decreasing infection rates, p1 + p2 is the initial infection rate; p1 is the eventual infection rate after human prevention and control; p3 is the hysteresis of the human intervention, and the larger the value indicates that the intensity of the human intervention is high and the infection rate is rapidly decreasing. On the other hand, the smaller the value, the lower the intensity of human intervention and the lower the infection rate. During periods of increasing infection rates, p1 + p5 is the eventual infection rate after loosening prevention and control; p6 is the relaxation hysteresis of the human intervention, and the larger the value indicates that the sooner the intervention is relaxed, the faster the infection rate will increase. Instead, it suggests that human intervention is being relaxed slowly and infection rates are increasing slowly. p4 and p6 is the inflection point of changing infection rates. When the virus mutates, p1 will not be the same in both periods because they do not have the same viral properties, and the ascending period curve changes like the dark purple dotted line in Figure 3.

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#### NO .-based non-pharmaceutical intervention model2 concentration

Dynamic models of infectious diseases are usually smooth curves. However, the actual curve with large fluctuations may be due to the inaccuracy of human detection on the one hand, and human intervention on the other. Researcher [25,26,27] NO reduction. observed all over the world2 concentration due to lockdowns and associated reduced human activities, especially reduced industrial and vehicle use. Besides, there is also a lot of research [28, 29] showed that a strong correlation between changes in NO2 concentration and COVID-19. NO2 concentration, due to exhaust gases from vehicle emissions and industrial production, can indirectly reflect human intervention to limit work, travel and human activities [8, 22]. The impact of human intervention is mainly reflected in the reduction in the number of people infected. Therefore, the parameter was introduced to express the reduction proportion and added to the infectious disease dynamics model. Specific upgrades are divided into SEAICR peningkatan upgradescircle model and IRcircle model, as in the following equation. (11, 12, 13 and 14). Parameters linearized by the difference between NO2 concentration and concentration without human intervention, as in Eq. 10.

$$\varepsilon ={\varepsilon}_0+{\varepsilon}_1\left(\overline{C}-C\right)$$

(10)

• SEACRcircle Model

$${S}_t={S}_1+\sum \limits_{j=1}^{t-1}\left(\eta {R}_j+\varepsilon {I}_j+\varepsilon {A}_j-\beta {S}_j\right)$$

(11)

$${A}_t={A}_1+\sum \limits_{j=1}^{t-1}\left(v \sigma {E}_j-\left(\theta +{\gamma}_A\right ){A}_j-\varepsilon {A}_j\right)$$

(12)

$${I}_t={I}_1+\sum \limits_{j=1}^{t-1}\left(\left(1-v \right)\sigma {E}_j-\left(\varphi +{\gamma}_I+{d}_I\right){I}_j-\varepsilon {I}_j\right)$$

(13)

• IRcircle Model

$${I}_t={I}_1+\sum \limits_{j=1}^{t-1}\left(\beta {I}_j-\gamma {I}_j-\varepsilon {I}_j\ right)$$

(14)

where is the moderating parameter of the epidemic curve, which mainly corresponds to uninfected persons protected due to human prevention and control. $$\overline{C}$$ is the average NO2concentration without human intervention, andC is daily NO2concentration in g/m3.