A mathematical prediction model of infectious diseases considering vaccine and temperature, and its prediction in Hong Kong

Abstract

Since 2020, COVID-19 has launched a rather difficult challenge in public health all over the world. For the current situation of COVID-19 outbreaks in various places, predicting the trend of COVID-19 plays a vital role in later policy-making, allocating medical materials, and developing the economy and society. Although there may be corresponding specific drugs soon, which can reduce the mortality of COVID-19 to a relatively low level, but the main popular Omicron BA.1 and BA.2 strain is highly infectious, which has a great impact on the elderly population, so COVID-19 cannot be equated with general influenza. This paper aims to predict the trend of the number of infections over time and the final number of infections from the fifth wave of epidemic in Hong Kong. To establish a more practical infection model, this paper introduces an infectious disease transmission model with the influence of temperature and vaccine. The model shows that the fifth wave of the epidemic in Hong Kong will end at the end of April 2022. At that time, the cumulative number of infections is expected to reach about 1.6 million. By formulating and implementing reasonable policies, the final number of infections can be controlled at about 1.1 million. Therefore, we hope that the policymakers and managers of COVID-19 in Hong Kong will formulate and adopt reasonable measures to control these epidemic.

Highlights

• •A mathematics model with vaccine and infection rate affected by temperature and randomness is proposed.
• •The epidemic situation in Hong Kong was assessed and predicted.
• •Reducing the contact rate and increasing the isolation rate have a significant impact on the results of the epidemic.

COVID-19; SIR model; Temperature and vaccine; Hong Kong.

1. Introduction

Since December 2019, China's Wuhan reported the first case of COVID-19 [1]. As the world's third-largest financial center, Hong Kong has experi-enced four major outbreaks. As of March 10,2022, the fifth outbreak in early 2022 had infected 617000 people, accounting for 8.3% of the permanent pop-ulation of Hong Kong. By establishing a reasonable prediction model, we can timely make a reasonable prediction of the development of the epidemic in the future. Through the prediction results, we can put forward correspond-ing management opinions to the government and reduce the impact of the virus on all aspects of society.

Scientists have been committed to developing better mathematical models to simulate the spread of infectious diseases. The SIR model is the most classical in infectious diseases and disease control. It was first proposed and studied by Kermack and McKendrick [2]. Subsequently, the SIR model and its variants, such as SIS and SEIR [3], were widely used in the direction of infectious disease prediction and control. At present, such models also play a huge role in predicting COVID-19. For example, K S. Sharov et al [4], using data from 15 European countries, created a SIR model to analyze its effectiveness. Joseph Wu and Kathy Leung [5] predicted the epidemic development of COVID-19 in Wuhan city and predicted the global spread of COVID-19 using the case number data exported by Wuhan International, and Liu et al [6] proposed an SEIR model considering the seasonal climate. We note that researchers generally make the deeper correction to the SEIR model. However, with the gradual variation of the virus, we found that the median incubation period tends to shorten gradually (initial strain: 5.1–6.4 days [7, 8], alpha: 5 days, delta: 4 days [9]). There are also relevant studies showing that the incubation period of Omicron is shorter than that of the delta [10, 11, 12]. Therefore, the number of people in the incubation period will be reduced accordingly. Therefore, this paper chooses to modify the basic model from the SIR model.

From the prediction results of an SEIR model considering the seasonal climate proposed by Liu et al⁶. it has been noted that the impact of temperature on infectivity cannot be ignored. In addition, we are considering the existing vaccination situation. In this study, a modified SIR model includ-ing realistic factors was designed to investigate the effects of temperature and vaccination on the spread of the epidemic. Considering the existing confirmed cases, the fifth wave of the epidemic situation in Hong Kong was evaluated and predicted. Finally, the impact of changes in the direction of daily vaccination number, isolation rate, and personnel mobility on the final situation of the epidemic is given, and some feasible suggestions and appeals are put forward for epidemic policymakers.

The rest of this paper has organized as follows: in Section 2, the improved SIR model is designed, and the system and algorithm are described. The modified SIR model is also compared with the traditional SIR model. In Section 3, the modified SIR model is applied to predict the epidemic situation in Hong Kong, and the parameters are estimated using the PSO algorithm. The final prediction results under different measures are given. Finally, in Section 4, some improvement directions of the model are given, and some feasible suggestions and appeals are made for epidemic decision-makers.

2. Mathematical models

2.1. SIR model

We first consider the basic form of the SIR model. It is assumed that the total population size is expressed as a constant N, without considering the heterogeneity of the population (the probability from the infected to the recovered is a constant value) and without considering the occurrence of secondary infection. Then the most basic SIR model can be expressed by the following equation [1] - [5]:

$$\frac{dS\left(t\right)}{dt}=\frac{-c\beta \left(t\right)I\left(t\right)S\left(t\right)}{N}$$ (1)

$$\frac{dI\left(t\right)}{dt}=\frac{c\beta \left(t\right)I\left(t\right)S\left(t\right)}{N}-\gamma I\left(t\right)$$ (2)

$$\frac{dR\left(t\right)}{dt}=\gamma I\left(t\right)$$ (3)

$$\beta \left(t\right)=\beta +\xi \left(t\right)$$ (4)

$$N\left(t\right)=S\left(t\right)+I\left(t\right)+R\left(t\right)$$ (5)

where S represents the number of susceptible people, I represents the number of infected people, and R represents the number of cured people; β is the probability of infection, γ represents the cure rate of infected people, ξ(t) is a Guassian white noise noise with zero-mean and unit variance, the intensity of, c represents the strength of personnel mobility. N represents the total population of the region. In epidemiology, scholars generally pay attention to the basic reproduction number R₀ of the infectious disease, while in the basic SIR model, R₀ can be expressed as $\frac{\beta }{\gamma }$.

2.2. Modified SIR model

It is worth noting that most of the current methods for predicting trends use differential equations based on SEIR and SEIRD models. But in a single city or region, especially for Hong Kong, a city with a high population density, the boundaries between different groups are not very clear. Therefore, this paper chooses to use the SIR model for improvement and prediction.

Compared with the classical model, the actual situation will be more complex. As of March 10, Hong Kong had received 14 million doses of vac-cine. In the traditional SIR model, the impact of vaccine is not taken into account, so the influencing factors of the vaccine on epidemic transmission are considered in this study; In addition, a considerable number of studies have shown that the infection rate of COVID-19 will be affected by external temperature and individual factors [13, 14, 15]. Therefore, this paper also makes some modifications to the infection rate β. Finally, the modified SIR model can be expressed by the equation [6] - [9]:

$$\frac{dS\left(t\right)}{dt}=-d\epsilon -\frac{c\beta \left(t\right)I\left(t\right)S\left(t\right)}{N}$$ (6)

$$\frac{dI\left(t\right)}{dt}=\frac{c\beta \left(t\right)I\left(t\right)S\left(t\right)}{N}-\gamma I\left(t\right)$$ (7)

$$\frac{dR\left(t\right)}{dt}=d\epsilon +\gamma I\left(t\right)$$ (8)

$$\beta \left(t\right)=m\left[1-c_1{\left(1-\exp \left(-kt\right)\right)}^{ms}\right]+\xi \left(t\right)$$ (9)

where d represents the number of vaccinations per day and ε represents the effective rate of the vaccine. In the β expression, m, c₁, k, and ms are used to control the influence of temperature on the infection rate, and ξ is the Gaussian white noise with the intensity of $\sqrt{2D}$, which can be understood as the difference of infection rate among different individuals. Unlike the classical model, γ here represents the removal rate of infected persons, including isolation rate, cure rate, and mortality rate.

2.3. The PSO algorithm and parameter estimation

Particle swarm optimization (PSO) is a well-known random parallel op-timization algorithm based on population. It simulates the behaviour of an intelligent population, such as the foraging process of insects, birds, and fish. In the PSO algorithm, each particle represents a bird. This algorithm can be understood as follows: starting from the random initialization of the particle position, for one iteration, each particle tracks its own best position and the best position of the population to update its position and speed. Considering a one-dimensional optimization problem, the velocity and position of particle i are defined as:

$$V_i\left(t+1\right)=V_i\left(t\right)+l_1{r}_1\left[P_{b,i}\left(t\right)-X_i\left(t\right)\right]+l_2{r}_2\left[P_g\left(t\right)-X_i\left(t\right)\right]$$ (10)

and

$$X_{i+1}\left(t+1\right)=X_i\left(t\right)+V_i\left(t+1\right)$$ (11)

where $X_i$ and $V_i$ are position of the particle i at the t-th iteration and velocity respectively. $l_1$ and $l_2$ are the learning factors. $r_1$ and $r_2$ are random numbers between 0 and 1. $P_{b,i}$ represents the optimal position of particle i at the t-th iteration, and $P_g\left(t\right)$ represents the optimal position of particles in the population at the t-th iteration. Then, Suppose that we meet a minimum optimization problem, the implementation steps of the PSO algorithm are summarized as follows:

Step 1

The position of each particle is randomly initialized.

Step 2

Calculate the fitness value $F\left(X_i\left(t\right)\right)$ of the particle i, and find the $P_{b,i}\left(t\right)$ and the $P_g\left(t\right)$.

Step 3

If $F\left(X_i\left(t\right)\right)\le F\left(P_{b,i}\left(t\right)\right)$ then replace the $P_{b,i}\left(t\right)$ by the $X_i\left(t\right)$.

Step 4

If $F\left(X_i\left(t\right)\right)\le F\left(P_g\left(t\right)\right)$, then replace the $P_g\left(t\right)$ by the $X_i\left(t\right)$.

Step 5

Update velocity and position of the particle $i$ according to Eqs. [10] and [11], respectively.

Step 6

Repeat the Steps 3∼5 until the termination criterion is satisfied.

The settings of PSO algorithm are set as: population of particle swarm NP = 40, learning factors $l_1=l_2=2$, maximal iteration T = 100. After that, the real data is used to import the population (N) of a city, the number of initial infections (I) and the number of cured people (R), and then the ordinary least squares is used to measure the quality of the estimated parameters.

2.4. Comparison between SIR model and modified SIR model

Firstly, we use the data of London [16] from March 1, 2020 to June 30, 2020 to compare the two models. Using the data of the first 50 days and the PSO method for parameter selection, Figure 1(a) and Figure 1(b) show the prediction results of the base SIR model and the modified SIR model for the epidemic. It can be clearly noted that the fitting effect of the modified SIR model is much better than that of the basic SIR model. This shows that the effect of introduction temperature on the infection rate is very effective. In addition, it is worth noting that in the revised SIR model, there is still a certain gap between the late prediction effect, because the late London has suffered a new wave of COVID-19, which leads to errors in the latter prediction.

Figure 1.

Simulation of epidemic situation in London by two models. Shaded areas are 95% confidence intervals.

Finally, using the infection data from February 21 to March 12 from the Hong Kong Department of health, using the modified SIR model and PSO algorithm to fit the parameters, this paper gives the epidemic prediction under the current management plan and the impact of Changing Epidemic management measures on the final direction of the epidemic.

3. Result

Figure 2 shows the predicted turning point of the epidemic in Hong Kong, the end date of the fifth wave of the epidemic, and the final number of infections based on the existing management methods. Fig.2(a) and (b) use a noise intensity of D = 0.03 and Fig.2(c) and (d) use a noise intensity of D = 0.05. In addition the number of new cases per day and the overall increase in the number of people are plotted respectively. It can be noted that according to the existing management methods, under different noise intensities, the final number of confirmed cases in Hong Kong will peak between April 20 and 30, and the peak number of infected people may reach 1.6 million. It is worth noting that the current population of Hong Kong is 7.4 million. In a sense, the fifth wave of the epidemic may cause 22% of people in Hong Kong to be infected. At the same time, according to the average mortality rate of 0.3% of COVID-19, this is unacceptable.

Figure 2.

Prediction results of the revised model for Hong Kong cases at noise intensities of 0.03 and 0.05. The shaded area is the 95% confidence interval.

In Figure 3, we increase the $\gamma$ of the existing removal rate by 10%–30%, respectively, and the number of final infections will decrease by 0.2–0.6 million. This shows that more strict isolation measures can be taken under the existing management for the confirmed positive patients so that the final infection situation will also be significantly improved.

Figure 3.

Prediction results of the modified model for Hong Kong cases at different removal rates. The first level of management is the existing management model, and the second, third, and fourth level of management are the existing management model with a 10%, 20%, and 30% increase in removal rate $\gamma$ in that order. Shaded areas are 95% confidence intervals.

At the same time, as shown in Figure 4(a), by adjusting the number of vaccine injections, it can be found that the peak infection and the number of final infections will decrease to a certain extent. In addition, as shown in Figure 4(b), we also consider the number of final infections that can be achieved by vaccination with a more effective vaccine. Both illustrate the effectiveness of vaccination on the final infection result. Although adjusting the number of vaccinations and choosing a more effective vaccine does not have the same significant impact on the final results as the removal rate, but many studies have shown that vaccination will significantly reduce the severe disease rate and mortality [17, 18], which can also significantly reduce the final death toll.

Figure 4.

Adjust the number of vaccinations per day and the impact of vaccine effectiveness on the final results. (a)Change only the number of vaccinations per day. (b)Change the number of vaccinations and vaccine efficiency every day. Shaded areas are 95% confidence intervals.

In Figure 5, we also consider the impact of changing the personnel contact rate on the final number of infected people and make the prediction results of the decrease of the contact rate by 10%–30%. The prediction results show that reducing the exposure rate also plays an important role in reducing the number of infections. Of course, when only considering the epidemic factors, the forced isolation of all people can achieve the best prevention effect, but many studies show that the forced isolation of all people is not the best [19]. Therefore, it requires policymakers to formulate corresponding management methods according to the actual situation.

Figure 5.

Prediction results of the modified model for Hong Kong cases at different exposure rates. The first level of management is the existing management model, and the second, third, and fourth level of management are the existing management model with a 10%, 20%, and 30% increase in exposure rates c in that order. Shaded areas are 95% confidence intervals.

4. Discussion and conclusion

Due to the alarming fifth wave of COVID-19 in Hongkong in January 2022, the main infectious strain is Omicron, which is highly contagious. It is noteworthy that on March 10, the number of new cases in a single day in Hong Kong reached 31400, and there are about 38.5 deaths per million people in a single day, the highest in the world. Therefore, in view of this situation, this study adopts short-term infection data and uses the modified SIR model for modeling. It is intended to use fewer data to evaluate and predict the epidemic situation in Hong Kong. It hopes that the prediction results will provide effective data support for policymakers to determine appropriate policies and strategies.

Our research shows that the spread of COVID-19 in Hong Kong, a city with a large population density, is extremely terrible. For the existing large-scale outbreak, We hope that the Hong Kong government will take a variety of measures to curb the spread of the epidemic (including but not limited to raising public awareness of maintaining social distance and home isolation, setting up hospitals for COVID-19 patients, arranging designated isolation places and isolating positive people, blocking some areas, reducing large-scale gathering activities, and calling on the public to vaccinate). The model's prediction shows that these measures can effectively reduce the final number of infections.

So far, the number of people infected in Hong Kong has reached 1.2 million, and the number of new people per day has dropped to less than 500, which shows that the fifth wave of the epidemic has basically passed. However, because the Hong Kong government did not always implement the initial isolation policy in the later stage, and did not detect all infected persons as much as possible, there may be a large error between the later reported data and the real situation.

In addition, through the simulation of the model in London from March to July 2020, it can be found that the impact of the temperature considered in this paper on the infection rate is very necessary. At the same time, due to the gradual increase of the vaccination rate of the vaccine all over the world, the impact of the vaccine on the epidemic situation can not be ignored. Of course, these two influencing factors can also be applied to the transmission of other infectious disease models with a wide range of portability. Finally, it should be noted that the spread of the virus needs to consider more factors in real life. The prediction results of this paper may not be completely consistent with the actual situation, and the decision-makers should make corresponding changes according to the actual situation.

Finally, it is worth pointing out that when considering the impact of vaccine, temperature and other factors on the epidemic situation, this paper uses a relatively simple situation, which may lead to a decrease in the accuracy of prediction. In subsequent studies, we can consider using more population classifications to explore the impact of different vaccines (including different vaccination times, vaccine types, and immune duration) on the epidemic trend. In addition, there are also relevant studies showing that with the variation of the virus, there may be the possibility of secondary infection. The infectious disease model with multiple infections can also be considered in subsequent studies.

Declarations

Author contribution statement

Hao Ai: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Qiubao Wang: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Wei Liu: Contributed reagents, materials, analysis tools or data.

Funding statement

This work was supported by the Natural Science Foundation of Hebei Province [A2021210011, A2020210005], the Department of Education of Hebei Province [ZD2021335], Hebei Postgraduate Demonstration Course Project-Numerical Solution of Ordinary Differential Equation [KCJSX2022073], Innovation and entrepreneurship plan for College Students [2021101070194] and the National Natural Science Foundation of China [Nos.11602151, 11872253, 12101418].

Data availability statement

Data associated with this study has been deposited at GitHub:(https://github.com/Only-magic-can-defeat-magic/the-PSO-algorithm-for-parameter-simulation-of-SIR-model.git). Epidemic data can be obtained through the following website: (https://github.com/CSSEGISandData/COVID-19[16]), (https://www.chp.gov.hk/en/media/116/index.html?page=1).

Declaration of interest’s statement

The authors declare no competing interests.

Additional information

No additional information is available for this paper.