How can I further analyze the results of an antibody test and a test for viral infection in COVID19
I don’t know how many IT specialists will be interested in this and want to go down from technological heights to “mortal” analytical solutions of simple differentials. equations.
But I found it funny that the results of tests for coronavirus – PCR, etc., and tests for antibodies can be quite easily linked together by a simple mathematical model, and the analysis of the model leads to interesting conclusions.
Such tests are now being massively carried out around the world in connection with COVID19, affecting, among other things, respected IT professionals. Moreover, they have the opportunity to apply their knowledge of mathematics to optimize this procedure. How is described below.
If you look more broadly, then for some reason I am sure that in the distant bright future, an outpatient card in a polyclinic will be replaced by an individual computer model of the body, with the help of which a person will be able to predict his condition, analyze management options, etc., that is, do everything what computer simulation models allow. If I were a billionaire, I would invest my billions in just such a model. It is a pity that Elon Musk has not done this yet, he is probably still too young.
Initial assumptions and derivation of equations
We consider two variables: n_{one} And n_{2}
n_{one} – the concentration of viral particles (virions) in some environment inside the body;
n_{2} – the concentration of antibodies at the same point.
We apply the wellknown, widely used firstorder kinetic equations to describe the increase in the number of virions as a result of their replication and the decrease in the number of virions as a result of their destruction by antibodies, assuming that both processes occur at a rate proportional to n_{one}. The rate of death of virions is also proportional to the concentration of antibodies n_{2} .
As a result, we get that the rate of change in the concentration of virions is equal to:
where the dot above denotes the derivative with respect to time, k_{one} – multiplication factor, k_{2} is the virion death rate.
Rate of change in antibody concentration n_{2 }is determined by two processes: the appearance of new antibodies produced by the immune system in response to a viral load, and a decrease in the number of antibodies in the process of interaction with virions. Both processes depend on the concentration of virions, so the difference between their rates can be considered proportional n_{one}:
;
k_{3} – coefficient, if k_{3} > 0 – the number of antibodies increases with time if k_{3 }< 0, then it decreases. At k_{3 }= 0 antibody concentration is constant.
Equation Analysis
Solutions of the system of equations (1)(2), that is, the dependencies n_{one} And n_{2} from time, it is convenient to consider on the phase plane, that is, the first quarter of the coordinate plane (n_{2} , n_{one}) with axes: horizontal n_{2} and vertical n_{one . }For brevity – Diagram. The point on the Diagram reflects the state of the “patient” at the current time. The state points for a certain period of time form a phase trajectory.
On Fig. 1 shows diagram for case k_{3 }= 0.
There are two areas on the Chart:
1) band, where 0 < n_{2} < k_{one }/k_{2} ;
2) halfplane, where n_{2} > k_{one}/k_{2} .
At all points in the first region n_{one} increases, which means the development of the disease, at all points of the second area n_{one} decreases to zero, which means recovery.
Numbers 1 and 2 denote examples of phase trajectories for these two regions.
Equality k_{3 }= 0 suggests a fine balance between antibody production and death, which is unlikely to be possible.
To find phase trajectories for any k_{3 }≠ 0, we divide Eq. (1) by Eq. (2), while for convenience we replace the variables:
x=n_{2} ; y=n_{one}
We get:
dy/dx = A – B x (3)
where
A=k_{one }/k_{3 }, B=k_{2}/k_{3, }k_{3 }≠ 0.
The solution of the differential equation (3) is a quadratic dependence:
y=y_{0} +A(xx_{0}) – B(x^{2}x_{0}^{2})/2 ; (4)
where (x_{0 },y_{0) }– coordinates of the starting point of the trajectory.
It follows from equation (4) that on the Diagram all phase trajectories of system (1)(2) are sections of parabolas.
You can make sure that the vertices of all parabolas (4) must be on the vertical axis x= k_{one }/ k_{2} .
At k_{3} > 0 the branches of all parabolas are directed downwards. See Fig. 2.
At k_{3 }< 0 the branches of all parabolas are directed upwards. See Fig.3.
On Fig. 2, trajectory 1 corresponds to the situation of an “unvaccinated patient”, in which the initial concentration of antibodies n_{2} is zero and there is an initial level of virus infection n_{one} = y_{one}. As new antibodies are produced, the infection rate first rises to a maximum, then decreases to 0, which means “recovery”.
Trajectory 2 in Fig. 2 corresponds to the situation of a “vaccinated patient”, in which the initial concentration of antibodies n_{2} is equal to x_{2} the initial level of infection is y_{2} . It can be seen that despite the higher initial infection rate y_{2} > y_{one} there is no increase in the concentration of the virus, recovery, obviously, is faster.
In general, at k_{3} > 0 “recovery” occurs at any initial level of antibodies n_{2} and any initial level of infection n_{one} .
On the contrary, at k_{3 }< 0 “recovery” does not always occur, only at initial values (n_{2}. , n_{one} ) lying to the right of curve 1 in Fig. 3. This curve is a branch of a parabola with vertex at ( k_{one }/k_{2} , 0 ).
In fact, the conditions for “recovery” are a fairly high initial level of antibodies n_{2} and relatively low initial infection rate n_{one}which is true for curve 2. Moreover, by itself, a high level of antibodies before the “disease” does not guarantee “recovery”, if there was a severe infection – a high initial n_{one} which can be seen from curve 3. After the initial phase of apparent “recovery”, antibodies are exhausted to a critically low concentration, then an unlimited increase in the concentration of virions begins.
Discussion
The results obtained allow us to propose a “vector” method for assessing the dynamics of the state of a person infected with coronavirus. The technique should include the simultaneous measurement of the level of antibodies – n_{2} . and the level of viral infection (viral load) – n_{one} at two points in time: initial – t_{0} and some subsequent – t_{one} . From these two pairs of values, one can construct on the plane (n_{2} , n_{one} ) a vector whose direction will indicate the nature of the change in the patient’s state, as shown in Fig. 4.
In Fig.4, the point (0,0) is the initial one, the point (1,1) is the current state. The transition to quadrant 4 indicates unambiguously positive dynamics (see curves 1 and 2 in Fig. 2). The transition to quadrant 2 indicates unambiguously negative dynamics (see the ascending part of curve 1 in Fig. 3). The transition to quadrant 1 indicates the dynamics of the disease with a high probability of recovery (see curve 1 in Fig. 2). The transition to quadrant 2 indicates dynamics with an uncertain probability of recovery depending on the initial state (curves 2 and 3 in Fig. 3).
Thus, if the number of antibodies increases over time (positive immune response), then the process of recovery proceeds. In the model, this corresponds to the case k_{3} > 0 .
If during the course of the disease the number of antibodies decreases over time (negative immune response), then recovery is possible only at a sufficiently low initial concentration of virions, not exceeding a critical level, depending on the initial concentration of antibodies. In the model, this corresponds to the case k_{3 }< 0, curve 2 in Fig.3. With a stronger infection, an undesirable development of the disease should be expected: curve 3 in Fig.3.
The possibility of a negative immune response to a viral infection is supported by data [ 1 ].
In this work, the authors found that inside the respiratory syncytial virus (RSV) virion, which causes severe forms of pneumonia in humans, there is a protein (RSV NS2 nonstructural protein) that can suppress the immunity of the host cell. RSV NS2 protein analogues are produced by many respiratory pathogens. It is known that most deaths from COVID19 and complications from it are associated with inflammation caused by other, nonviral infections. The development of these inflammations [ 1 ] bind to a protein similar to RSV NS2, which is found inside coronavirus virions.
Study authors [ 1 ] worked with mutant versions of the RSV virus, in some of which the concentration of NS2 molecules was reduced, and in others this protein was completely removed. Observations of the reproduction of the resulting virus strains in lung cell cultures showed that the removal of NS2 from virions had an extremely negative effect on the rate of reproduction of RSV. Lung cells began to produce significantly more antiviral interferon proteins and destroy the emerging virions with the help of enzymes responsible for processing intracellular “garbage” and selfdestruction of infected cells.
Thus, the situation with k_{3 }< 0 looks quite real.
conclusions
1. It seems surprising, but from the analysis of the model it is clear that a high initial level of antibodies achieved before infection due to, for example, vaccination, does not in itself guarantee recovery.

The development of the disease depends on the sign of the body’s immune response to viral infection. This sign can be determined using existing tests for antibody levels and viral load levels.

An antibody test and a coronavirus test, preferably with viral load, should be done together, i.e. couple. In case of infection, the presence of even two paired tests will allow us to assess the dynamics of the disease.
Literature
one. A.S.M. Journals mBio Ahead of Print January 18, 2022
Kim Chok, Svechha M. Pokharel, Indira Mohanty et al.