Infectious disease model
Overview
Since 2019, the COVID-19 have been spreading widely around the world, causing significant economic recession in many countries. Among them, the nucleic acid detection in undeveloped regions remains insufficient, and the detection methods have certain limitations. Therefore, they cannot significantly help the disease control. Based on this phenomenon, we developed the following infectious disease model (SAIR-Q-D) to deduce the characteristics that an ideal detection method should have. After that, we also designed a G-quadruplex-based colorimetric virus detection system.
Figure 1. The SAIR-Q-D model.
We used the ordinary differential equation (ODE) to assist us in simulating the effects of different detection methods on the transmission process of infectious diseases in both developed and undeveloped regions. In this structure, susceptible people (S) are infected by asymptomatic carriers (A) and infectious people (I), who become A. A becomes infectious people (I) after a period of time. S is misdiagnosed due to a false-positive kit, and some people are mistakenly quarantined to Q1. During this time, if they are tested again and found to be uninfected by the virus, they revert to S. However, if they are detected to be infected, they enter Q2 for quarantine, during which time they are not infected and gradually revert to recovered population (R). Due to false positives of the kit, a fraction of R people will be mistakenly quarantined to Q3. In reality, due to virus mutation or low antibody concentration, the recovered population may not have corresponding immunity to the COVID-19 and will be re-infected with the virus afterwards. Therefore, there is a certain possibility that the recovered population (R) will become susceptible again (S).
Inspired by the project of Leiden University last year, the quarantine rates S1, S2, S3 and S4 depend on
1) the probability of being tested Pi
2) the accuracy of the test. (False positives and false negatives)
Combining the probability of being tested with the false positive rate (FP) and the false negative rate (FN) in the diagnostic tests, the following quarantine rates were obtained.
We then simulated a situation where the disease spreads when the total number of people is 4.5 million. Based on the graph, the number of susceptible people first kept decreasing over time, then slightly increased, and finally stabilized. The number of recovered people in general keeps increasing over time with potential slight fluctuations. This is due to the fact that some of the recovered individuals became susceptible again because of virus mutations or the decrease of antibodies in their bodies over time. The number of asymptomatic transmitters increased and then gradually decreased. Over time, a proportion of asymptomatic transmitters become infected and another proportion of them are tested positive and then quarantined, so that the peak number of infected persons is lower than that of asymptomatic transmitters. Eventually, the epidemic will enter a normalization phase over time due to the asymptomatic transmission of the COVID-19 as well as its susceptibility to mutations.
Figure 2. Relationship among the numbers of susceptible (S), asymptomatic infected (A), infected (I), recovered (R) and quarantined (Q) people over time.
Application
Detection-Quarantine-Treatment
In order to slow down the spread of the virus, it is crucial to diagnose and quarantine the patient for treatment in a timely manner during the asymptomatic transmission phase. This is due to the fact that when a patient is quarantined, he is not exposed to susceptible people in the outside world, which makes it more difficult to spread the disease in the population. This is reflected in the model by a reduction in the number of contacts c1. In addition when the patient is treated in quarantine, he generally receives targeted treatment for recovery. Thus in the model q < itr (where q means that the rate of getting out of quarantine; itr means recovery rate of infectious population). The model results were as follows.
Figure 3. Relationship between the number of infections over time under quarantine and treatment after detection within 3 days and 10 days for asymptomatic transmitters
Figure 4. Relationship between the number of deaths over time under quarantine and treatment after detection within 3 days and 10 days for asymptomatic transmitters
The results of the model were also consistent with what we suspected. During the asymptomatic transmission phase, the number of infections and deaths were significantly lower when people are diagnosed and quarantined for treatment after 3 days compared with 10 days. This represents an effective control of disease transmission and demonstrates the significant role of timely detection for CDC. Based on this, we analyzed three different assays (nucleic acid test, antibody test and antigen test) that are commonly used today.
Different detection methods
We then explored the impact of three different methods (nucleic acid test, antibody test and antigen test) on the transmission of infectious diseases.
1. The genetic material amplifies rapidly after pathogen infection.
Since nucleic acid detections determine the presence of genetic materials of pathogens, it can be used to detect the early stages of infectious diseases. Therefore, a nucleic acid test has the characteristics of early diagnosis, sensitivity and specificity, and is the "gold standard" of the confirmative diagnosis of the COVID-19.
However, there are disadvantages in nucleic acid tests:
1) High demanding for the testing equipment or platform;
2) Relatively long test time, usually up to 24 hours to report the results.
| Developed regions | Undeveloped regions | |
|---|---|---|
| Detection time D2 (h) | 12 | 36 |
| False negative rate | 15% | 25% |
Average test time of developed regions is shorter than that of undeveloped regions, due to the difference in the promptness of sample transport and availability of PCR instrument. Additionally, medical personnel in developed regions are more professionally trained in sampling methods and have better sample preservation than undeveloped regions. These factors expectedly lead to low false negative rates in developed regions versus undeveloped regions.
Figure 5. Relationship between the number of infections over time under PCR testing conditions in developed and undeveloped regions
2. COVID-19 serum antibody test
Serum-specific antibodies are gradually produced from the 7th day after the onset of the COVID-19 pneumonia. The biggest advantage of the serological test is its convenience and short test time.
However, there are disadvantages in serum antibody tests:
1) False positives: Some individuals’ blood samples contain rheumatoid factor, heterophilic antibodies or autoantibodies, as well as drugs and tumor cells, which can easily cross-react with the test reagents and consequently cause false positive results;
2) False negatives: Due to the existence of serum antibody in a certain time window, the method may not always show positive results for infected individuals. In addition, the sensitivity of different test kits may significantly vary.
| Developed regions | Undeveloped regions | |
|---|---|---|
| False negative rate | 1% | 2% |
| False positive rate | 3% | 7% |
In antibody detection, developed regions typically have relatively low substance interference and thus produce less false positive results than undeveloped region. In addition, availability of more advanced test kits with high detection sensitivity in developed regions also reduces the rates of false negatives.
Figure 6. Relationship between the numbers of infections over time under antibody detection conditions in developed and undeveloped regions
3. COVID-19 antigen test
COVID-19 antigen test can directly detect the presence of the SARS-CoV-2 in human samples, with rapid diagnosis, high accuracy and low requirements for equipment and personnel.
However, there are disadvantages in antigen tests.
1) False negatives: Antigen detection has relatively low sensitivity, and requires relatively high virus titers in the samples. Since SARS-CoV-2 mainly invades the lower respiratory tract, such as alveoli, sampling from the upper respiratory tract, such as the nasopharynx and oropharynx, may not always reach the pathogen, or the virus number in the sample is too small to be detected. These factors can cause false negative detection.
2) Cumbersome and time-consuming sample preparation: This may restrict the use of the antigen test in undeveloped regions, due to their lack of professional or well-trained medical personnel. The deficiencies in the science and technology result in the retarded development of antigen detection kits in these regions.
| Developed regions | Undeveloped regions | |
|---|---|---|
| R&D Time D1 /Month | 2 | 3 |
| False negative rate | 25% | 35% |
On the contrary, promoted by advanced science and technology in developed regions, recombinant antigens and monoclonal antibodies can be quickly generated. Therefore, the progresses in antigen test are relatively smooth and rapid, and the development time for new products is generally shorter than that in undeveloped regions. Additionally, the availability of professionals and well-trained medical personnel in developed regions significantly reduces the rates of false test results.
Figure 7. Relationship between the numbers of infections over time under antigen detection conditions in developed and undeveloped regions.
Figure 8. Relationship between the time after passing nucleic acid, antibody and antigen detection tests and the numbers of infections in developed and undeveloped regions.
The results showed that
1. The use of PCR for nucleic acid detection keeps the numbers of both the infected and the increase rates of the infected relatively low in multiple countries. This may help us to contain the epidemic. However, the PCR test methods require advanced equipment and take relatively long time, usually up to 24 hours to report the results.
2. The number of the infected is higher when antibody test alone is used than that when the nucleic acid test alone is performed. This is due to the fact that antibody tests are subject to certain false positives and false negatives, and there is a window of time during which the disease cannot be detected.
3. The use of antigen testing alone has resulted in a high maximum number of the infected due to its long development time and its undetectability during this period. In addition, antigen test is relatively demanding in terms of technical development. Therefore, the spread of an infectious disease varies greatly between developed and undeveloped regions.
Based on the model results, the existing detection method is not very ideal for the disease control in undeveloped areas. An ideal test protocol should have the following characteristics.
1. Retains the advantages of the "gold standard", such as the nucleic acid assay, but avoids its dependence on instrument and professionals (for PCR assays)
2. Avoid false positives and false negatives based on the assay principle (for antibody assays)
3. High developability, short development time, and rapid and high throughput detection of the assay (for antigen detection)
In summary, we developed an infectious disease model. Both the advantages and disadvantages of the commonly used nucleic acid detection methods for disease prevention and control have been evaluated. Based on the previous analysis, we designed a G-quadruplex-based colorimetric virus detection system, which is well-suited to the characteristics of an ideal surveillance scheme and reflects the feedback of the modeling for wet laboratory research.
Differential Equations used in the Model
Table. Parameters for simulating the dynamics of an Ebola epidemic
| Parameter | Description | Value | Unit |
|---|---|---|---|
| c1 | Number of people accessible to the infected | 5 | contacts/day |
| c2 | Number of people accessible to asymptomatic transmitters | 7 | contacts/day |
| a1 | Infected person disease transmission rate | 0.05 | 1/contact |
| a2 | Disease transmission rate of asymptomatic transmitters | 0.03 | 1/contact |
| d | Death rate | 0 | 1/day |
| ati | Per-capita rate of progression to infectious population | 0->0.2 | 1/day |
| itr | per-capita recovery rate of infectious population | 0->0.1 | 1/day |
| s1 | Probability of tested when susceptible | 0.15 | |
| s2 | Probability of tested when exposed | 0.15 | |
| s3 | Probability of tested when infectious | 0.85 | |
| s4 | Probability of tested when recovered | 0.15 | |
| D1 | R&D Time | 60 | days |
| D2 | Detection time | 12 | hours |
| FPR | False positive rate of test kit | 0.03 | (ratio) |
| FNR | False negative rate of test kit | 0.01 | (ratio) |
| q | Get out of quarantine rate | 0.07 | 1/day |
| N0 | Number of simulators | 4500000 | |
| h | Re-infection rate | 0.1 | ratio |
| m1 | Q1 to Q2 conversion rate | 0.8 | ratio |
Molecular kinetic models
In this section, we describe how we developed a semi-empirical model to simulate our detection scheme. We used reaction kinetic equations to model each of the three processes that include:
1) Recombinase polymerase amplification (RPA)
2) Double-stranded DNA (dsDNA) cleavage to form single-stranded DNA (ssDNA)
3) Rolling circle amplification (RCA) and catalytic color development
The parameters in these models were optimized to match our experimental observations. Finally, we integrated three separate models to obtain a model describing our entire G-quadruplex-based colorimetric virus detection system.
Recombinase polymerase amplification (RPA) model
RPA is a reaction that amplifies a specific dsDNA sequence, flanked by a specific set of primers.[1] The recombinase binds to the primers to form a protein-DNA complex that searches for homologous sequences in the dsDNA. Once a primer locates the matched sequence, a strand exchange reaction occurs to form and initiate DNA synthesis, which exponentially amplifies the target region on the template. The RPA reaction produces target dsDNA amplicons and the rate of its production is limited by the amount of a given resource.
The entire process proceeds very quickly, with detectable levels of amplification products typically obtained within thirty minutes.
Model equation:
dsDNA-double stranded DNA
t-time1
r1-reaction coefficient
Cmax-maximum concentration
Figure 9. Calculation of dsDNA concentration versus time by molecular dynamics
Based on this figure, the rate of RPA amplification kept dropping with the time, and finally the amount of dsDNA reached a relatively constant level of 2×10-7.
dsDNA cleavage to form ssDNA
dsDNA undergoes single strand break in the presence of a nickase, followed by strand displacement in the presence of a DNA polymerase. The products of these two reactions are ssDNA, and its amount is determined by the substrate dsDNA concentration and the reaction time. a, b and c are three coefficients, among which c is the most closely related to time.Therefore, we consider two variables, the amount of dsDNA and the reaction time, to jointly determine the amount of ssDNA.
Model equation:
dsDNA-double stranded DNA
ssDNA-single stranded DNA
t-time2
h-displacing coefficient
Figure 10. Calculation of dsDNA concentrations versus time by modeling the relationship of dsDNA.
Based on this figure, as the dsDNA content increases with time, the ssDNA levels also climbs.
Rolling circle amplification (RCA) model
In our project, we used RCA to generate the reporter, G-quadruplex (G4).[2] G4 catalyzes the onset of the color development reaction, where the colored substrate ABTS discolored the solution. [3] The production of G4 is characterized by measuring the absorbance of the solution. The primers required for the RCA are the ssDNA products from the previous reaction. Therefore, we analyzed the relationship between absorbance values and the amount of ssDNA primers by the model.
Model equation:
Abs- absorbance
r2-reaction coefficient
Amax- theoretical absorbance maximum value
Figure 11. Parameters were fitted by molecular dynamics parametric equations and the variation of absorbance (Abs) versus the amount of ssDNA primers was plotted
According to the data obtained from the experiment, the differential equation was fitted to the parameters, and the function curve was plotted as above. The absorbance increased rapidly with the primer content rising from 1×10-7 to 3×10-7, and then the increase rate gradually leveled off. Based on this observation, we further evaluated and chose Abs=0.3 as the actually used distinctive color point.
Sequential integration of models
At the end of the molecular dynamics model, we derived from extensive experiments that the color development was very pronounced when the absorbance of the end product was greater than 0.3. This can facilitate the judgment of the test results using our device under non-laboratory or environmental conditions. Therefore, we performed a backpropagation based on this result to determine the time required for each step of the reaction to complete. The results further validate the advantage of our assay in terms of relatively low time consuming in nucleic acid detection.
According to our test, the color of the products can be clearly identified when the absorbance is above 0.3. Therefore, based on our calculation using the RCA differential equation, the color development is obvious when the ssDNA primer reaches 1.275×10-7 M, as shown in the figure below.
Subsequently, according to this set of equations, the required ssDNA primer levels can be reached when the second part of the reaction proceeds to 3820 s. For this purpose, the required dsDNA levels need to be around 1.635×10-7 M. The data are shown in the figures below. (A 3D image was expanded in two dimensions for ease of observation.)
Finally, using the differential equation in the first step, we can determine that when the reaction time is 977s, a sufficient amount of dsDNA can be amplified to participate in the subsequent reaction, and finally achieve an ideal color development of the test result. The data are shown in the figure below.
In summary, the set of established molecular kinetic models provided us a better way to utilize the experimental data. The simulation results were applied to the optimization of the assay protocol to achieve the best detection results within the shortest time.
| Parameter | Value |
|---|---|
| r1 | 0.01619 |
| Cmax | 2×10-7 |
| a | 6250000 |
| b | 200730690 |
| c | 0.9266927167171597 |
| h | 0.78 |
| r2 | 16738486 |
| Amax | 0.6775 |
References
- DeShields, Joseph B , Moroz, et al. Recombinase Polymerase Amplification (RPA) for the Rapid Isothermal Detection of Spongospora subterranea f. sp. subterranea and Potato Mop-Top Virus.
- Tian Y , Yu H , Prof C M . Cascade Signal Amplification for DNA Detection[J]. Chembiochem, 2010, 7(12):1862-1864.
- Xw A , Xc A , Cc A , et al. Naked-eye detection of site-specific ssRNA and ssDNA using PAMmer-assisted CRISPR/Cas9 coupling with exponential amplification reaction[J]. Talanta, 233.
